#### Title

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

#### Document Type

Article

#### Publication Date

12-16-2006

#### Abstract

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 work 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 this be so?

In order to justify this observation, there have been many attempts to create 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 the parametric fits are not unique and because they exhibit fundamental physical 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 between *Z* and *R*apply in *statistically homogeneous* rain. (Note that statistical homogeneity does not mean that the rain is spatially uniform.) In contrast, nonlinear power laws are empirical fits to correlated, but statistically inhomogeneous data. This conclusion is proven theoretically after developing a ‘generalized’ *Z–R* relation based upon physical consideration of *R* and *Z*as random variables. This relation explicitly incorporates details of the drop microphysics as well as the variability in measurements of *Z* and *R*. In statistically homogeneous rain, this generalized expression shows that the coefficient relating *Z* and *R* is a constant resulting in a linear *Z–R* relation. In statistically inhomogeneous rain, however, the coefficient varies in an unknown fashion so that one must resort to statistical fits, often power laws, in order to relate the two quantities empirically over widely varying conditions. This conclusion is independently verified using Monte Carlo simulations of rain from earlier work and is also corroborated using disdrometer observations. Thus, the justification for nonlinear power‐law *Z–R* relations is not physical, but rather statistical, in that they provide convenient parametric fits for estimating mean *R* from measured mean *Z* in statistically 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 scales. The examples also offer hope that data collected over disparate sampling‐volumes and sampling‐frequencies can still be combined to yield meaningful estimates. Although additional testing is required, this allows us to write programs which combine estimates of *R* using remote‐sensing techniques with sparse but direct rainfall observations.

#### Publication Title

Quarterly Journal of the Royal Meteorololgical Society

#### Recommended Citation

Jameson, A. R.,
&
Kostinski, A.
(2006).
Reconsideration of the physical and empirical origins of Z–R relations in radar meteorology.
*
Quarterly Journal of the Royal Meteorololgical Society,
127*(572), 517-538.
http://doi.org/10.1002/qj.49712757214

Retrieved from: https://digitalcommons.mtu.edu/physics-fp/242

## Publisher's Statement

Copyright © 2001 John Wiley & Sons, Ltd. Publisher’s version of record: https://doi.org/10.1002/qj.49712757214