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{\bf Optimality of empirical $Z$--$R$ relations}\\
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\normalsize
\begin{center}
Ziad S.\ Haddad \\
{\em Jet Propulsion Laboratory, California Institute of Technology}\\
{\em Pasadena, California} \\[\baselineskip]
and \\[\baselineskip]
Daniel Rosenfeld \\
{\em The Hebrew University of Jerusalem}\\
{\em Jerusalem, Israel} 
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\today        
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\pagebreak

\vspace*{.5in}

\begin{center}
{\bf Summary}
\end{center}
This paper attempts to justify mathematically the two empirical approaches to 
the problem of deriving $Z$--$R$ relations from $(Z,R)$ measurements, namely the 
power-law regression and the ``probability matching method'' (PMM). 
The basic mathematical assumptions that apply in each case are explicitly identified.
In the first case, the appropriate assumption is that
the scatter in the $(Z,R)$ measurements reflects exactly the randomness in the
connection between $Z$ and $R$ due to a lack of sufficient a priori
information about either of them. In the second case, the assumption is that
the measurements have been classified into categories a priori, in a way that allows 
one to expect a nearly one-to-one correspondence between $Z$ and $R$ in each
catergory, the scatter in the measurements being due to residual noise.
The paper then shows how the assmuptions naturally lead, in the first case, to a 
``conditional-mean'' $Z$--$R$ relation of which the power-laws
are regression-based approximations, and, in the second case, to
a ``probability-matched'' relation.

\pagebreak

\section{Introduction}

\indent There are two essentially different approaches to establishing 
radar-reflectivity $\leftrightarrow$ rain-rate relations from experimental data.
The first (and original) technique to estimate rainfall using radar measurements 
is based on 
the physical relation between the rain parameters (rain rate $R$, drop size
density function) and the radar reflectivity coefficient $Z$ (see e.g.\ Ryde
1946, Marshall and Palmer 1948). Originally, a simple power law $Z = aR^b$ was used.
Subsequent regression analyses of measured data consisting of simultaneous
observations of rain intensities and radar reflectivities have produced a plethora 
of power-law $Z$--$R$ relations, showing large variations in the value of the 
coefficient $a$ and the exponent $b$ (see e.g.\ Battan 1973). To be fair, it
should be stated that most of these relations were calculated from 
disdrometer-measured drop size histograms, typically using sample moments to estimate 
the corresponding values of $Z$ and $R$. In addition to the non-stationarity of 
the drop size distribution itself, the main sources of randomness in the 
resulting values of $Z$ and $R$ include errors in the drop fall velocity function
(which only affect $R$), and the different bias effects of the disdrometer
measurement errors and finite sample size on the estimation of the corresponding
moment ($6^{\rm th}$ in the case of $Z$, $\simeq 3.67^{\rm th}$ in the case of $R$)
using notoriously misbehaved sample moments, these bias effects depending
strongly on the actual shape of the drop size density function.
Several authors have
proposed subjective classification criteria to reduce these ambiguities (see
e.g.\ Austin 1987). Typical subjective classifying categories include drizzle, 
thunderstorm rain, wide-spread rain, convective cell rain, etc. 

An altogether different second approach to relate $Z$ to $R$ and reduce these 
ambiguities was originally proposed by Calheiros and Zawadski (1987), and 
later developed and extended by Atlas and Rosenfeld (see e.g.\ Atlas et al 1990,
Rosenfeld et al, 1994 and 1995b). In 
its present form, the resulting ``probability matching method'' (PMM) seeks to 
classify the type of rain
at hand using objective criteria before attempting to derive the 
appropriate $Z$--$R$ relation. Rather than expressing the $Z$--$R$ relation in
terms of a given set of rain parameters, the method first classifies the rain
regime according to quantitatively robust criteria, then derives the appropriate 
$Z$--$R$ relation directly from {\em marginal} $Z$ and $R$
data, collected from sample events governed by that particular regime.
The resulting relations are therefore no longer given by analytic closed-form 
expressions,
but they are just as efficiently computable as the parametric closed-form relations 
described above. 

This paper attempts to justify mathematically the two empirical approaches to 
the problem of deriving $Z$--$R$ relations: the power-law regression and the
PMM. In each case, we identify explicitly the basic mathematical assumptions 
that apply, and we show in what way they lead to a power-law relation, in the 
first case, and a ``probability-matched'' relation in the second. 
It turns out that
the starting mathematical hypotheses appropriate to each case are
\vspace*{-.2in}
\begin{itemize}
\item[1)] With little or no a priori climatological or physical information about
the rain event, how should one use
simultaneous $(Z,R)$ measurements to derive the optimal $Z$--$R$ relation which
minimizes the variance associated to this (perhaps quite large) class of rain
events.
\item[2)] With careful a priori classification of the rain event according to
climatological, physical and geometric considerations, how should one use $Z$-
and $R$- measurements to derive the exact $Z$--$R$ relation associated with the
particular rain regime at hand.
\end{itemize}
\vspace*{-.2in}
The main difference is in the amount 
of effort one decides to expend a priori in classifying the rain events according to
the relevant climatological/physical/geometric considerations. In each of the two
cases, the guiding principle is to make sure that the $Z$--$R$ relation derived 
is optimal according to the mathematical assumptions made.
This ensures that, given a particular rain event, the estimates obtained
using either one of these two approaches can be rigorously justified without
resorting to extraneous assumptions.

The first set of assumptions leads to a conditional-mean $Z$--$R$ relation,
the optimal relation in this case. It is discussed in section 2, where it is then 
shown how the power laws $Z = aR^b$ are more or less reasonable approximations 
of the optimal conditional-mean $Z$--$R$ relation. In section 3, we show how 
the second set of assumptions naturally leads to the PMM. 
In this case, the PMM is itself the optimal relation.

We take this opportunity to point out that other approaches, such as the
``parametric'' (for lack of a better description) method described in Haddad et
al (1996a) and applied to estimate rain profiles in Haddad et al (1996b), can be
used. We shall confine this discussion to the two empirical methods described
above.  

\section{$Z$--$R$ using no a priori classification}

\indent Assume that a large number of careful simultaneous $(Z,R)$
measurements were collected using disdrometers or radar-and-rain-gauges
during one or more typical rain events, and 
that a sampled joint probability density function ${\cal P}_{(Z,R)}$ has been
compiled.
Without accounting for horizontal and vertical inhomogeneities in the drop
distribution, which can cause
non-uniformities in the $Z$- and $R$- measurements, or for the
possible changes in the stratiform/convective nature of the rain event as it 
evolves in
time, $Z$ and $R$ will still be correlated, but there
will be no one-to-one correspondence between measured reflectivity levels and
single-point rain-rate figures. In this case therefore, it is not reasonable to
look for a deterministic $Z$--$R$ relationship. Rather, the best one can do under
these assumptions is to look for that function $R = f(Z)$ which makes, on average,
the smallest error as calculated from the observed $(Z,R)$ measurements. It is
a basic result of probability theory that $f(z)$ is
the conditional mean ${\cal E}\{R \,|\, Z = z \}$ of $R$ given that $Z=z$,
\begin{equation}
f(z) \,=\, {\cal E}\{R \,|\, Z = z \} \,=\, { \int r \, {\cal P}_{(Z,R)} (z,r) dr 
\over \int \,{\cal P}_{(Z,R)} (z,r) dr } 
\label{deal30}
\end{equation}

Yet, in this case, traditionally, power-law relations of the form $Z = a R^b$
have been used. How are they related to the statistically optimal approach ? 
Practically, the optimal formula above would lead one to infer directly the 
appropriate relation using the sample conditional mean. However if the conditional 
mean of $R$ given that $Z = z$ does turn out to be a power-law function of $z$, then 
one should obtain the same values for $a$ and $b$, 
whether one calculates the power-law parameters $a$ and $b$ directly from the
sample conditional mean or by performing a regression on the data.
Three problems arise: first and most obvious, if the conditional mean of 
$R$ given $Z$ is not exactly a power-law function of $Z$, the power-law regression 
would produce a correspondingly inaccurate fit. Figure 1a reproduces an example 
due to Short et al (1993), showing one instance in which no power law can adequately
describe the governing $Z$--$R$ relation. Figure 1b shows the optimal
conditional-mean relation for this data, obtained using (\ref{deal30}), along with the
power-law regression curves corresponding to the two more or less distinct rain 
regimes governing different segments of the data. 

A second drawback of using unclassified data to infer a $Z$-$R$ relation, 
whether one chooses to use a power-law approximation or the sample
conditional mean, is that one then needs a number of
simultaneous and co-located $(Z,R)$ measurements that is large enough to build a 
suitably accurate
joint density function ${\cal P}_{(Z,R)}$. In practice, it is quite difficult to
make measurements of the pair $(Z_e,R)$ that can be considered simultaneous,
$Z_e$ denoting the radar-estimated $Z$ over the rain gauge and $R$ the
rain-gauge-measured rain intensity. Moreover, collecting a sufficiently large 
number of samples can affect the homogeneity of the resulting $(Z,R)$ population, 
as the example of Fig.\ 1a illustrates. 

Finally, while the conditional-mean approach
does produce a $Z$--$R$ relation which, among all possible formulas 
one can use to relate $Z$ and $R$, makes the smallest r.m.s.\ error when the data
is unclassified, this error is not necessarily small. In fact, the error can 
be estimated directly from the simultaneous $(Z,R)$ measurements: it is given 
by the square root of the (conditional) sample variance, i.e.\ by the scatter 
in the joint $(Z,R)$ measurements. When this scatter is large, the usefulness 
of the resulting $Z$--$R$ relation is uncertain. Figure 1c shows a plot of the 
relative r.m.s.\ uncertainty (i.e.\ of the square root of the conditional variance,
divided by the conditional mean), as a function of the radar reflectivity, for 
the data of Fig.\ 1a. In this case, as $Z$ increases, the relative r.m.s.\ 
uncertainty in the optimally-estimated $R$ gradually decreases to about 20\% (the
subsequent decrease when $Z$ is greater than 45 dBZ is due mainly to the
extremely small sample size). At 31 dBZ, one standard deviation still amounts to
an uncomfortably large 50\%.

\section{$Z$--$R$ using comprehensive a priori classification}

\indent In order to avoid having a $Z$--$R$ relation based on measurements 
with a relatively large variance, and, therefore, to avoid making a correspondingly 
large r.m.s.\ error in the estimation of $R$, one might try to carefully 
classify one's data a priori. In this section, we shall therefore consider a 
radically different starting hypothesis: assume that a careful climatological classification
allows one to characterize the rain events as belonging to well-defined 
categories such that within each category $R$ and $Z$ can be determined from one
another exactly (with probability 1), i.e.\ such that for each category there exists 
a monotone function $f$ which satisfies $R=f(Z)$. This function 
is assumed to be derivable from physical (rather than probabilistic) considerations
appropriate to each of the rain categories within our hypothetical classification.
Thus, unlike the function of the same name in the previous case, this function $f$ 
has so far nothing to do with probabilities. Suppose now that one wants to determine
$f$ not from physical considerations, but rather from measurements of a carefully 
compiled
and classified catalogue of $Z$ and $R$ measurements. How can one use probabilities
to find $f$ from such a collection of samples ?

It turns out that, in this case, simultaneous measurements are not needed.
Indeed, suppose that we have identified the rain regime of interest, and that we
have enough $Z$- and $R$- measurements from events that fall within this regime
to construct their respective probability density functions ${\cal P}_Z$ and
${\cal P}_R$ (note that this requires much fewer samples than in the previous
approach). Indeed, since $f(Z)=R$,
\begin{equation}
\int_0^z {\cal P}_Z (x) dx \: = \: \int_0^{z} {\cal P}_R (f(x)) f'(x) dx  
\: = \: \int_0^{f(z)} {\cal P}_R (t) dt  \:.
\label{deal2}
\end{equation}
must hold for any $z \geq 0$. (\ref{deal2}) shows how matching the percentiles of
the density functions ${\cal P}_Z$ and ${\cal P}_R$ allows one to compute $f(z)$.
This result is quite different from that of the conditional-mean 
approach of the previous section, because
the fundamental a priori hypotheses are different in the two cases (indeed,
in this second case, the joint distribution of $(Z,R)$ is entirely supported on
the 1-dimensional curve $R=f(Z)$ in the $ZR$-plane).

In practice, however, it will rarely be possible to classify the rain events so
comprehensively as to end up with rain regimes each having an exact 1--1
$Z$--$R$ relation. A small amount of uncertainty, due to intrinsic
ambiguity in one's measurements, will inevitably remain. 
How does that affect the accuracy and applicability of the $Z$--$R$ relation 
obtained using formula (\ref{deal2}), derived under idealized assumptions ?

To address this question, suppose that instead of an exact relation $f(Z)
= R$, one postulates the existence of a relating function $F$ which produces a
``noisy'' $Z$--$R$ relation. Mathematically, the assumption is that there exists an
increasing function $F$ such that 
\begin{equation}
F(Z_d) = R_d + N
\label{deal3}
\end{equation}
where $N$ is a random variable representing additive noise, and where, to justify
the additivity of this source of residual randomness, we used the dB-variables
$Z_d = 10 \log_{10} (Z)$ and $R_d = 10 \log_{10} (R)$, instead of $Z$ and $R$
themselves. Given actual data, if one now computes a relating function $\hat{F}$
according to formula (\ref{deal2}), i.e.\ a function which satisfies
\begin{equation}
\int_{- \infty}^z {\cal P}_{Z_d} (x) dx \: = \: \int_{- \infty}^{\hat{F}(z)} 
{\cal P}_{R_d} (t) dt  \:,
\label{deal4}
\end{equation}
how close will this $\hat{F}$ come to the actual (optimal) $Z$--$R$ function $F$
? Under the above hypothesis (\ref{deal3}), 
\begin{eqnarray}
\int_{- \infty}^z {\cal P}_{Z_d} (x) dx & = & {\rm pr}\{  Z_d < z \} \\
& = & {\rm pr}\{  F(Z_d) < F(z) \} \\
& = & {\rm pr}\{  R_d + N < F(z) \} \\
& = & \int_{- \infty}^{F(z)} {\cal P}_{R_d +N} (t) dt \:.
\label{deal5}
\end{eqnarray}
Putting (\ref{deal4}) and (\ref{deal5}) together, this implies that $\hat{F}$ and
$F$ are related by
\begin{equation}
\int_{- \infty}^{\hat{F}(z)} {\cal P}_{R_d} (t) dt  \:=\:
\int_{- \infty}^{F(z)} {\cal P}_{R_d +N} (t) dt \:.
\label{deal6}
\end{equation}
To get a quantitative answer, let us make the simplifying assumption that $R_d$
and $N$ are independent, and that they are both Gaussian, with $N$ having 0 mean.
To be more exact, one could consider the presence of an additional
reflectivity-noise term, i.e.\ replace (\ref{deal3}) with
$F(Z_d+N_Z) = R_d + N$, where $N_Z$ and $Z$ are independent. For simplicity, we
shall
assume that the radar-reflectivity measurements are accurate enough that $N_Z$
would be small compared to the other terms and may therefore be ignored.
Equation (\ref{deal6}) then implies that  
\begin{equation}
\hat{F}(z) \: =\:
F(z) \,-\, (1 - {1 \over \sqrt{1+2\alpha}}) (F(z) - {\cal E}\{R_d\}) \,,
\label{deal7}
\end{equation}
where $\alpha = 0.5 \sigma_N^2 / \sigma_{R_d}^2$ is one-half the ratio of the
variances of $N$ and $R_d$, and where ${\cal E}\{R_d\}$denotes the average dBR in
this rain regime. Thus $\hat{F}$ under-estimates $F$ at above-average rain rates,
and over-estimates $F$ at below-average rain rates, by amounts that are
proportional to the ratio $\alpha$ of the noisy variation to the true variation.
Converting back from logarithmic quantities, if we write $F = 10 \log_{10}(f)$
and $\hat{F} = 10 \log_{10}(\hat{f})$, and rewriting (\ref{deal7})
in terms of the ideal $f$ and the retrieved $\hat{f}$, one finds that the relative
error is 
\begin{equation}
{\hat{f} - f \over f } 
\:=\: \left( {\overline{R} \over f} \right)^{1-{1\over \sqrt{1+2\alpha}}}
\,-\, 1 
\label{deal40}
\end{equation}
where $\overline{R}$ is the average rain rate in this rain regime. Thus, if
the noisy variation is 20\% as big as the true variation, i.e.\ if
$\sigma_N = 0.2 \sigma_{R_d}$ (so that $\alpha = 0.02$), in order for the
relative error to exceed 5\%, the ratio $\overline{R} / f$ must either fall below
0.08, or it must exceed 11. This means that even with a random variation that is
20\% as big as the true variation in the rain rate, the relative error incurred
in using the $Z$--$R$ relation given by the ``probability-matching'' formula 
(\ref{deal2}) will not exceed 5\%
at rain rates that lie in the interval $[ \,0.1\,\overline{R}\,,\,10\,\overline{R}\,]$
about the average rain rate.

This approach is the one adopted by Rosenfeld et al (1994) to implement the
``probability matching method'' (PMM). The classification criteria used in the 
PMM are 
\vspace*{-.2in}
\begin{itemize}
\item[a)] the effective efficiency, i.e.\ the relative
difference between the cloud top and cloud bottom vapor saturation mixing ratios
\item[b)] the bright band fraction, i.e.\ the fraction of the radar echo area in
which the maximal reflectivity occurs within $\pm 1.5$ km of the $0^{\circ}$C-isotherm
\item[c)] the horizontal reflectivity gradients
\item[d)] the freezing level itself.
\end{itemize}
\vspace*{-.1in}
The $Z$--$R$ relation obtained as described
above using these classification criteria has yielded quite accurate estimates of the
near-surface rain rate for tropical rain systems near Darwin, Australia, as
well as for winter convective rain systems in Israel. A detailed discussion of
the results is beyond the scope of this paper. We refer the interested reader to 
(Rosenfeld et al, 1995a and 1995b) for a detailed discussion of the 
classification criteria and the resulting relations.
Here, we briefly illustrate the method using the example of Fig.\ 1. 
Since the sample size is small, rather than use a comprehensive objective
classification process, we start by
(subjectively) classifying the data and retaining the data subset consisting of
those measurements which correspond to stratiform rain. Figure 2a
shows the conditional-mean estimate for the stratiform-rain data subset, obtained
using (\ref{deal30}) of the previous section, along with the two regression curves
$Z=400R^{1.47}$ and $Z=170R^{1.47}$. Figure
2b shows the corresponding relative r.m.s.\ uncertainty. Recall that
the relation shown in Fig.\ 2 is based on the assumption that the scatter in the
joint $(Z,R)$ samples exactly expresses the looseness of the correlation between $Z$
and $R$: as a result, this conditional-mean relation is visibly too ``conservative'' 
at the low-$Z$ and high-$Z$ extremes. Note also that the r.m.s.\ uncertainty
increases sharply for low reflectivities. This is quantitatively consistent with
Monte Carlo tests of the r.m.s.\ uncertainty in the PMM illustrated in Fig.\ 2
of Rosenfeld et al (1994).
In the process of extracting this data subset, 
however, we were assuming
that the classification was tight enough to allow us to expect a nearly one-to-one
$Z$--$R$ relation. Our assumption is that the residual noise in the data is 
due to some intrinsic residual ambiguity and to residual geometric effects, not
to a true lack of perfect correlation between $Z$ and $R$.
Under these assumptions, the optimal estimate is the one given by the 
probability-matching 
formula (\ref{deal2}). Figure 3 shows the resulting PMM relation, with the
stratiform-rain power-law regression and the corresponding conditional-mean
estimate overlaid. The difference is not great because the sample size is very
small indeed. But the trend is the one to be expected for larger sample sizes.

Figure 4a shows a contour plot of the percentage error that
the probability-matching relation would be making, as a function of a) the ratio 
$\sigma_N / \sigma_R$ of the noisy variation in the rain measurements to the true 
variation of the rain, and b) the ratio $R/\overline{R}$ of the rain rate to the 
mean rain rate. As observed above, as long as
$\sigma_N / \sigma_R$ is less than 20\%, the error is less than 5\% for $0.1 \leq
R/\overline{R} \leq 10$. Figure 4b shows the ratio of the r.m.s.\ scatter that is
due to residual noise to the true r.m.s.\ variation in the rain. Figure 4c shows
the difference between 4a and 4b, i.e.\ the reduction of error when using the
PMM.

Since the probability-matching relation is only an approximation
to the optimal relation when $\sigma_N$ is non-zero, one might well ask how
well the power-law relation would do if it were used in this case instead. Since
the power-law regressions are themselves only approximations of the conditional-mean
$Z$--$R$ relation, let us examine what the latter does in this case, {\em
assuming that we are able to obtain simultaneous co-located $(Z,R)$ measurements
without introducing any additional error beyond the assumption of equation}
(\ref{deal3}), i.e.\ assuming that the scatter in the joint $(Z,R)$ samples is
due entirely to (\ref{deal3}). This last condition is important because, on the 
one hand, it is practically impossible to realize, and because it is not a condition 
that one needs to worry about when using the probablity-matching relation: in that 
case, the individual marginal densities are all that is needed.
Adopting the approach of section 2, (\ref{deal3}) then implies that the joint
density ${\cal P}_{(Z_d,R_d)}$ is given by
\begin{equation} {\cal P}_{(Z_d,R_d)}(z,r) = {\cal P}_{R_d}(r) \cdot
{\cal P}_N(F(z)-r) F'(z)
\end{equation}
With the simplifying assumption that $R_d$ and $N$ are independent Gaussian with
$N$ having 0 mean, this would then imply that the conditional density
function ${\cal P}_{R_d | Z_d=z}$ is itself Gaussian with conditional mean 
\begin{equation}
{\cal E}\{R_d | Z_d=z \} \:=\: { F(z) \,+\, (\sigma_N^2 / \sigma_R^2) 
{\cal E}\{R_d\} \over 1 \,+\, (\sigma_N^2 / \sigma_R^2) }
\end{equation}
and standard deviation $\sigma_N / \sqrt{1+(\sigma_N/\sigma_{R_d})^2}$. Thus the
approach of section 2 would produce the $Z$--$R$ relation given by the
conditional mean
\begin{equation}
\hat{F}_{cm}(z) \:=\: { F(z) \,+\, 2 \alpha {\cal E}\{R_d\} 
                   \over 1 \,+\, 2 \alpha }
\label{deal19}
\end{equation}
where we have used $\alpha$ as in (\ref{deal7}) to denote 
one half the ratio $\sigma_N^2 / \sigma_{R_d}^2$ of the
variances of $N$ and $R_d$. Equation (\ref{deal19}) shows the bias between
the conditional-mean relation $\hat{F}_{cm}$ and the actual relation $F$, while
the r.m.s.\ uncertainty in the relation $\hat{F}_{cm}$ is actually
$\sigma_N / \sqrt{1+(\sigma_N/\sigma_{R_d})^2}$.
Upon converting back from logarithmic quantities, the relative bias between
this relation and the underlying actual relation $f$ is
\begin{equation}
{\hat{f}_{cm} - f \over f } \:\simeq\: \left( {\overline{R} \over f} \right)^{2\alpha
\over 1+2\alpha} \,-\, 1 \,,
\label{deal18}
\end{equation}
which is always bigger in absolute value than the relative error
(\ref{deal40}) in the probability-matching approach (because $2\alpha
/(1+2\alpha)$ is always bigger than $1 \,-\, 1/\sqrt{1+2\alpha}$). Of course, the
error in the conditional-mean $Z$--$R$ relation will increase further if one
replaces this relation by its corresponding regression-based power-law
approximation. Still, it may seem surprising that the conditional-mean approach
does not choose the relation $f$ itself as the way to estimate $R$. Rather, it
uses a ``bias'' as shown in (\ref{deal19}).
The reason for this apparent discrepancy is the following: when the noise variance
is zero, (\ref{deal19}) confirms that the conditional-mean method does use $f$
to obtain its estimate of the rain, as expected; however, as the noise increases,
the conditional-mean approach biases its
estimate towards the average ${\cal E}\{R\}$, until, in the limit, when the
noise variance tends to infinity, it (properly) rejects the $Z$--$R$ relation
altogether and prefers to estimate $R$ by its a-priori-known mean ${\cal E}\{R\}$
without any regard to the useless value of $Z$. This bias is responsible
for the fact that the resulting relative difference between $\hat{f}_{cm}$ and the
underlying $f$ is greater than the difference between the probability-matched
$\hat{f}$ and $f$. In the power-law regressions, it would translate into an
exaggerated underestimate of the rain-rate associated to higher reflectivities,
and an overestimate of the rain-rate associated to lower reflectivities.

\section{Conclusions}

\indent The two empirical methods of deriving $Z$--$R$ relations, power-law
regressions and the PMM-based approach, are not directly comparable because they
start with fundamentally different underlying mathematical assumptions about
the nature of the randomness in the data. The power laws are approximations to the
optimal relations when the original data is largely uncategorized
(in that case, the seldom-used conditional-mean method actually gives the
optimal relation).
The PMM is the optimal relation when the original data is classified a
priori in such a way that one may reasonably expect a one-to-one correspondence
between $Z$ and $R$. The PMM remains a better approximation than the
conditional-mean method (and, a fortiori, better than the regression-based power
laws) if a relatively small amount of residual randomness is still present after
classification, i.e.\ if, up to some measurement white noise that is small
compared to the rain intensity, one can still expect that one's classification 
produces categories each admitting a one-to-one correspondence between $Z$
and $R$.

\section{Acknowledgements}

\indent We are very grateful to David Atlas for many fruitful discussions about
empirical methods of deriving $Z$--$R$ relationships. We are also grateful to Otto
Thiele and the TRMM ground truth program, and in particular to Tom Keenan,
David Short and David Wolff for providing the data. The first author's
work was performed at the Jet Propulsion Laboratory, California Institute of
Technology, under contract with the National Aeronautics and Space Administration.

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Classification of Rain Regimes by the Three-dimensional Properties of
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\item[] D.\ Rosenfeld, E.\ Amitai and D.B.\ Wolff, 1995b: 
Improved accuracy of radar WPMM estimated rainfall upon application of objective
classification criteria, {\em J.\ Appl.\ Meteor.}, {\bf 34}, 212-223
\item[] D.\ Rosenfeld, D.B.\ Wolff and E.\ Amitai, 1994: 
The window probability matching method for rainfall measurements with radar,
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\end{itemize}

\pagebreak

\begin{center}
{\bf Figure captions}
\end{center}

\begin{itemize}
\item[Figure 1a:] Convective and stratiform rain regimes of tropical squall lines
in Darwin, Australia -- Reflectivity factor versus rain rate observed on 26 January
1989, between 1800 and 2400 local time (after Short et al, 1993).
\item[Figure 1b:] Conditional-mean relation for the data of figure 1a. The dashed
lines correspond to the two power-law regressions $Z = 170 R^{1.47}$ and $Z = 400
R^{1.47}$ corresponding to the two rain regimes governing different segments of
the data.
\item[Figure 1c:] Relative r.m.s.\ uncertainty in the conditional-mean relation
of figure 1b, in \%, as estimated by the ratio of the conditional standard 
deviation to the conditional mean.
\item[Figure 2a:] Conditional-mean relation for the subset of the data of figure 1a
corresponding to stratiform rain.
\item[Figure 2b:] Relative r.m.s.\ uncertainty in the conditional-mean relation
of figure 2a, in \%.
\item[Figure 3:] Probability-matched relation for the subset of the data of figure 1a
corresponding to stratiform rain. The dashed lines correspond to the power-law
regression $Z = 400 R^{1.47}$ and the conditional-mean relation.
\item[Figure 4a:] Contour plot of the relative error made by PMM,
in \%, as a function of the ratio $\sigma_N/\sigma_R$ of noisy variation in the
measruements to true variation in the rain (in \%), and of the ratio 
$10\log_{10}(R/\overline{R})$ of the rain rate to the mean rain rate (in dB).
\item[Figure 4b:] Contour plot of the ratio $\sigma_N/\sigma_R$ of noisy 
variation in the measruements to true variation in the rain, in \%
\item[Figure 4c:] Contour plot of the difference between the relative r.m.s.\ 
variation of the noise and the relative error made by PMM, in \%
\end{itemize}

\end{document}

