## Introduction

To apply radar reflectivity data in hydrometeorology, one needs to determine a *Z*_{m}–*R*_{p} relationship to convert radar-measured reflectivities *Z*_{m} into the radar-predicted rainfall intensities *R*_{p}. The process of adjusting this conversion based on a data sample usually is referred to as calibration (estimation, tuning) of a *Z–R* relationship. As a result of numerous data analyses, an abundance of quite different reflectivity–rainfall relationships has been obtained. In a comprehensive monograph by Battan (1973), a summary of the power-law *Z*–*R* relationships encountered in many studies is presented. For example, this summary reports estimated *Z–R* exponent values in a broad range from 0.75 up to 3.0. The diversity of the obtained *Z–R* relationships is attributed typically to large and systematic climatologic variability of the microphysics of precipitation systems. The differences that could originate from other reasons, including different estimation schemes, observational uncertainties, and/or *Z–R* estimation objectives, still are not understood fully. These differences are, in a limited scope, the main focus of this study. A large impact of radar characteristics and data processing on selected statistics of rain fields has been demonstrated by Krajewski et al. (1996). In the present study, we continue their pursuit to try to get more insight into the reflectivity to rain-rate conversion aspect of the larger problem.

Two goals of joint radar and rain gauge rainfall data analysis can be distinguished. The first goal is to produce reliable radar-based predictions of rainfall intensities. Radar products represent different approximations of this quantity because their derivation can be subject to different user-oriented criteria. The second goal is to determine possible physical dependency (i.e., relationship) between surface reflectivity and rainfall intensity in a specific precipitation system. This dependency should be an objective property of the precipitation regime and, thus, it should be independent of the observational uncertainties and of the scheme applied to its identification. So far, not much attention has been paid to this distinction between the two goals, and usually they are considered as one task of “*Z–R* relationship estimation.” In this study, we propose a simple nonlinear model of joint radar–rain gauge measurements to show that these two objectives are not equivalent. We apply it to investigate and to conceptualize the possible impact of large measurement errors of both reflectivity and rainfall sensors on our inference based on the joint radar–rain gauge data.

Specifically, we compare results of three popular *Z–R* estimation methods that often are applied in research and/or operational practice of radar hydrometeorology. The schemes analyzed here are two approaches based on nonlinear regression, and the probability matching method (PMM). We show that the three techniques produce different parameters of the estimated *Z–R* relationships. The estimated parameters are three different functions of the assumed measurement errors and the intrinsic *Z–R* relationship of the model. These results suggest that the diversity of *Z–R* relationships encountered in the literature perhaps can be attributed not only to their climatological variability but also to observational uncertainties and different estimation methods. The model presented here is intended to provide insight into the impact of those factors. We also use it to discuss briefly some possible methods that could account for the uncertainties and help to separate the observational and natural causes of the *Z–R* variability.

The model that we define in the next section is idealized and is not designed to mimic accurately all aspects of the real radar–rain gauge measurements. An advantage of the specific framework assumed here is that the model is tractable analytically and the pursued solutions can be derived in the form of closed mathematical formulas. They explicitly demonstrate the behavior of our rainfall observation system in a general and synthetic way. However, their direct practical applicability might be limited due to the assumed simplifications of the model. Also, no discussion of the superiority or inferiority of any of the three estimation methods is attempted here. This discussion is a matter of using specific application-oriented criteria and is beyond the scope of this study.

This paper is organized as follows. In the next section we define and discuss the model framework. In section 3, solutions for the three *Z–R* estimation methods are derived. In section 4, the solutions are compared and discussed. A summary and conclusions section closes the study.

## Definition of the model

The model developed in this study consists of simple parameterizations of the radar and rain gauge observation errors and of a fixed one-to-one dependency between the surface reflectivity and rainfall intensity. We assume that this physical dependency can exist only between quantities that are associated with the same spatial domain. The domain is determined by the radar resolution and can be defined as a single radar grid or a projection of a radar volume bin onto the surface. The differences between the surface reflectivity averaged over this grid area and the actually measured radar reflectivity can originate from numerous sources (Battan 1973; Zawadzki 1984). In our model they all are treated synthetically and are described as one error factor. The second error component applies to the differences between the rainfall intensity averaged over the same grid and its rain gauge sampling. Large differences of the spatial resolution between the quasi–point rain gauge sampling and the grid-averaged radar measurements result from extreme spatial variability of rainfall fields. They prevent any straightforward interpretation of the radar–rain gauge comparisons (Zawadzki 1975; Kitchen and Blackall 1992). In fact, there is evidence that the area–point effects might even dominate the comparisons (Ciach and Krajewski 1997, 1999). This possibility is not surprising once it is realized that the difference between radar grid area and rain gauge collection area may be as big as eight orders of magnitude (this scale difference is comparable to the difference between a radar grid and a continent). Again, in our model the area–point differences and other rain gauge errors are described synthetically as one error factor.

*R*

_{a}and its corresponding surface reflectivity

*Z*

_{a}averaged over the same area. They are associated with each other through a commonly known power-law relationship:

*Z*

_{a}

*AR*

^{b}

_{a}

*A*traditionally is called the multiplier of the

*Z–R*relationship and parameter

*b*is called its exponent. The observables associated with the two physical quantities are corrupted with large measurement errors. In this model, the errors are assumed to be multiplicative and independent of the truth. Thus, the rain gauge–measured rainfall intensity

*R*

_{m}and the radar-measured reflectivity

*Z*

_{m}are linked with their corresponding physical quantities defined in (1) through the following observation equations:

*R*

_{m}

*R*

_{a}

*E*

_{R}

*Z*

_{m}

*Z*

_{a}

*E*

_{Z}

*E*

_{R}and

*E*

_{Z}are the measurement errors for the rainfall and reflectivity, respectively. As already mentioned,

*E*

_{R}represents the difference between rainfall intensity averaged over a radar grid and its quasi–point rain gauge sampling. The factor

*E*

_{Z}describes the difference between surface reflectivity averaged over the same area and the actually measured radar reflectivity.

*R*

_{a}is defined by its mean

*μ*

_{R}and coefficient of variation CV

_{R}, and

_{ER}

_{EZ}

*Z*

_{a}distribution are determined by the model parameters that already have been defined. Equations (1), (2), and (3) completely define the mathematics of our model. Now, we will briefly discuss its assumptions.

We assumed lognormal probability distributions of all the random variables involved, and the power-law and multiplicative forms of the nonlinear interactions between them. This makes the model framework mathematically consistent because the lognormality is invariant only to the power-law and multiplication operations. Also, each set of the true and measured values is an independent sample and the model does not account for the temporal structure of the rainfall process. Similarly, the spatial dependencies in the rain fields are not addressed in an explicit way. Instead, differences between the true grid-averaged rainfall intensity and its quasi–point rain gauge sampling are synthetically described as a multiplicative, lognormal measurement error *E*_{R}. All these structural features make the model analytically tractable, as shown in the next section. We need to reiterate that the model describes a highly idealized picture of the radar–rain gauge reality and only some of its assumptions can be supported by the published results. Other features have been chosen so that they do not contradict the existing experimental evidence, are intuitively sensible, and are plausible.

Another assumption is the fixed *Z*_{a}–*R*_{a} relationship [(1)] between the true physical variables. Although a real precipitation system is usually a combination of different regimes, many researchers support the idea of fixed (or almost fixed) *Z–R*s, at least for specific rainfall regimes (Atlas and Chmela 1957; Joss and Waldvogel 1970; Rosenfeld et al. 1995). Thus, a physical meaning of this assumption may be that the model applies to such a specific situation.

The lognormality of the rainfall intensities has been proposed and tested by Kedem et al. (1994, 1997). These results, however, are based on rain gauge data only. There is no experimental evidence as to whether they also can apply to the rainfall intensities averaged over a radar grid area, which is about a hundred million times larger than the rain gauge collection area. In regard to the measurement errors as they are defined in this study, there is virtually no data on their functional and statistical structure because relevant experiments have not even been designed yet. The multiplicative form [used also by Krajewski and Georgakakos (1985)] reflects a general feature of positively defined physical quantities:the fact that the absolute measurement errors tend to be bigger for large values of the measured variables than for small values. The lognormality of the errors is assumed first of all for the mathematical consistency of the model. It is unlikely that real errors have exactly this distribution, although we can expect that they are positively defined and highly variable because of the extreme variability of the factors involved. These arguments suggest that our lognormal multiplicative error model is at least a physically sensible approximation.

Last, one should note that the expected values of the errors in (3b) and (3c) are assumed to be equal to 1.0, which is equivalent to the lack of overall measurement biases in our model. The main reason for this lack of bias is our choice of the focus of this work, which concentrates on random errors and their impact on the estimated *Z–R* relationship. Of course, systematic biases also are important and could be analyzed within our model, but to keep this study concise we decided that they are beyond its scope.

## Solutions for three *Z–R* estimation schemes

*Z*

_{m}–

*R*

_{p}relationship, which converts radar-measured reflectivities into radar-predicted rainfall intensities, using a specific radar–rain gauge data sample. The three schemes are direct nonlinear regression (from

*Z*to

*R*), reverse nonlinear regression (from

*R*to

*Z*), and PMM. They will be defined rigorously below. In our model, because of the lognormality of all random variables and the fact that only power-law transformations preserve it, the

*Z*

_{m}–

*R*

_{p}relationship will be also in the class of power-law functions. There are two mathematically equivalent forms of this relationship:

*α*and

*β*in (4a), or

*c*=

*α*

^{−1/β}and

*d*= 1/

*β*in (4b), need to be estimated based on available measurement data. Equation (4a) defines the relationship in a traditional way, after Marshall and Palmer (1948), as reflectivity being a function of rainfall. For the purpose of radar rainfall prediction, an inverse [(4b)] of this form is used. We will refer to the traditional form [(4a)] to compare directly the estimated parameters with the parameters of the assumed intrinsic

*Z–R*defined by (1).

*β*only and skip derivation of the multiplier

*α*in (4). From a physical point of view, the multiplier can be treated as an adjustment that removes the long-term overall bias of radar rainfall predictions. It can be determined uniquely by

*β*and the model parameters through the following comparison of the

*R*

_{p}and

*R*

_{m}ensemble means:

### Direct nonlinear regression

*Z*

_{m}–

*R*

_{p}relationship as conditional expectation of the measured rainfall, conditioned on a fixed value of the measured reflectivity:

*R*

_{p1}

*z*

_{m}

*E*

*R*

_{m}

*Z*

_{m}

*z*

_{m}

*c*

_{1}

*z*

^{d1}

_{m}

*α*

^{−1/β1}

_{1}

*z*

^{1/β1}

_{m}

*Z*

_{m}–

*R*

_{p}estimation method considered in this study. Hereinafter, according to common convention in statistics, we use capital letters to denote random variables and corresponding lowercase letters for their measured values. It has been proven rigorously (Karr 1993) that conditional expectation is a unique and most general solution that minimizes the mean square prediction error.

*R*

_{p1}

*z*

_{m}

*E*

*R*

_{m}

*Z*

_{m}

*z*

_{m}

*E*

*R*

_{a}

*E*

_{R}

*Z*

_{m}

*z*

_{m}

*E*

*R*

_{a}

*Z*

_{m}

*z*

_{m}

*E*

*E*

_{R}

*E*

*R*

_{a}

*Z*

_{m}

*z*

_{m}

*R*

_{a}and

*Z*

_{m}have a bivariate Gaussian distribution characterized by five parameters:

*R*

_{a}

*Z*

_{m}

*N*

*μ*

_{lnR}

*μ*

_{lnZm}

*σ*

^{2}

_{lnR}

*σ*

^{2}

_{lnZm}

*r*

_{lnR;lnZm}

*μ*

_{lnR}and

*σ*

^{2}

_{lnR}

*R*

_{a},

*μ*

_{lnZm}

*σ*

^{2}

_{lnZm}

*Z*

_{m}, and

*r*

_{lnR;lnZm}

*R*

_{a}and ln

*Z*

_{m}. The parameters in this distribution can be expressed through the parameters of our model based on the close relation between the lognormal and normal distributions (e.g., Crow and Shimizu 1988). If random variable

*X*is lognormally distributed, then ln

*X*is normally distributed and there exist the following links between their means and variances, which will be used in our derivations:

_{Z}is related to CV

_{R}through

^{2}

_{Z}

*b*

^{2}

^{2}

_{R}

*R*

_{a}conditioned on a fixed

*Z*

_{m}=

*z*

_{m}(or equivalently on ln

*Z*

_{m}= ln

*z*

_{m}) has to be lognormally distributed. Applying standard textbook formulas (e.g., Hogg and Craig 1995), one obtains the following expressions for the conditional mean and variance of the ln

*R*

_{a}:

*R*

_{a}conditioned on

*z*

_{m}, we now can obtain easily the sought expression for the conditional mean of

*R*

_{a}itself by applying (9c):

*z*

_{m}(the other terms affect the

*Z*

_{m}–

*R*

_{p1}multiplier only), one finally obtains

*d*

_{1}(and consequently of its inverse

*β*

_{1}) is

*D*

_{EZ}

*D*

_{EZ}

### Reverse nonlinear regression

*Z*

_{m}–

*R*

_{p}parameters in (4) using a regression procedure in which the reflectivity is a response variable and the measured rainfall

*R*

_{m}is a predictor. This approach is quite common and probably stems from the traditional form of

*Z–R*relationship [(1)]. In this arrangement, reflectivity is a parametric power-law function of rainfall and it seems natural to take the reflectivity as a response variable in the regression. The reverse regression can be defined generally as the conditional mean of radar reflectivities conditioned on a fixed value of rain gauge–measured rainfall:

*Z*

_{p}

*r*

_{m}

*E*

*Z*

_{m}

*R*

_{m}

*r*

_{m}

*α*

_{2}

*r*

^{β2}

_{m}

*Z*

_{p}is the reflectivity predicted for measured rainfall equal to

*r*

_{m}. The estimated parameters

*α*

_{2}and

*β*

_{2}are then applied to predict rainfall from measured reflectivity:

*R*

_{p2}

*z*

_{m}

*α*

^{−1/β2}

_{2}

*z*

^{1/β2}

_{m}

*β*

_{2}can be achieved from the previous result for

*d*

_{1}[see (17a)] by utilizing a symmetry between (6) and (17). Note that the physics of the model defined by (1) and (2) is invariant to the following transformation of the variable and parameter space:

*E*

_{R}with

*E*

_{Z}, etc.). Specifically, the parameter

*d*

_{1}in (6) is replaced by

*β*

_{2}in (17), and the straightforward consequence of this symmetry is the solution

*D*

_{ER}

*D*

_{EZ}

### Probability matching method

*Z*

_{m}–

*R*

_{p}conversion function,

*R*

_{p3}

*z*

_{m}

*α*

^{−1/β3}

_{3}

*z*

^{1/β3}

_{m}

*R*

_{p3}and the rain gauge–measured rainfalls

*R*

_{m}have the same (matched to each other) marginal probability distributions:

*R*

_{p3}

*Z*

_{m}

*R*

_{m}

*Z–R*tuning criterion perhaps is stimulated by the desire to make the radar rainfall products mimic statistical behavior of the rain gauge data with which the hydrometeorology community has been familiar for decades. In our model, estimates of the two parameters

*α*

_{3}and

*β*

_{3}that fulfill (23) exist and are unique because of the fact that the lognormal distribution also is determined uniquely by its two parameters. If two variables have the same distribution, then their functions also are distributed identically. Thus, we can rearrange (22) and (23) as follows:

*β*

_{3}:

## Discussion of the results

*Z*

_{m}–

*R*

_{p}relationship, which could be used to convert radar-measured reflectivities into radar-predicted rainfall. The three

*Z*

_{m}–

*R*

_{p}exponents originating from these procedures are

*b,*which in our model plays the role of an exponent of physical

*Z*

_{a}–

*R*

_{a}dependency. One can see that, based on direct nonlinear regression, the estimated exponent

*β*

_{1}always exceeds the assumed

*b.*This result is due to the errors in measurements of the surface reflectivity, which are numerous (Zawadzki 1984) and undoubtedly large enough to make the coefficient of the radar measurement error

*D*

_{EZ}

*β*

_{2}is always smaller than

*b*because of the extreme variability of rainfall intensity fields and the resulting area–point differences. In both cases the departure of the estimated exponent from the original

*b*depends on the error factor of the regressor variable only. This dependence reflects a general fact that regression can filter out the random measurement errors of the response variable (Seber 1989). In contrast to the regression techniques, the

*β*

_{3}that results from the PMM depends on a combined effect of both radar and rain gauge error factors. From the point of view of correct retrieval of the intrinsic

*Z*

_{a}–

*R*

_{a}relationship, these factors work in opposite direction in (27c) and can compensate for each other. This compensation can happen only in favorable conditions when the coefficients

*D*

_{EZ}

*D*

_{ER}

*Z*

_{m}–

*R*

_{p}exponent is a geometric mean of the two solutions given by the direct and reverse nonlinear regressions. At present, we cannot tell whether this feature is characteristic for our model only or perhaps has more general meaning.

The dependence of the estimated *Z–R* exponent on the estimation method and on the uncertainties can be explained based on statistical theory of the “error-in-variable” problem (Fuller 1987). In a nutshell, errors in both the predictor and the response variables make the estimation of an underlying dependency a mathematically ill-posed problem. In practice, this fact leads to biased parameter estimates of a functional dependency between the true variables for any standard estimation method. The theory also shows that the problem cannot be solved without additional information that can make the mathematical system complete. This requirement poses a practical question about the conditions under which one could solve fully the system described by our radar–rain gauge model in terms of unbiased estimation of all the model parameters. An advantage of this complete solution would be identification of the underlying physical system and its separation from the distorting effects caused by the observation and data analysis tools (Krajewski et al. 1996). The problem is not trivial, especially for a nonlinear system like that in our model (Carroll et al. 1995; Seber 1989). Additional information on statistical properties of small-scale rainfall, which could be collected from specially designed rain gauge networks (Krajewski et al. 1998), probably could help in answering such questions. However, further pursuit of this direction is beyond the scope of this study.

## Summary and conclusions

In this study, an idealized and analytically tractable model of joint radar–rain gauge rainfall observations was developed. It was applied to analyze three common *Z–R* estimation schemes. We showed that under large observational uncertainties the three techniques result in quite different estimates of the *Z–R* exponent. All three also can be significantly different from the value of the intrinsic *Z–R* exponent assumed in the model. The analysis indicates a strong impact of the magnitude of radar and rain gauge measurement errors on the estimation results. This impact suggests that, even for the same estimation method and the same rainfall regime, significant differences of the estimated *Z–R* relationships still might exist among different radar sites. These differences are caused by the fact that the sites often have their specific technical characteristics that can impact the *D*_{EZ}*D*_{ER}*Z–R* estimates that might not be associated with any physical *Z–R* differences among the regimes, if the estimates are obtained using reverse regression or the PMM. We believe that all these effects add to the abundance of the reflectivity–rainfall relationships encountered in the literature (Battan 1973). We also suspect that, apart from the problems investigated in this study, several other commonly applied procedures such as arbitrary manipulations with data samples or nonlinear variable transformations can affect the estimated *Z–R*s in a way that often is not understood fully. Perhaps more specific information on the applied data analysis methods, together with conceptual tools that we started to develop here, could facilitate systematic interpretation of the literature results.

Different *Z–R* relationships used in hydrometeorological practice imply different properties of the resulting radar–rainfall products. To assess the utility of these products from the user point of view, appropriate criteria have to be applied. We do not address this subject in this study, and we refrain from discussing the relative value of the three estimation methods for radar-based rainfall prediction. Instead, we advocate an explanatory potential of the simple model developed here to provide insight into the impact of observational uncertainties on the radar estimates.

The possibly large differences between the estimated and the actual *Z–R* relationships discussed here can be important from both the physical and practical points of view. This study shows that distinguishing the natural variability of the *Z–R* exponent from the observational effects is not a trivial task. Statistical literature on the error-in-variable problem suggests that, apart from joint radar–rain gauge data, additional information on the measurement errors is required to solve it. We hope that the new experimental efforts under way, such as the Tropical Rainfall Measuring Mission (Simpson et al. 1988) field campaigns, geared toward collecting small-scale rainfall data, will provide the needed information.

## Acknowledgments

This work was supported by NASA Grant NAG 5-2084 and by the United States Agency for International Development Grant HRN-5600-G-00-2037-00. G. J. Ciach also was supported by NASA under Graduate Student Fellowship in Global Change Research, NASA Reference 4146-GC93-0225 (Award NGT 30160). This support is gratefully appreciated.

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