The Relation between the Area-Average Rain Rate and the Rain Cell Size Distribution Parameters

A review of the literature on the rain cell diameter distribution (RCDD) is first presented. It shows that RCDD, for a given rain-rate threshold τ, is accurately described by an exponential distribution with an exponential parameter λ weakly variable from one place to another and almost independent of τ. Then it is demonstrated that integrating the RCDD enables the writing of interesting and simple relations between the moments of the RCDD and useful area-averaged rain field parameters, notably the average cell diameter and F(τ), the fractional area occupied by the rain cells having a rain rate above τ. From F(τ), an equation relating the area-average rain rate 〈R〉 to the second moment of the RCDD is obtained. Now, an explicit knowledge of the RCDD is not necessary to obtain 〈R〉. RCDD can be replaced by the distribution of the chords of the rain cells. Indeed, the moments of the rain cell chord distribution are analytically related to the moments of the RCDD. Thus, an equation showing that 〈R〉 is a function of the only area-averaged chord length is written. Based on this relation, a very simple method for the measurement of precipitation from the random intersect of the rain cell population by a scanning remote sensor is proposed. In this method, 〈R〉 is inferred from the simple estimation of the area-averaged chord length observed, for example, with a radar sampling, in a binary way, a rain cell distribution thresholded at a mean rain rate τ. The interest of this method is a reduced requirement for the measuring time and for the range of R to be sensed. Besides, it is suggested that this method can be implemented for area-average rain-rate estimates from a spaceborne microwave sensor. Some tests and simulations using ground-based radar data are presented in order to support the validity of the proposed relation.