Parametrization of drop size distribution with rain rate

Because the rain rate R is easier to observe than the drop size distribution (DSD), it is useful to express the DSD as a function of R only, i.e. N(D,R). Disdrometer data from four sites in West Africa are used to compare observed DSDs with fitting to a lognormal function in which the three parameters Xi, with i = 1, 2, 3, are expressed as functions of R only. Observed DSDs are found well represented by such N(D,R) function. Writing the N(D,R) moments for a calculation of useful physical quantities, such as water content M, rain rate, or radar reflectivity factor Z, raises constraints of self consistency of the various equations. Modifying the analytical form of DSD in order to take into account the constraint of self-consistency is not found to increase significantly the quality of the DSD fitting with respect to the direct use of Xi(R) in the distribution, ignoring the lack of self-consistency. Then Z–R relations obtained by direct regression of observed data are compared with the relations derived analytically from N(D,R) and the expressions Xi(R) observed in each site. The two groups of relations compare very well.