Reconsideration of the physical and empirical origins of Z-R relations in radar meteorology

The rainfall rate, R, and the radar reflectivity factor, Z, are represented by a sum over a finite number of raindrops. It is shown here and in past w ork that these variables should be linearly related. Yet observations show that correlations between R and Z are often more appropriately described by nonlinear power laws. In the absence of measurement effects, why should th is be so? In order to justify this observation, there have been many attempts to crea te physical 'explanations' for power laws. However, the present work argues that, because correlations do not prove causation (an accepted fact in the statistical sciences), such explanations are suspect, particularly since t he parametric fits are not unique and because they exhibit fundamental phys ical inconsistencies. So why, then, do so many correlations fit power laws when physical arguments show that Z and R should be related linearly? It is shown in the present work that physically based, linear, relations be tween Z and R apply in statistically homogeneous rain. (Note that statistic al homogeneity does not mean that the rain is spatially uniform.) In contra st, nonlinear power laws are empirical fits to correlated, but statisticall y inhomogeneous data. This conclusion is proven theoretically after develop ing a 'generalized' Z-R relation based upon physical consideration of R and Z as random variables. This relation explicitly incorporates details of th e drop microphysics as well as the variability in measurements of Z and R. In statistically homogeneous rain, this generalized expression shows that t he coefficient relating Z and R is a constant resulting in a linear Z-R rel ation. In statistically inhomogeneous rain, however, the coefficient varies in an unknown fashion so that one must resort to statistical fits, often p ower laws, in order to relate the two quantities empirically over widely va rying conditions. This conclusion is independently verified using Monte Car lo simulations of rain from earlier work and is also corroborated using dis drometer observations. Thus, the justification for nonlinear power-law Z-R relations is not physical, but rather statistical, in that they provide con venient parametric fits for estimating mean R from measured mean Z in stati stically inhomogeneous rain. Finally examples based upon disdrometer data suggest that such generalized relations between two variables defined by such sums are potentially useful over a wide range of remote-sensing problems and over a wide range of scal es. The examples also offer hope that data collected over disparate samplin g-volumes and sampling-frequencies can still be combined to yield meaningfu l estimates. Although additional testing is required, this allows us to wri te programs which combine estimates of R using remote-sensing techniques wi th sparse but direct rainfall observations.