One, no one, and one hundred thousand: The paradigm of the Z–R relationship

The Z–R relationship is a scaling-law formulation, Z 5 ARb, connecting the radar reflectivity Z to the rain rate R. However, more than 100 Z–R relationships, with different values of the parameters, have been reported in literature. This abundance of relationships is in itself a strong indication that no one ‘‘physical’’ relationship exists, a state of affairs that we find similar to that of the protagonist of Luigi Pirandello’s novel One, No One and One Hundred Thousand. Nevertheless the ‘‘elevation’’ of a simple linear fit in the (logR, logZ) space to the role of ‘‘scaling law’’ is such a widespread tenet in literature that it eclipses the simple realization that the abundance of different intercepts and slopes reflects the inhomogeneous nature of rain, and, in ultimate analysis, the statistical variability existing between the number of drops and drop size dis-tribution. Here, we ‘‘eliminate’’ the contribution of the number of drops by rescaling both reflectivity and rainfall rate to per unit drop variables, (Z, R) → (z, r), so that the remaining variability is due only to the variability of the drop size distribution. We use a worldwide database of disdrometer data to show that for the rescaled variables (z, r) only ‘‘one,’’ albeit approximate, scaling law exists.