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Advances in Water Resources Vol. 22, No. 6, pp 585±595, 1999
Ó 1999 Elsevier Science Ltd
Printed in Great Britain. All rights reserved
0309-1708/99/$ ± see front matter
On the estimation of radar rainfall error variance
Grzegorz J. Ciach* & Witold F. Krajewski
Iowa Institute of Hydraulic Research, The University of Iowa, Iowa City, IA 52242, USA
(Received 25 May 1998; revised 2 September 1998; accepted 15 September 1998)
One of the major problems in radar rainfall (RR) estimation is the lack of accurate reference data on area-averaged rainfall. Radar±raingauge (R±G) comparisons are commonly used to assess and to validate the radar algorithms, but
large dierences of the spatial resolution between raingauge and radar measurements prevent any straightforward interpretation of the results. We assume that
the R±G dierence variance can be partitioned into the error of the radar areaaveraged rainfall estimate, and the area-point background originating from the
resolution dierence. A robust procedure to decompose these components, named
the error separation method (ESM), is proposed, discussed, and demonstrated. If
applied to a suciently large sample, it allows the estimation of the radar error
part and description of the uncertainties of hydrological radar products in rigorous statistical terms. An extensive data set is used to illustrate the ESM application. Proportion of the error components in the R±G dierence variance is
studied as a function of rainfall accumulation time. The intervals from 5 min
through 4 days are considered, and the radar grid resolution of 4 ´ 4 km is assumed. The results show that the area-point component is a dominant part of the
R±G dierence at short time scales, and remains signi®cant even for the 4-day
accumulations. Ó 1999 Elsevier science Limited. All rights reserved
tics, the error variance. We call the approach presented
here the error separation method (ESM), because its key
concept is based on partitioning of the radar±raingauge
(R±G) dierence variance into the RR estimation error
variance and the raingauge sampling (representativeness) error variance. The method is to be applied to the
®nal products of the radar data processing, which are
gridded rainfall accumulation ®elds. The accumulation
time and the grid size depend upon the application in
which the products are to be used.
The de®nition of the RR estimation error implies
that, to conceptualize it, one needs to refer to some
objectively existing true values of rainfall accumulation
averaged over the radar grid area. They might be unknown or unobservable in every detail with the present
day technology, but one has to know that at least some
of the characteristics of the real rainfall accumulation
®elds can be measured with sucient accuracy. At
present, the only readily available source of direct information on rainfall accumulations are the commonly
used raingauges. Although they de®nitely collect real
rainwater (in contrast to the remote sensors that can
only detect signals indirectly related to rainfall), their
1 INTRODUCTION
A common goal of radar rainfall (RR) estimation
techniques is to produce grid-averaged rainfall accumulations that are as close to the truth as possible. The
dierence between the estimates and the true rainfall
accumulation over the grid will be called hereafter the
RR estimation error, or shortly the RR error. Two aspects of quantitative use of radar observations can be
distinguished: validation and estimation. We de®ne the
validation problem as quanti®cation and statistical
characterization of the RR error. The estimation problem refers to building algorithms that impose certain
desired properties on the error. From this perspective,
reliable error assessment starts to play a key role in rational quantitative utilization of the radar remote sensing information.
In this study, a statistical method is proposed that,
under certain conditions, allows practical assessment of
one of the important RR estimation error characteris-
*
Corresponding author.
585
586
G. J. Ciach, W. F. Krajewski
collection area is of the order of 100 cm2 only. On the
other hand, area resolution of remote sensing products
is de®ned by the size of the grid over which the estimated
rainfall is averaged. In the case of a hydrological radar,
the averaging area is of the order of 1±10 km2 . If recent
developments on the extreme spatiotemporal rainfall
variability are taken into account (Lovejoy and Schertzer,19 Crane,7 Over and Gupta,20 and references therein), it becomes clear that direct comparisons of data
from the two sensors are problematic. The large resolution dierence of as much as 9 orders of magnitude (in
area) must cause large dierences of the statistical
sampling properties of the extremely variable rainfall
process. This is a source of fundamental diculties resulting in the fact that raingauge data cannot be directly
treated as a ground truth reference for the area-averaged
rainfall. Although comparisons of radar data with the
corresponding raingauge accumulations are often carried out by hydrometeorologists to assess the quality of
the RR estimates, many of them realize that the R±G
dierence cannot be treated as RR estimation error
because raingauges do not measure spatially averaged
rainfall accumulations (Zawadzki,26 Harrold et al.,10
Krajewski,15 Kitchen and Blackall13). However, the
consequences of this fact for the estimation/validation
problems are still far from being fully recognized and
understood. Also, no analysis of those consequences,
using systematic statistical apparatus, has been oered
so far. The ESM formulated in this paper is an attempt
to partially ®ll this gap.
From the statistical point of view, the RR estimation
error can be described in terms of a spatiotemporal
stochastic process that contains all the information
necessary to obtain the statistical distribution of the
dierence between the estimated and the actual rainfall
accumulations for any required area and time interval.
In practice, complete quanti®cation of this stochastic
process is not possible. To make the problem tractable,
one has to con®ne interest to some key characteristics
only, and to apply reasonable models of the rainfall
statistical structure and of the measurement processes.
In this study, we focus our attention on one of the point
statistics of the RR estimation error process, its variance
as a function of rainfall accumulation time. This is done
for two reasons. First, it is a standard and commonly
used characteristic of any random variable, describing
its variability around the mean value, to which one can
apply the apparatus of variance partitioning that is well
established in statistics (see e.g. Johnson and Wichern12). Second, the RR estimation error variance seems
to be the most important error feature, after the estimation error overall long-term bias. The bias, although
an important problem in itself, is much easier to deal
with. De®ned as a ratio of long-term radar and raingauge rainfall averages, it can be fairly reliably determined and removed from the RR products as shown, for
example, by Steiner et al.25 or Ciach et al.6 The random
component of the RR estimation error, represented here
by its variance, is much more dicult to assess and to
control. Building a systematic apparatus for its extraction from the R±G joint statistics is the purpose of this
work.
2 THE CONCEPT OF THE ESM
2.1 Partitioning of R±G dierence variance
Let us consider an algorithm that produces RR estimates for a given accumulation time interval and over a
speci®ed area (a radar grid, for example). The variance
of the R±G dierence can be expressed as follows:
VarfRr ÿ Rg g Varf Rr ÿ Ra ÿ Rg ÿ Ra g
VarfRr ÿ Ra g ÿ 2 Cov fRr ÿ Ra ; Rg ÿ Ra g
VarfRg ÿ Ra g;
1
where Ra is the true area-averaged rainfall accumulation
for the speci®ed time and area, Rr its radar estimate, Rg
a reading of the raingauge which is located within a
radar grid, and Var{á} and Cov{á, á} are respectively
unconditional (including zeros) variance and covariance
of random variables. If one can assume (this assumption
will be discussed in Section 4) that the dierences between the truth and the radar, and between the truth
and the raingauge, are uncorrelated, then the above R±
G variance can be partitioned as follows:
VarfRr ÿ Rg g VarfRr ÿ Ra g VarfRg ÿ Ra g:
2
Recognizing the importance of the area-point dierence, we propose to estimate the second part of the
right-hand term in eqn (2), and consequently to obtain
an estimate of the RR estimation error variance as:
VarfRr ÿ Ra g VarfRr ÿ Rg g ÿ VarfRg ÿ Ra g:
3
Formula (3) enables the assessment of one of the
important RR estimation error attributes through subtracting from the overall R±G variability the part caused
by the raingauge sampling error. It is based on the
partitioning of the R±G dierence variance and that is
why we call this approach the ESM.
The RR estimation error based on eqn (3) is determined as a dierence of two terms and in some cases
might be subjected to large estimation errors, i.e. when
the terms are much bigger than their dierence, and
when they cannot be assessed with sucient accuracy.
The evaluation of the R±G dierence variance is fairly
straightforward:
VarfRr ÿ Rg g
N
1X
Rr i ÿ Rg i2 ;
N i1
4
and its precision depends mainly on the size N of the
data sample. This, however, does not apply to the
On the estimation of radar rainfall error variance
second term in eqn (3). Not much work has been done
so far on the problems of the area-point dierence statistics, especially at the scales comparable with typical
radar grid size. Thus, estimation of this term must be
treated carefully and will be discussed in the next section.
In eqn (4) and further in this study, it is assumed that
the RR estimation algorithm and the raingauge measurements are free of an overall bias. This means that
the unconditional ensemble means of the true area-averaged accumulations, the raingauge accumulations and
the radar accumulations are the same:
EfRg g EfRr g EfRa g;
5
where operator E{á} denotes expectation of a random
variable. In practice, the ensemble means have to be
replaced by the unconditional (including zero values)
sample averages of radar and rainfall accumulations,
and the RR estimates are adjusted for eqn (5) to be
ful®lled. This is in agreement with our focus on the
random component of the RR estimation error, and
with the fact that the sample RR bias can be removed
without much diculty prior to the ESM application.
Further agreement of the measured accumulations averaged over the sample with the true rainfall, can only be
assured by calibration of the raingauges and careful
correction of their systematic errors which will be discussed in Section 4.
The concept of statistical error separation described
above is not new and it has already been proposed by
Barnston2 in an empirical investigation. His study is
also based on variance partitioning, and thus, close to
our statistical formulation, although no reference to this
mathematical apparatus appears therein. However,
there are several major dierences between Barnston2
and this study. Barnston2 deals with relatively large
domain sizes of the order of 100 km, while our study
focuses on the scales of a single radar product grid. He
con®nes his analysis to rainfall intensities, whereas we
are interested in accumulation time scales ranging from
minutes to days. Barnston2 also proposes to estimate
the error induced by the near-point raingauge sampling
in a fairly heuristic way, by using the adjusted RR estimates to approximate the spatial structure of the
gauge-measured rainfall ®eld. It has been recognized
since then, that dierences between the spatial statistics
of near-point rainfall and the radar estimates can be
large and cannot be corrected in a simple way
(Krajewski et al.17). We believe that to achieve reasonable precision of the RR error variance assessed using
eqn (3), the area-point component has to be analyzed
more thoroughly by applying the statistical apparatus
presented below.
The error decomposition at the scales of a single radar grid was investigated by Kitchen and Blackall13 in
their observation-driven simulation. They used a dense
network of 16 raingauges scattered over an area of 9
587
km2 and assumed that their averages behave like some
perfect radar rainfall estimates. Kitchen and Blackall13
perform their analysis on the level of the logarithms of
the investigated random variable ratios, which makes
application of the rigorous variance partitioning technique problematic. Nevertheless, their results give a
preliminary insight into the importance of the raingauge
representativeness problem for the R±G comparisons.
According to their estimates, the impact of the raingauge representativeness error on the R±G dierence
can be as big as 50±80% for instantaneous and hourly
rainfalls and the grid size of 3 ´ 3 km.
2.2 Area-point dierence
The importance of spatial rainfall variability and its
implications for raingauge network sampling design has
been recognized for decades (Hendrick and Comer,11
Zawadzki,27 Harrold et al.10). The relevance of areapoint dierences to the R±G comparison problems was
discussed by Zawadzki28 who, based on Taylor hypothesis considerations, suggested raingauge accumulation times, which in a sense correspond to the rainrate
spatial averages. The eect of spatial averaging on the
rainfall variability was extensively studied by Rodriguez-Iturbe and Mejia23 and applied to hydrological
design (Rodriguez-Iturbe and Mejia,22 Bras and Rodriguez-Iturbe,4 Bras and Rodriguez-Iturbe5). These
studies, however, are almost exclusively focused on the
impact of spatial averaging on the variance of the
rainfall accumulations. Their primary goal was to establish simple engineering guidance for the evaluation of
the exceedence probabilities of extreme rainfalls over a
catchment area, based on the reduced variance estimates
and fairly crude distributional assumptions. The areapoint dierences of the rainfall accumulations were also
investigated by Bras and Rodriguez-Iturbe3 and Silverman et al.24 Those studies were concerned with spatial
scales of the order of tens and hundreds of kilometers,
typical for the operational hydrological networks. Statistics of the area-point dierences at the scales of
modern weather radar resolution, and their consequences for the RR estimation/validation questions,
have not been addressed in the literature yet.
One can describe rainfall mathematically as a spatiotemporal, complex stochastic process in a four-dimensional real number domain. The part of the process
that is of major interest in hydrometeorology concerns
surface rainfall accumulations for given areas and time
intervals. Statistically, surface rainfall can be characterized by its point distribution, which is a function of
place and time, and by its spatiotemporal dependency
structure. In this study, we will consider only the twodimensional spatial aspects of the rainfall accumulation
®elds, treating their time samples as separate, although
statistically not independent, realizations from a statistical ensemble. The basic relation between the
588
G. J. Ciach, W. F. Krajewski
near-point raingauge measurement process and the true
rainfall averages over some area A is assumed to be:
Z
1
Ra
Rg xdx2 ;
6
A A
where x is the location vector. Although natural and
simple, this relation plays a fundamental role in further
derivations and in fact de®nes the raingauge measurement process as a high-resolution sampling of the rainfall accumulation ®elds. Accepting this link allows us
omit the notion of the strictly point rainfall process and
the conceptual diculties associated with its mathematical de®nition. An implicit assumption here is that
the raingauge data are of good quality and properly
calibrated, as will be discussed in Section 4.
If one can assume that, within the domain considered, the spatial rainfall process is second order homogeneous, so that the expected values and variances of the
point rainfall ®eld are equal:
EfRg xg lg ;
7a
VarfRg xg r2g ;
7b
for each point x within the area A, then also the mean of
the area-averaged rainfall is the same:>
Z
1
Rg xdx2
EfRa g E
A A
Z
Z
1
1
EfRg xgdx2
l dx2 lg :
8
A A
A A g
From the practical point of view, it is dicult to say
what is the upper limit of the area size for the homogeneity assumption to hold. However, it is safe to say,
that the terrain-induced and synoptic scale in¯uences are
of more concern, rather than the rainfall type, since the
climatological averages usually include all regimes.
The area-point dierence variance can be expressed
as:
VarfRg ÿ Ra g VarfRg g ÿ 2 CovfRg ; Ra g
VarfRa g;
9
where the ®rst term is given by eqn (7b). The second
term can be written as:
CovfRg ; Ra g Ef Rg xg ÿ lg Ra ÿ lg g;
VarfRa g Ef Ra ÿ lg 2 g
Z
Z
1
1
2
2
Rg x ÿ lg dx
Rg yÿlg dy
E
A A
A A
Z Z
1
EfRg x ÿ lg Rg y ÿ lg gdx2 dy 2
2
A A A
Z Z
r2g
2
q x; ydx2 dy 2 :
12
A A A
Substituting eqns (7b), (11) and (12) into eqn (9), the
®nal form of the area-point dierence variance is obtained:
Z
2
2
q xg ; xdx2 dy 2
VarfRg ÿ Ra g rg 1 ÿ
A A
Z Z
1
2
2
13
q x; ydx dy ;
2
A A A
which will be used in further applications to derive estimates of the area-point part for the error separation
eqn (2) and RR error variance estimation eqn (3). According to eqn (13), the only additional information
required is a suciently detailed rainfall correlation
structure at small scales. The data have to characterize
the correlation function of the rainfall accumulation
®elds at the distances below the resolution of the RR
products.
3 RESULTS OF THE ESM APPLICATION
Below, ®rst results of the ESM implementation on a real
radar and raingauge data sample are demonstrated. At
this stage, we want this application to serve as an example rather than a full exploration of the ESM, which
will be a subject of future studies. The results are based
on a simpli®ed analysis and are limited to one radar grid
resolution 4 ´ 4 km that is characteristic for the WSR88D precipitation products (Klazura and Imy14). All the
results presented here are estimated statistics in function
of accumulation time, with the time interval lengths
ranging from 5 min to 4 days.
10
where xg is the raingauge location within the radar grid.
Substituting Ra with eqn (6) and changing the order of
expectation and integration, one obtains the explicit
form of the covariance term for the locally homogeneous ®eld:
Z
1
Ef Rg xg ÿ lg Rg x ÿ lg gdx2
CovfRg ; Ra g
A A
Z
r2g
q xg ; xdx2 ;
11
A A
where q(á, á) is the ®eld correlation function. The third
term in eqn (9), the variance of the area-averaged rainfall, can be expressed the same way, through the point
®eld variance and the correlation function:
3.1 Data sample and z±r conversion
At present, accurate high-resolution information on the
small scale rainfall accumulation correlation structure is
not easily available. Nevertheless, one can use the existing possibilities, together with additional assumptions
regarding scales smaller than the raingauge distances, to
produce rough estimates of the RR estimation error and
of the proportions between the R±G dierence variance
parts in eqn (2). In this study, we use good quality radar
data from the Darwin radar in Australia located at
12.458 S latitude and 130.920 E longitude. The radar
data are completed with data from a group of ®ve relatively close raingauges. This data sample was prepared
On the estimation of radar rainfall error variance
and carefully scrutinized by NASA, and was a part of
the NASA TRMM Ground Validation experiment
(Ciach et al.6). It was selected for this study due to its
completeness and thorough quality control.
The raingauge group is located at the distance of
about 40 km East from the radar, and the distances
between the gauges range from about 2 km to about 7
km. The geometry of this small network is shown in
Fig. 1 together with a background of the 4 ´ 4 km radar
grid used here. The distances between gauges cover the
range that is fairly close to the typical radar grid resolution, although more information from the scales of
hundreds of meters would de®nitely improve the reliability of the error estimates. The data sample covers a
Fig. 1. Schematic of the raingauge network used to illustrate
the ESM application. The ®ve gauges are shown on the
background of the 4 ´ 4 km radar grid. The center of the region is 40 km East and 14 km South from the radar.
589
period of 20 days from December 24, 1993 to January
12, 1994. During that period each gauge collected on the
average about 200 mm of rainfall.
The radar data sample for this period consists of 2778
volume re¯ectivity scans with regular 10-minute temporal spacing. The raw radar re¯ectivities from the base
scan are converted to rainfall rates using a simple
power-law Z±R relationship Z A á Rbg . The value of
parameter b 1.26 is adjusted to minimize the mean
square R±G dierence eqn (4) between the single scan
radar rainrates and the raingauge rainrates averaged
over 5-minute intervals. The parameter A is adjusted to
remove the overall sample bias, so that the unconditional averages of RR and raingauge accumulations for
the whole analyzed period are equal as in eqn (5). The
resulting value of A obtained this way depends on a
particular radar calibration. Then, the coordinate
transformation from polar to rectangular grid with the
resolution 4 ´ 4 km is performed. The radar accumulations are created by simple averaging of the single
radar rainfall grid-values that are falling within a given
time interval. There are only a few short gaps in the
series of the radar scans and they do not signi®cantly
aect the results.
In Fig. 2, scatterograms of synchronous radar and
raingauge accumulations are presented for hourly and
daily intervals. They illustrate the behavior of the data
at the two timescales. The scatterograms also give an
idea about the large amount of variability in the measurements, which is the fundamental problem in this
study.
3.2 Estimation of the small scale correlation structure
For each accumulation time considered here, the statistical dependencies between the gauges have to be de-
Fig. 2. Scatterograms of radar and raingauge rainfall accumulations for the hourly (a) and daily (b) timescales.
590
G. J. Ciach, W. F. Krajewski
scribed through the rain®eld correlation structure in
order to estimate the area-point part eqn (13) in the
error separation formula eqn (2). To express the small
scale rain®eld decorrelation in a parsimonious way, the
correlation function model is assumed to be exponential
with the so-called nugget eect:
d
;
14
q d q0 exp ÿ
d0
where q0 < 1 is the immediate correlation jump (nugget)
parameter, d0 the long range correlation distance, and
the argument d is the separation distance. The expression 1 ) q0 gives a value of the ®eld correlation drop at
very small distances and will be called hereafter a decorrelation parameter. The nugget eect model is often
used in applied spatial statistics to describe very small
scale variability of a stochastic ®eld (Cressie8). The exponential model has been used frequently to describe the
long range correlation structure of the rainfall ®elds
(Bras and Rodriguez-Iturbe5). The assumed correlation
function is isotropic, and thus, it depends only on the
separation distance between two rain®eld points. We
believe that the model eqn (14) is ¯exible enough to
describe eects essential for the estimation of the areapoint error part eqn (13), and also that, due to very
limited information on the small scale rain®eld correlation structure, searching for a more accurate model
would be problematic.
The estimation of the two parameters in eqn (14) can
be carried out in many ways, several of them are presented and compared in a study by Zimmerman and
Zimmerman29. For the purpose of this example, we used
at ®rst a simple best-®t to the correlograms as described
in a simulation study by Krajewski and Duy16. The
preliminary analysis resulted in the distance scales d0 in
eqn (14) of the order of 30±70 km, which is much bigger
than the grid size considered in this analysis. Within the
4 km distance, the exponential part in eqn (14) is close
to unity, and so the correlation structure at the scales of
our interest is practically determined by the nugget eect
parameter q0 . This simpli®es the estimation procedure
by con®ning it to this physically signi®cant parameter.
The assumption of second order homogeneity in the
small area domain considered here implies equal rainfall
variances for the ®ve gauges and the same correlation
between them equal to q0 . This allows us building an
estimator of q0 that is based on the method of moments
and utilizes the comparison of two easily computed
statistics, the single gauge rainfall variance, and the
variance of their average:
(
)
n
1X
Rg i
VarfRav g Var
n i1
1
n VarfRg g n n ÿ 1 CovfRg i; Rg jg
n2
1
15
VarfRg g n ÿ 1 VarfRg gq0 ;
n
^
Fig. 3. Decorrelation parameter 1 ÿ q0 estimated based on
(16), as function of rainfall accumulation interval.
where n is the number of gauges in the group (n 5) and
Rav the rainfall average over those gauges. From that,
after substituting the variances with their sample estimates, one can obtain the following expression for the
q0 estimator:
^
^
q0
^
n VarfRav g ÿ VarfRg g
^
;
16
n ÿ 1VarfRg g
which is used for the analyses presented in this section.
^
The estimates of 1 ÿ q0 are shown in Fig. 3 as function
of the accumulation time. Regular drop of the decorrelation with the time scale is evident. Its value is about
0.3 for the ®ve-minute accumulation interval. Up to
about daily accumulations, the drop is roughly powerlaw (close to linear in log±log scale) with the exponent
coecient equal to about )0.4. In the range of longer
accumulation periods (from 1 to 4 days) the drop is
again approximately power-law, but is much faster and
the exponent coecient is equal to about )0.9 in this
region. However, this is only a rough description of the
results and should not be treated as a parametrization of
the curve in Fig. 3.
3.3 The ESM in action
For each time scale, the results on the small scale rain®eld correlation structure described above are applied to
approximate the area-point dierence variance eqn (13).
As before, due to the fact that the correlation distance d0
in the exponential part of eqn (14) is much bigger than
the distances considered, the integrals in eqn (13) can be
also simpli®ed:
Z
2
jxg ÿ xj
2
dx2
q exp ÿ
VarfRg ÿ Ra g rg 1 ÿ
A A 0
d0
On the estimation of radar rainfall error variance
1
A2
Z Z
A
A
jx ÿ yj
dx2 dy 2
q0 exp ÿ
d0
r2g 1 ÿ 2q0 q0 r2g 1 ÿ q0 :
17
This leads to the following estimate of the area-point
part in eqn (2):
^
^
^
VarfRg ÿ Ra g VarfRg g 1 ÿ q0 ;
18
where the point rainfall variance is approximated by the
sample variance of the raingauge accumulations.
Using this formula, one can estimate the raingauge
representativeness error variance. The overall R±G difference variance can be estimated using eqn (4). Finally,
after obtaining the radar error variance based on
eqn (3), the ESM application is completed. The basic
estimates are shown in Fig. 4 as functions of accumulation time. In part (a) of this ®gure, the solid line presents the R±G sample variance eqn (4) in
nondimensional terms ± its square root normalized by
the rainfall accumulation sample mean (averaged unconditionally over the whole sample period). The dashed
line describes the point rainfall variability in terms of its
sample coecient of variance, which is the square root
of the raingauge accumulation variance used in eqn (18)
divided by the same sample mean. Both statistics are
regularly decreasing with time, but the R±G dierence
variability drops much faster than the raingauge rainfall
variability. For example, the former factor is about 1.5
times smaller than the later for short accumulation
times, whereas for longer periods it is as much as ®ve
times smaller (the proportion of the respective variances
is about 25).
The two lines in part (b) of Fig. 4 show respectively
the standard errors of the RR estimates (solid line), and
591
the standard errors of the raingauge measurements
(dashed line) normalized by the rainfall average as for
the statistics in part (a). These errors can be obtained by
appropriate combining of the information presented in
Fig. 3 and Fig. 4(a). Both decrease monotonously with
accumulation time, however, for short periods the
raingauge error drops much faster than the radar error.
The standard error of the RR estimation presented here
is signi®cantly smaller than the RMS error traditionally
obtained from the raw R±G comparisons. However, for
the simple Z±R conversion used here, it is still fairly
large for short accumulation periods. In relative terms, it
is about 240% for the shortest times, drops to about
100% for the 9-hour accumulations, and remains at the
level of about 20% for 4-day periods. There is still plenty
of room for improvement of the RR estimates either by
better data analysis and/or by utilizing other concurrent
information.
The results shown in Fig. 4(b) are also presented in
Fig. 5 in terms of the proportions of the error variances
for better understanding of the behavior of the error
components produced using the ESM. The percentage
of the radar part within the overall R±G dierence
variance is presented in Fig. 5(a) as function of the accumulation time. The ratio of the raingauge to radar
error variances is in Fig. 5(b). One can see that the relative impact of the raingauge representativeness
background noise is more than twice as big as the RR
error variance for short accumulation periods, which
con®rms the results of Kitchen and Blackall13 whose
simulations suggest similar proportions for the instantaneous rainrates. The proportion drop rate in Fig. 5(b)
is very fast at the beginning which means that, when the
accumulation time increases, the raingauge errors ®lter
Fig. 4. The ESM estimates as function of accumulation time. In panel (a), the RMS of the R±G dierence normalized by the rainfall
sample mean (solid line), and the coecients of variance of the raingauge accumulations (dashed line), are presented. In panel (b),
normalized standard errors of the RR estimates (solid line), and normalized standard errors of the raingauge measurements (dashed
line), are shown.
592
G. J. Ciach, W. F. Krajewski
Fig. 5. The ESM results in terms of the proportions of the error variance components, as function of accumulation time. The
percentage of the radar part within the R±G dierence variance is in panel (a), and the ratio of the raingauge to radar error variances
is in panel (b).
out much faster than the radar errors. Above the daily
timescales, the raingauge part is about 2±3 times smaller
than the RR estimation error variance, however, the
proportion drop rate becomes very slow in this region
suggesting that the area-point raingauge noise still remains a signi®cant part of the R±G comparisons. This
behavior of the long-term raingauge accumulations
might be an evidence of some persistent small scale
rain®eld variability that cannot be eectively ®ltered out
by time averaging.
Finally, we want to make an observation returning to
the rain®eld decorrelations in Fig. 3. Around the threeday accumulation periods the value of the decorrelation
parameter is about 0.01, and thus, the correlation coecient of the point and area-averaged rainfall accumulations is about 0.99. The correlation seems high
enough so that no big area-point dierences at this time
scales are to be expected. On the other hand, results in
Fig. 4 and Fig. 5 show still a fairly high level of this
noise variance. It can be explained by relatively high
level of the rainfall variance (see Fig. 3(a)) which combines with the decorrelation in eqn (18). This also
demonstrates a more general fact that, once extremely
variable processes are concerned, even very high correlation levels still leave room for large dierences between
the random variables, and that in such situations, considering the correlations only can be misleading.
4 DISCUSSION OF THE ESM ASSUMPTIONS
This section elaborates on the assumptions and requirements that are important for the eectiveness of the
mathematical framework that has been outlined above.
Many of them have been stated already, but without
discussing their merits essential to the ESM reliability
when applied to real world data.
4.1 Zero covariance assumption
The basic assumption that allows the variance partitioning as in eqn (2) is that the RR estimation error and
the area-point rainfall dierence are uncorrelated random variables:
CovfRg ÿ Ra ; Rr ÿ Ra g 0;
19
or that at least the covariance term is negligible:
CovfRg ÿ Ra ; Rr ÿ Ra g VarfRr ÿ Rg g:
20
Intuitively, it can be argued that the RR estimation
error is basically associated with the properties of the
radar measurement, whereas the area-point dierence
originates from the spatial rainfall variability. Radar
re¯ectivity depends on the overall amount, size and
electrical properties of the scattering particles within the
radar product grid, and does not depend strongly on
their speci®c distribution within the grid (only to the
extent that nonuniform beam pattern weighing is involved). In contrast, the area-point dierences depend
solely on the particular spatial distributions of the
rainfall within the radar grid. This suggests that the
covariance term in eqn (1) might be indeed negligible for
practical purpose of the ESM, but this is a heuristic
argumentation rather than a proof. One can prove
mathematically that the condition eqn (19) is ful®lled in
an average sense, when the covariance term is averaged
over all possible positions of the raingauge within the
radar grid. It is obvious that, for each realization of the
On the estimation of radar rainfall error variance
rain®eld as a stochastic process, only Rg in the covariance term depends on the raingauge location. Both areaaveraged rainfall and the radar estimate are associated
with the grid area as a whole and do not change when
dierent possible gauge positions are considered. Thus,
one can area-average the covariance term as follows:
Z
1
CovfRg x ÿ Ra ; Rr ÿ Ra gdx2
A A
Z
1
Cov
Rg xdx2 ÿ Ra ; Rr ÿ Ra
A A
CovfRa ÿ Ra ; Rr ÿ Ra g
Covf0; Rr ÿ Ra g 0:
21
This result indicates a feasible practical way out, if for
a ®xed raingauge position the covariance term happens
to be signi®cant. Estimating the R±G dierence variance
eqn (4), one needs to construct a number of radar grids
from the raw radar data, so that the gauge positions
within them cover the grid area uniformly. Derivation
eqn (21) implies that the covariance term cancels out
from the variance averaged over all the positions. Of
course, to use eqn (3) for the RR error estimation, the
same averaging has to be applied to the area-point term
eqn (13). This technique is computationally demanding
and has not been implemented is this study, as its purpose is the presentation and general discussion of the
ESM.
In many practical circumstances, one does not have
the ¯exibility to arbitrarily change the radar grid positions around the raingauge. This happens when RR
estimate products in a ®xed grid are only available, for
example, if the ESM is to be applied to evaluate the
WSR-88D (Klazura and Imy14) products. In such situations, the averaging eqn (21) of the covariance term
cannot be used, but one can use a network of many
raingauges under the radar umbrella instead. Typically,
the raingauges are located randomly within each radar
grid and this natural gauge position randomization reduces the impact of the covariance term on the RR error
variance estimates. This further suggests that the correlation term of the partitioned random variables in
eqn (1) is most likely negligible in practice. However, it
seems that the most convincing validation of this conclusion would be a special experiment which will be
discussed later.
4.2 Other assumptions and application questions
It is also assumed that the random variables Rr and Rg
are free of an overall bias, which is equivalent to eqn (5)
being ful®lled. This implies an appropriate calibration of
the RR estimates as mentioned in previous sections.
However, it also requires that raingauge measurements
are bias-free, ®ne resolution samples of the rainfall accumulation ®elds which, in general, does not have to be
true. The raingauges are subject to serious undercatch-
593
ment errors that have to be investigated and corrected
(Robinson and Rodda,21 Lindroth18). Here, we assume
good quality of the raingauge data and our concern is
with the random component of RR estimation error.
An important assumption that is necessary for the
derivation of the area-point dierence variance eqn (13)
is the local homogeneity of the rainfall accumulation
®eld. It requires the second order stationarity of the
stochastic ®eld within the area of a radar grid. This includes the point mean, variance and correlation function, which have to be constant within the grid, as
demonstrated by the derivation eqn (8), for example. If
a single radar grid of the size of a few kilometers is
considered, this assumption is not very demanding, except of the areas with strong orographic eects. The
homogeneity assumption might become more demanding, if several raingauges scattered under the radar
umbrella have to be combined for the sample to be
suciently big. In this situation, the large-scale rain®eld
homogeneity can be simply assumed, if realistic, or the
ESM has to be extended to cope with the ®eld nonhomogeneity. The latter is not trivial and is beyond the
scope of this paper. For the example presented in Section 4, homogeneity in the whole domain is assumed
because only a group of close gauges is considered. In a
more general situation, the grids that are candidates for
the ESM application should be scrutinized for the absence of local orographic eects, and/or statistical tests
for consistency of the rainfall sample means and variances should be applied. Elaboration on such tests is
beyond the scope of this paper.
Our discussion has to be completed with consideration of several statistical problems, if obtaining realistic
estimates is the goal. The question of sample size, which
has to be suciently large for computation of second
order ®eld statistics, is a standard problem of applied
spatial statistics (see e.g. Cressie8) and will not be analyzed here. An appropriate simulation model can be an
excellent tool to investigate sampling properties of the
EMS and to determine its data demands for dierent
rainfall ®eld characteristics. The spatiotemporal ®eld
ergodicity that allows the estimation of the ensemble
statistics through their spatiotemporal sample averages
is a deep and dicult question and scienti®c literature
on it is abundant. Of course, the rainfall process ergodicity is implicitly assumed here by attributing to the
sample statistics a meaningful interpretation. There is,
however, another point of practical importance that is
hardly mentioned in the literature. It concerns the fact
that, if one uses a rainfall data sample that includes
raingauges from a large area of hundreds of kilometers
and several months of observations that include many
types of precipitation systems, then treating it as a
statistically homogeneous sample might be a crude
simpli®cation. For example, one might ask what is the
meaning and the value of a spatial correlation function
which is evaluated from a mixture of rainfall ®elds that
594
G. J. Ciach, W. F. Krajewski
have dierent correlation distances (most likely a quite
common case). Can it still be used reliably to estimate
the errors using the ESM described here? These questions of applied statistics certainly should be thoroughly
investigated and will be included in our future studies on
the method.
5 SUMMARY AND CONCLUSIONS
In this study, we have outlined a statistical method that
allows a realistic quantitative assessment of the RR estimation error variance, an important characteristic of
the radar product uncertainties. The method is based on
partitioning of the R±G dierence variance into the error of the radar grid-averaged estimate, and the areapoint background originating from the large discrepancy between the raingauge measurement resolution and
the grid size of the RR products. The ESM can also be
applied to other remote sensing rainfall estimation
techniques (e.g. satellites) because the apparatus developed here is quite general. For practical application, the
method requires information on the rain ®eld correlation structure at the scales below the resolution of the
remote sensing rainfall products. To assure satisfactory
precision of the results, it has to be applied to a suciently large data sample.
An extensive data set is used to illustrate the ESM
application. The proportions of the error components in
the R±G dierence variance are studied as functions of
rainfall accumulation time. The times from 5 min
through 4 days are considered, and the radar grid resolution of 4 ´ 4 km, which is close to the WSR-88D
product grid, is assumed. The results suggest that the
area-point component is a dominant part of the R±G
dierence at short time scales, and remains signi®cant
even for the 4-day accumulations. The time behavior of
the RR estimation errors obtained in Section 3 is the
major objective of the ESM developed in this study. It
can be regarded as a statistically supported uncertainty
characteristic of hydrological radar products, and is
de®nitely more informative than the raw R±G comparisons which are contaminated with the raingauge representativeness noise.
The question is how to verify the ESM assumptions
and its range of applicability. Simulation models of
space-time rainfall and RR measurement (FoufoulaGeorgiou and Krajewski,9 Krajewski et al.,17 Anagnostou and Krajewski1) can be the most eective to
study some problems like the method sampling properties or its extensions to nonhomogeneous and nonstationary situations. For testing some other questions,
like the zero-covariance assumption eqn (19), the models might be less useful because they are based on other
assumptions that might be in direct relation to the ones
that are to be tested. This is aggravated by the fact that
there are practically no data on the rainfall structure at
the scales of the order of tens and hundreds of meters to
calibrate such models.
It seems that the only way to get more insight into the
impact of the small scale rainfall variability on the remote sensing techniques is through analysis of suciently detailed experimental evidence. Carefully
designed, very high resolution networks covering several
single radar grids could probably tell us more on the
subject than hundreds of raingauges scattered within the
whole radar observation ®eld. To approach directly the
radar estimation/validation problems, one would like to
be able to accurately measure the actual rainfall accumulations with adequate spatial resolution. Most likely,
if this ground truth were to be based on a dense raingauge network, its spatial resolution should be at least
one order of magnitude better than the radar product
resolution. This brute force solution is unfeasible on the
operational basis, however, a few experiments would
give information necessary to parameterize the problems
involved and to develop more ecient methods. Such
experiments based on small scale network designs are
currently being implemented at Iowa City Municipal
Airport in Iowa, at the Washita Basin in Oklahoma, and
near Warsaw in Poland. Thus, repeating our analysis
with another data and obtaining more accurate information on the radar and raingauge uncertainties will be
possible in the near future.
Regarding operational exploitation of the ESM and
other RR estimation and validation methods, new designs of the operational raingauge networks are required
to initiate substantial progress in this area. We want to
conclude by suggesting a concept of a hierarchical cluster network designed to collect the information on small
scale rain®eld variability, together with the usual rainfall
accumulations. Its essence is in installation of clusters of
two or more raingauges at a site. Such small clusters,
with the gauges separated by a 1±2 m, would improve
the reliability of the data collection. Separating these
elementary clusters with distances ranging from tens to
hundreds of meters would provide information on the
small scale variability on a routine basis. Such designs
could be quite inexpensive nowadays and could also
help in solving other numerous dilemmas of the remote
sensing rainfall estimation.
ACKNOWLEDGEMENTS
This work was supported by NASA grant NAG 52084. We would like to thank the NASA TRMM oce
for the preparation of a good quality data sample. We
also appreciate helpful comments provided by C.G.
Collier on an earlier version of this paper.
On the estimation of radar rainfall error variance
REFERENCES
1. Anagnostou, E. N. and Krajewski, W. F., Simulation of
radar re¯ectivity ®elds: algorithm formulation and evaluation. Water Resour. Res., 1997, 33(6), 1419±1429.
2. Barnston, A. G., An empirical method of estimating
raingauge and radar rainfall measurement bias and resolution. J. Appl. Meteor., 1991, 30, 282±296.
3. Bras, R. L. and Rodriguez-Iturbe, I., Evaluation of mean
square error involved in approximating the areal average
of a rainfall event by a discrete summation. Water Resour.
Res., 1976, 12, 181±184.
4. Bras, R. L. and Rodriguez-Iturbe, I., Network design for
the estimation of area mean of rainfall events. Water
Resour. Res., 1976, 12, 1185±1195.
5. Bras, R. L. and Rodriguez-Iturbe, I., Random Functions
and Hydrology. Dover, New York, 1993.
6. Ciach, G. J., Krajewski, W. F., Anagnostou, E. N., Smith,
J. A., Baeck, M. L., McCollum, J. R. and Kruger, A.,
Radar rainfall estimation for ground validation studies of
the Tropical Rainfall Measuring Mission. J. Appl. Meteor., 1997, 36(6), 735±747.
7. Crane, R. K., Space-time structure of rain rate ®eld. J.
Geophys. Res., 1990, 95, 2011±2020.
8. Cressie, N. A. C., Statistics for Spatial Data. Wiley, New
York, 1993.
9. Foufoula-Georgiou, E. and Krajewski, W. F., Recent
advances in rainfall modeling, estimation, and forecasting.
In Reviews of Geophysics, US National Report to International Union of Geodesy and Geophysics 1991±1994,
1995, pp. 1125±1137.
10. Harrold, T. W., English, E. J. and Nicholass, C. A., The
accuracy of radar derived rainfall measurements in hilly
terrain. Q. J. R. Met. Soc., 1974, 100, 331±350.
11. Hendrick, R. L. and Comer, G. H., Space variations of
precipitation and its implications for rain gauge network
design. J. Hydrology, 1970, 10, 151±163.
12. Johnson, R.A. and Wichern, D.W., Applied Multivariate
Statistical Analysis. Prentice-Hall, Englewood Clis, NJ,
1992.
13. Kitchen, M. and Blackall, R. M., Representativeness
errors in comparisons between radar and gauge measurements of rainfall. J. Hydrology, 1992, 134, 13±33.
14. Klazura, G. E. and Imy, D. A., A description of the initial set
of analysis products available from the NEXRAD WSR88D system. Bull. Am. Meteor. Soc., 1993, 74, 1293±1311.
595
15. Krajewski, W. F., Co-kriging of radar-rainfall and rain
rage data, J. Geophys. Res., 1987, 92(D8), 9571±9580.
16. Krajewski, W. F. and Duy, C.J., Estimation of homogeneous isotropic random ®elds structure: a simulation
study. Computers and Geosciences, 1988, 14, 113±122.
17. Krajewski, W. F., Anagnostou, E. N. and Ciach, G. J.,
Eects of the radar observation process on inferred rainfall
statistics. J. Geophys. Res., 1996, 101, 26, 493±502.
18. Lindroth, A., Reduced loss in precipitation measurements
using a new wind shield for raingauges. J. Atmos. Oceanic
Technol., 1991, 8, 444±451.
19. Lovejoy, S. and Schertzer, D., Fractals, raindrops, and
resolution dependence of rain measurements. J. Appl.
Meteor., 1990, 29, 1167±1170.
20. Over, T. M. and Gupta, V. K., Statistical analysis of
mesoscale rainfall: dependence of a random cascade
generator on large-scale forcing. J. Appl. Meteor., 1994,
33, 1526±1542.
21. Robinson, A. C. and Rodda, J. C., Rain, wind and the
aerodynamic characteristics of rain gauges. Meteor. Mag.,
1969, 98, 113±120.
22. Rodriguez-Iturbe, I. and Mejia, J. M., The design of
rainfall network in time and space. Water Resour. Res.,
1974, 10, 713±728.
23. Rodriguez-Iturbe, I. and Mejia, J. M., On the transformation of point rainfall to areal rainfall. Water Resour.
Res., 1974, 10, 729±736.
24. Silverman, B. A., Rogers, L. K. and Dahl, D., On the
sampling variance of raingauge networks. J. Appl. Meteor., 1981, 20, 1468±1478.
25. Steiner, M., Houze, Jr., R. A. and Yuter, S. E., Climatological characterization of three-dimensional storm structure from operational radar and rain gauge data. J. Appl.
Meteor., 1995, 34, 1978±2007.
26. Zawadzki, I., Errors and ¯uctuations of raingauge
estimates of areal rainfall. J. Hydrology., 1973, 18,
Advances in Water Resources Vol. 22, No. 6, pp 585±595, 1999
Ó 1999 Elsevier Science Ltd
Printed in Great Britain. All rights reserved
0309-1708/99/$ ± see front matter
On the estimation of radar rainfall error variance
Grzegorz J. Ciach* & Witold F. Krajewski
Iowa Institute of Hydraulic Research, The University of Iowa, Iowa City, IA 52242, USA
(Received 25 May 1998; revised 2 September 1998; accepted 15 September 1998)
One of the major problems in radar rainfall (RR) estimation is the lack of accurate reference data on area-averaged rainfall. Radar±raingauge (R±G) comparisons are commonly used to assess and to validate the radar algorithms, but
large dierences of the spatial resolution between raingauge and radar measurements prevent any straightforward interpretation of the results. We assume that
the R±G dierence variance can be partitioned into the error of the radar areaaveraged rainfall estimate, and the area-point background originating from the
resolution dierence. A robust procedure to decompose these components, named
the error separation method (ESM), is proposed, discussed, and demonstrated. If
applied to a suciently large sample, it allows the estimation of the radar error
part and description of the uncertainties of hydrological radar products in rigorous statistical terms. An extensive data set is used to illustrate the ESM application. Proportion of the error components in the R±G dierence variance is
studied as a function of rainfall accumulation time. The intervals from 5 min
through 4 days are considered, and the radar grid resolution of 4 ´ 4 km is assumed. The results show that the area-point component is a dominant part of the
R±G dierence at short time scales, and remains signi®cant even for the 4-day
accumulations. Ó 1999 Elsevier science Limited. All rights reserved
tics, the error variance. We call the approach presented
here the error separation method (ESM), because its key
concept is based on partitioning of the radar±raingauge
(R±G) dierence variance into the RR estimation error
variance and the raingauge sampling (representativeness) error variance. The method is to be applied to the
®nal products of the radar data processing, which are
gridded rainfall accumulation ®elds. The accumulation
time and the grid size depend upon the application in
which the products are to be used.
The de®nition of the RR estimation error implies
that, to conceptualize it, one needs to refer to some
objectively existing true values of rainfall accumulation
averaged over the radar grid area. They might be unknown or unobservable in every detail with the present
day technology, but one has to know that at least some
of the characteristics of the real rainfall accumulation
®elds can be measured with sucient accuracy. At
present, the only readily available source of direct information on rainfall accumulations are the commonly
used raingauges. Although they de®nitely collect real
rainwater (in contrast to the remote sensors that can
only detect signals indirectly related to rainfall), their
1 INTRODUCTION
A common goal of radar rainfall (RR) estimation
techniques is to produce grid-averaged rainfall accumulations that are as close to the truth as possible. The
dierence between the estimates and the true rainfall
accumulation over the grid will be called hereafter the
RR estimation error, or shortly the RR error. Two aspects of quantitative use of radar observations can be
distinguished: validation and estimation. We de®ne the
validation problem as quanti®cation and statistical
characterization of the RR error. The estimation problem refers to building algorithms that impose certain
desired properties on the error. From this perspective,
reliable error assessment starts to play a key role in rational quantitative utilization of the radar remote sensing information.
In this study, a statistical method is proposed that,
under certain conditions, allows practical assessment of
one of the important RR estimation error characteris-
*
Corresponding author.
585
586
G. J. Ciach, W. F. Krajewski
collection area is of the order of 100 cm2 only. On the
other hand, area resolution of remote sensing products
is de®ned by the size of the grid over which the estimated
rainfall is averaged. In the case of a hydrological radar,
the averaging area is of the order of 1±10 km2 . If recent
developments on the extreme spatiotemporal rainfall
variability are taken into account (Lovejoy and Schertzer,19 Crane,7 Over and Gupta,20 and references therein), it becomes clear that direct comparisons of data
from the two sensors are problematic. The large resolution dierence of as much as 9 orders of magnitude (in
area) must cause large dierences of the statistical
sampling properties of the extremely variable rainfall
process. This is a source of fundamental diculties resulting in the fact that raingauge data cannot be directly
treated as a ground truth reference for the area-averaged
rainfall. Although comparisons of radar data with the
corresponding raingauge accumulations are often carried out by hydrometeorologists to assess the quality of
the RR estimates, many of them realize that the R±G
dierence cannot be treated as RR estimation error
because raingauges do not measure spatially averaged
rainfall accumulations (Zawadzki,26 Harrold et al.,10
Krajewski,15 Kitchen and Blackall13). However, the
consequences of this fact for the estimation/validation
problems are still far from being fully recognized and
understood. Also, no analysis of those consequences,
using systematic statistical apparatus, has been oered
so far. The ESM formulated in this paper is an attempt
to partially ®ll this gap.
From the statistical point of view, the RR estimation
error can be described in terms of a spatiotemporal
stochastic process that contains all the information
necessary to obtain the statistical distribution of the
dierence between the estimated and the actual rainfall
accumulations for any required area and time interval.
In practice, complete quanti®cation of this stochastic
process is not possible. To make the problem tractable,
one has to con®ne interest to some key characteristics
only, and to apply reasonable models of the rainfall
statistical structure and of the measurement processes.
In this study, we focus our attention on one of the point
statistics of the RR estimation error process, its variance
as a function of rainfall accumulation time. This is done
for two reasons. First, it is a standard and commonly
used characteristic of any random variable, describing
its variability around the mean value, to which one can
apply the apparatus of variance partitioning that is well
established in statistics (see e.g. Johnson and Wichern12). Second, the RR estimation error variance seems
to be the most important error feature, after the estimation error overall long-term bias. The bias, although
an important problem in itself, is much easier to deal
with. De®ned as a ratio of long-term radar and raingauge rainfall averages, it can be fairly reliably determined and removed from the RR products as shown, for
example, by Steiner et al.25 or Ciach et al.6 The random
component of the RR estimation error, represented here
by its variance, is much more dicult to assess and to
control. Building a systematic apparatus for its extraction from the R±G joint statistics is the purpose of this
work.
2 THE CONCEPT OF THE ESM
2.1 Partitioning of R±G dierence variance
Let us consider an algorithm that produces RR estimates for a given accumulation time interval and over a
speci®ed area (a radar grid, for example). The variance
of the R±G dierence can be expressed as follows:
VarfRr ÿ Rg g Varf Rr ÿ Ra ÿ Rg ÿ Ra g
VarfRr ÿ Ra g ÿ 2 Cov fRr ÿ Ra ; Rg ÿ Ra g
VarfRg ÿ Ra g;
1
where Ra is the true area-averaged rainfall accumulation
for the speci®ed time and area, Rr its radar estimate, Rg
a reading of the raingauge which is located within a
radar grid, and Var{á} and Cov{á, á} are respectively
unconditional (including zeros) variance and covariance
of random variables. If one can assume (this assumption
will be discussed in Section 4) that the dierences between the truth and the radar, and between the truth
and the raingauge, are uncorrelated, then the above R±
G variance can be partitioned as follows:
VarfRr ÿ Rg g VarfRr ÿ Ra g VarfRg ÿ Ra g:
2
Recognizing the importance of the area-point dierence, we propose to estimate the second part of the
right-hand term in eqn (2), and consequently to obtain
an estimate of the RR estimation error variance as:
VarfRr ÿ Ra g VarfRr ÿ Rg g ÿ VarfRg ÿ Ra g:
3
Formula (3) enables the assessment of one of the
important RR estimation error attributes through subtracting from the overall R±G variability the part caused
by the raingauge sampling error. It is based on the
partitioning of the R±G dierence variance and that is
why we call this approach the ESM.
The RR estimation error based on eqn (3) is determined as a dierence of two terms and in some cases
might be subjected to large estimation errors, i.e. when
the terms are much bigger than their dierence, and
when they cannot be assessed with sucient accuracy.
The evaluation of the R±G dierence variance is fairly
straightforward:
VarfRr ÿ Rg g
N
1X
Rr i ÿ Rg i2 ;
N i1
4
and its precision depends mainly on the size N of the
data sample. This, however, does not apply to the
On the estimation of radar rainfall error variance
second term in eqn (3). Not much work has been done
so far on the problems of the area-point dierence statistics, especially at the scales comparable with typical
radar grid size. Thus, estimation of this term must be
treated carefully and will be discussed in the next section.
In eqn (4) and further in this study, it is assumed that
the RR estimation algorithm and the raingauge measurements are free of an overall bias. This means that
the unconditional ensemble means of the true area-averaged accumulations, the raingauge accumulations and
the radar accumulations are the same:
EfRg g EfRr g EfRa g;
5
where operator E{á} denotes expectation of a random
variable. In practice, the ensemble means have to be
replaced by the unconditional (including zero values)
sample averages of radar and rainfall accumulations,
and the RR estimates are adjusted for eqn (5) to be
ful®lled. This is in agreement with our focus on the
random component of the RR estimation error, and
with the fact that the sample RR bias can be removed
without much diculty prior to the ESM application.
Further agreement of the measured accumulations averaged over the sample with the true rainfall, can only be
assured by calibration of the raingauges and careful
correction of their systematic errors which will be discussed in Section 4.
The concept of statistical error separation described
above is not new and it has already been proposed by
Barnston2 in an empirical investigation. His study is
also based on variance partitioning, and thus, close to
our statistical formulation, although no reference to this
mathematical apparatus appears therein. However,
there are several major dierences between Barnston2
and this study. Barnston2 deals with relatively large
domain sizes of the order of 100 km, while our study
focuses on the scales of a single radar product grid. He
con®nes his analysis to rainfall intensities, whereas we
are interested in accumulation time scales ranging from
minutes to days. Barnston2 also proposes to estimate
the error induced by the near-point raingauge sampling
in a fairly heuristic way, by using the adjusted RR estimates to approximate the spatial structure of the
gauge-measured rainfall ®eld. It has been recognized
since then, that dierences between the spatial statistics
of near-point rainfall and the radar estimates can be
large and cannot be corrected in a simple way
(Krajewski et al.17). We believe that to achieve reasonable precision of the RR error variance assessed using
eqn (3), the area-point component has to be analyzed
more thoroughly by applying the statistical apparatus
presented below.
The error decomposition at the scales of a single radar grid was investigated by Kitchen and Blackall13 in
their observation-driven simulation. They used a dense
network of 16 raingauges scattered over an area of 9
587
km2 and assumed that their averages behave like some
perfect radar rainfall estimates. Kitchen and Blackall13
perform their analysis on the level of the logarithms of
the investigated random variable ratios, which makes
application of the rigorous variance partitioning technique problematic. Nevertheless, their results give a
preliminary insight into the importance of the raingauge
representativeness problem for the R±G comparisons.
According to their estimates, the impact of the raingauge representativeness error on the R±G dierence
can be as big as 50±80% for instantaneous and hourly
rainfalls and the grid size of 3 ´ 3 km.
2.2 Area-point dierence
The importance of spatial rainfall variability and its
implications for raingauge network sampling design has
been recognized for decades (Hendrick and Comer,11
Zawadzki,27 Harrold et al.10). The relevance of areapoint dierences to the R±G comparison problems was
discussed by Zawadzki28 who, based on Taylor hypothesis considerations, suggested raingauge accumulation times, which in a sense correspond to the rainrate
spatial averages. The eect of spatial averaging on the
rainfall variability was extensively studied by Rodriguez-Iturbe and Mejia23 and applied to hydrological
design (Rodriguez-Iturbe and Mejia,22 Bras and Rodriguez-Iturbe,4 Bras and Rodriguez-Iturbe5). These
studies, however, are almost exclusively focused on the
impact of spatial averaging on the variance of the
rainfall accumulations. Their primary goal was to establish simple engineering guidance for the evaluation of
the exceedence probabilities of extreme rainfalls over a
catchment area, based on the reduced variance estimates
and fairly crude distributional assumptions. The areapoint dierences of the rainfall accumulations were also
investigated by Bras and Rodriguez-Iturbe3 and Silverman et al.24 Those studies were concerned with spatial
scales of the order of tens and hundreds of kilometers,
typical for the operational hydrological networks. Statistics of the area-point dierences at the scales of
modern weather radar resolution, and their consequences for the RR estimation/validation questions,
have not been addressed in the literature yet.
One can describe rainfall mathematically as a spatiotemporal, complex stochastic process in a four-dimensional real number domain. The part of the process
that is of major interest in hydrometeorology concerns
surface rainfall accumulations for given areas and time
intervals. Statistically, surface rainfall can be characterized by its point distribution, which is a function of
place and time, and by its spatiotemporal dependency
structure. In this study, we will consider only the twodimensional spatial aspects of the rainfall accumulation
®elds, treating their time samples as separate, although
statistically not independent, realizations from a statistical ensemble. The basic relation between the
588
G. J. Ciach, W. F. Krajewski
near-point raingauge measurement process and the true
rainfall averages over some area A is assumed to be:
Z
1
Ra
Rg xdx2 ;
6
A A
where x is the location vector. Although natural and
simple, this relation plays a fundamental role in further
derivations and in fact de®nes the raingauge measurement process as a high-resolution sampling of the rainfall accumulation ®elds. Accepting this link allows us
omit the notion of the strictly point rainfall process and
the conceptual diculties associated with its mathematical de®nition. An implicit assumption here is that
the raingauge data are of good quality and properly
calibrated, as will be discussed in Section 4.
If one can assume that, within the domain considered, the spatial rainfall process is second order homogeneous, so that the expected values and variances of the
point rainfall ®eld are equal:
EfRg xg lg ;
7a
VarfRg xg r2g ;
7b
for each point x within the area A, then also the mean of
the area-averaged rainfall is the same:>
Z
1
Rg xdx2
EfRa g E
A A
Z
Z
1
1
EfRg xgdx2
l dx2 lg :
8
A A
A A g
From the practical point of view, it is dicult to say
what is the upper limit of the area size for the homogeneity assumption to hold. However, it is safe to say,
that the terrain-induced and synoptic scale in¯uences are
of more concern, rather than the rainfall type, since the
climatological averages usually include all regimes.
The area-point dierence variance can be expressed
as:
VarfRg ÿ Ra g VarfRg g ÿ 2 CovfRg ; Ra g
VarfRa g;
9
where the ®rst term is given by eqn (7b). The second
term can be written as:
CovfRg ; Ra g Ef Rg xg ÿ lg Ra ÿ lg g;
VarfRa g Ef Ra ÿ lg 2 g
Z
Z
1
1
2
2
Rg x ÿ lg dx
Rg yÿlg dy
E
A A
A A
Z Z
1
EfRg x ÿ lg Rg y ÿ lg gdx2 dy 2
2
A A A
Z Z
r2g
2
q x; ydx2 dy 2 :
12
A A A
Substituting eqns (7b), (11) and (12) into eqn (9), the
®nal form of the area-point dierence variance is obtained:
Z
2
2
q xg ; xdx2 dy 2
VarfRg ÿ Ra g rg 1 ÿ
A A
Z Z
1
2
2
13
q x; ydx dy ;
2
A A A
which will be used in further applications to derive estimates of the area-point part for the error separation
eqn (2) and RR error variance estimation eqn (3). According to eqn (13), the only additional information
required is a suciently detailed rainfall correlation
structure at small scales. The data have to characterize
the correlation function of the rainfall accumulation
®elds at the distances below the resolution of the RR
products.
3 RESULTS OF THE ESM APPLICATION
Below, ®rst results of the ESM implementation on a real
radar and raingauge data sample are demonstrated. At
this stage, we want this application to serve as an example rather than a full exploration of the ESM, which
will be a subject of future studies. The results are based
on a simpli®ed analysis and are limited to one radar grid
resolution 4 ´ 4 km that is characteristic for the WSR88D precipitation products (Klazura and Imy14). All the
results presented here are estimated statistics in function
of accumulation time, with the time interval lengths
ranging from 5 min to 4 days.
10
where xg is the raingauge location within the radar grid.
Substituting Ra with eqn (6) and changing the order of
expectation and integration, one obtains the explicit
form of the covariance term for the locally homogeneous ®eld:
Z
1
Ef Rg xg ÿ lg Rg x ÿ lg gdx2
CovfRg ; Ra g
A A
Z
r2g
q xg ; xdx2 ;
11
A A
where q(á, á) is the ®eld correlation function. The third
term in eqn (9), the variance of the area-averaged rainfall, can be expressed the same way, through the point
®eld variance and the correlation function:
3.1 Data sample and z±r conversion
At present, accurate high-resolution information on the
small scale rainfall accumulation correlation structure is
not easily available. Nevertheless, one can use the existing possibilities, together with additional assumptions
regarding scales smaller than the raingauge distances, to
produce rough estimates of the RR estimation error and
of the proportions between the R±G dierence variance
parts in eqn (2). In this study, we use good quality radar
data from the Darwin radar in Australia located at
12.458 S latitude and 130.920 E longitude. The radar
data are completed with data from a group of ®ve relatively close raingauges. This data sample was prepared
On the estimation of radar rainfall error variance
and carefully scrutinized by NASA, and was a part of
the NASA TRMM Ground Validation experiment
(Ciach et al.6). It was selected for this study due to its
completeness and thorough quality control.
The raingauge group is located at the distance of
about 40 km East from the radar, and the distances
between the gauges range from about 2 km to about 7
km. The geometry of this small network is shown in
Fig. 1 together with a background of the 4 ´ 4 km radar
grid used here. The distances between gauges cover the
range that is fairly close to the typical radar grid resolution, although more information from the scales of
hundreds of meters would de®nitely improve the reliability of the error estimates. The data sample covers a
Fig. 1. Schematic of the raingauge network used to illustrate
the ESM application. The ®ve gauges are shown on the
background of the 4 ´ 4 km radar grid. The center of the region is 40 km East and 14 km South from the radar.
589
period of 20 days from December 24, 1993 to January
12, 1994. During that period each gauge collected on the
average about 200 mm of rainfall.
The radar data sample for this period consists of 2778
volume re¯ectivity scans with regular 10-minute temporal spacing. The raw radar re¯ectivities from the base
scan are converted to rainfall rates using a simple
power-law Z±R relationship Z A á Rbg . The value of
parameter b 1.26 is adjusted to minimize the mean
square R±G dierence eqn (4) between the single scan
radar rainrates and the raingauge rainrates averaged
over 5-minute intervals. The parameter A is adjusted to
remove the overall sample bias, so that the unconditional averages of RR and raingauge accumulations for
the whole analyzed period are equal as in eqn (5). The
resulting value of A obtained this way depends on a
particular radar calibration. Then, the coordinate
transformation from polar to rectangular grid with the
resolution 4 ´ 4 km is performed. The radar accumulations are created by simple averaging of the single
radar rainfall grid-values that are falling within a given
time interval. There are only a few short gaps in the
series of the radar scans and they do not signi®cantly
aect the results.
In Fig. 2, scatterograms of synchronous radar and
raingauge accumulations are presented for hourly and
daily intervals. They illustrate the behavior of the data
at the two timescales. The scatterograms also give an
idea about the large amount of variability in the measurements, which is the fundamental problem in this
study.
3.2 Estimation of the small scale correlation structure
For each accumulation time considered here, the statistical dependencies between the gauges have to be de-
Fig. 2. Scatterograms of radar and raingauge rainfall accumulations for the hourly (a) and daily (b) timescales.
590
G. J. Ciach, W. F. Krajewski
scribed through the rain®eld correlation structure in
order to estimate the area-point part eqn (13) in the
error separation formula eqn (2). To express the small
scale rain®eld decorrelation in a parsimonious way, the
correlation function model is assumed to be exponential
with the so-called nugget eect:
d
;
14
q d q0 exp ÿ
d0
where q0 < 1 is the immediate correlation jump (nugget)
parameter, d0 the long range correlation distance, and
the argument d is the separation distance. The expression 1 ) q0 gives a value of the ®eld correlation drop at
very small distances and will be called hereafter a decorrelation parameter. The nugget eect model is often
used in applied spatial statistics to describe very small
scale variability of a stochastic ®eld (Cressie8). The exponential model has been used frequently to describe the
long range correlation structure of the rainfall ®elds
(Bras and Rodriguez-Iturbe5). The assumed correlation
function is isotropic, and thus, it depends only on the
separation distance between two rain®eld points. We
believe that the model eqn (14) is ¯exible enough to
describe eects essential for the estimation of the areapoint error part eqn (13), and also that, due to very
limited information on the small scale rain®eld correlation structure, searching for a more accurate model
would be problematic.
The estimation of the two parameters in eqn (14) can
be carried out in many ways, several of them are presented and compared in a study by Zimmerman and
Zimmerman29. For the purpose of this example, we used
at ®rst a simple best-®t to the correlograms as described
in a simulation study by Krajewski and Duy16. The
preliminary analysis resulted in the distance scales d0 in
eqn (14) of the order of 30±70 km, which is much bigger
than the grid size considered in this analysis. Within the
4 km distance, the exponential part in eqn (14) is close
to unity, and so the correlation structure at the scales of
our interest is practically determined by the nugget eect
parameter q0 . This simpli®es the estimation procedure
by con®ning it to this physically signi®cant parameter.
The assumption of second order homogeneity in the
small area domain considered here implies equal rainfall
variances for the ®ve gauges and the same correlation
between them equal to q0 . This allows us building an
estimator of q0 that is based on the method of moments
and utilizes the comparison of two easily computed
statistics, the single gauge rainfall variance, and the
variance of their average:
(
)
n
1X
Rg i
VarfRav g Var
n i1
1
n VarfRg g n n ÿ 1 CovfRg i; Rg jg
n2
1
15
VarfRg g n ÿ 1 VarfRg gq0 ;
n
^
Fig. 3. Decorrelation parameter 1 ÿ q0 estimated based on
(16), as function of rainfall accumulation interval.
where n is the number of gauges in the group (n 5) and
Rav the rainfall average over those gauges. From that,
after substituting the variances with their sample estimates, one can obtain the following expression for the
q0 estimator:
^
^
q0
^
n VarfRav g ÿ VarfRg g
^
;
16
n ÿ 1VarfRg g
which is used for the analyses presented in this section.
^
The estimates of 1 ÿ q0 are shown in Fig. 3 as function
of the accumulation time. Regular drop of the decorrelation with the time scale is evident. Its value is about
0.3 for the ®ve-minute accumulation interval. Up to
about daily accumulations, the drop is roughly powerlaw (close to linear in log±log scale) with the exponent
coecient equal to about )0.4. In the range of longer
accumulation periods (from 1 to 4 days) the drop is
again approximately power-law, but is much faster and
the exponent coecient is equal to about )0.9 in this
region. However, this is only a rough description of the
results and should not be treated as a parametrization of
the curve in Fig. 3.
3.3 The ESM in action
For each time scale, the results on the small scale rain®eld correlation structure described above are applied to
approximate the area-point dierence variance eqn (13).
As before, due to the fact that the correlation distance d0
in the exponential part of eqn (14) is much bigger than
the distances considered, the integrals in eqn (13) can be
also simpli®ed:
Z
2
jxg ÿ xj
2
dx2
q exp ÿ
VarfRg ÿ Ra g rg 1 ÿ
A A 0
d0
On the estimation of radar rainfall error variance
1
A2
Z Z
A
A
jx ÿ yj
dx2 dy 2
q0 exp ÿ
d0
r2g 1 ÿ 2q0 q0 r2g 1 ÿ q0 :
17
This leads to the following estimate of the area-point
part in eqn (2):
^
^
^
VarfRg ÿ Ra g VarfRg g 1 ÿ q0 ;
18
where the point rainfall variance is approximated by the
sample variance of the raingauge accumulations.
Using this formula, one can estimate the raingauge
representativeness error variance. The overall R±G difference variance can be estimated using eqn (4). Finally,
after obtaining the radar error variance based on
eqn (3), the ESM application is completed. The basic
estimates are shown in Fig. 4 as functions of accumulation time. In part (a) of this ®gure, the solid line presents the R±G sample variance eqn (4) in
nondimensional terms ± its square root normalized by
the rainfall accumulation sample mean (averaged unconditionally over the whole sample period). The dashed
line describes the point rainfall variability in terms of its
sample coecient of variance, which is the square root
of the raingauge accumulation variance used in eqn (18)
divided by the same sample mean. Both statistics are
regularly decreasing with time, but the R±G dierence
variability drops much faster than the raingauge rainfall
variability. For example, the former factor is about 1.5
times smaller than the later for short accumulation
times, whereas for longer periods it is as much as ®ve
times smaller (the proportion of the respective variances
is about 25).
The two lines in part (b) of Fig. 4 show respectively
the standard errors of the RR estimates (solid line), and
591
the standard errors of the raingauge measurements
(dashed line) normalized by the rainfall average as for
the statistics in part (a). These errors can be obtained by
appropriate combining of the information presented in
Fig. 3 and Fig. 4(a). Both decrease monotonously with
accumulation time, however, for short periods the
raingauge error drops much faster than the radar error.
The standard error of the RR estimation presented here
is signi®cantly smaller than the RMS error traditionally
obtained from the raw R±G comparisons. However, for
the simple Z±R conversion used here, it is still fairly
large for short accumulation periods. In relative terms, it
is about 240% for the shortest times, drops to about
100% for the 9-hour accumulations, and remains at the
level of about 20% for 4-day periods. There is still plenty
of room for improvement of the RR estimates either by
better data analysis and/or by utilizing other concurrent
information.
The results shown in Fig. 4(b) are also presented in
Fig. 5 in terms of the proportions of the error variances
for better understanding of the behavior of the error
components produced using the ESM. The percentage
of the radar part within the overall R±G dierence
variance is presented in Fig. 5(a) as function of the accumulation time. The ratio of the raingauge to radar
error variances is in Fig. 5(b). One can see that the relative impact of the raingauge representativeness
background noise is more than twice as big as the RR
error variance for short accumulation periods, which
con®rms the results of Kitchen and Blackall13 whose
simulations suggest similar proportions for the instantaneous rainrates. The proportion drop rate in Fig. 5(b)
is very fast at the beginning which means that, when the
accumulation time increases, the raingauge errors ®lter
Fig. 4. The ESM estimates as function of accumulation time. In panel (a), the RMS of the R±G dierence normalized by the rainfall
sample mean (solid line), and the coecients of variance of the raingauge accumulations (dashed line), are presented. In panel (b),
normalized standard errors of the RR estimates (solid line), and normalized standard errors of the raingauge measurements (dashed
line), are shown.
592
G. J. Ciach, W. F. Krajewski
Fig. 5. The ESM results in terms of the proportions of the error variance components, as function of accumulation time. The
percentage of the radar part within the R±G dierence variance is in panel (a), and the ratio of the raingauge to radar error variances
is in panel (b).
out much faster than the radar errors. Above the daily
timescales, the raingauge part is about 2±3 times smaller
than the RR estimation error variance, however, the
proportion drop rate becomes very slow in this region
suggesting that the area-point raingauge noise still remains a signi®cant part of the R±G comparisons. This
behavior of the long-term raingauge accumulations
might be an evidence of some persistent small scale
rain®eld variability that cannot be eectively ®ltered out
by time averaging.
Finally, we want to make an observation returning to
the rain®eld decorrelations in Fig. 3. Around the threeday accumulation periods the value of the decorrelation
parameter is about 0.01, and thus, the correlation coecient of the point and area-averaged rainfall accumulations is about 0.99. The correlation seems high
enough so that no big area-point dierences at this time
scales are to be expected. On the other hand, results in
Fig. 4 and Fig. 5 show still a fairly high level of this
noise variance. It can be explained by relatively high
level of the rainfall variance (see Fig. 3(a)) which combines with the decorrelation in eqn (18). This also
demonstrates a more general fact that, once extremely
variable processes are concerned, even very high correlation levels still leave room for large dierences between
the random variables, and that in such situations, considering the correlations only can be misleading.
4 DISCUSSION OF THE ESM ASSUMPTIONS
This section elaborates on the assumptions and requirements that are important for the eectiveness of the
mathematical framework that has been outlined above.
Many of them have been stated already, but without
discussing their merits essential to the ESM reliability
when applied to real world data.
4.1 Zero covariance assumption
The basic assumption that allows the variance partitioning as in eqn (2) is that the RR estimation error and
the area-point rainfall dierence are uncorrelated random variables:
CovfRg ÿ Ra ; Rr ÿ Ra g 0;
19
or that at least the covariance term is negligible:
CovfRg ÿ Ra ; Rr ÿ Ra g VarfRr ÿ Rg g:
20
Intuitively, it can be argued that the RR estimation
error is basically associated with the properties of the
radar measurement, whereas the area-point dierence
originates from the spatial rainfall variability. Radar
re¯ectivity depends on the overall amount, size and
electrical properties of the scattering particles within the
radar product grid, and does not depend strongly on
their speci®c distribution within the grid (only to the
extent that nonuniform beam pattern weighing is involved). In contrast, the area-point dierences depend
solely on the particular spatial distributions of the
rainfall within the radar grid. This suggests that the
covariance term in eqn (1) might be indeed negligible for
practical purpose of the ESM, but this is a heuristic
argumentation rather than a proof. One can prove
mathematically that the condition eqn (19) is ful®lled in
an average sense, when the covariance term is averaged
over all possible positions of the raingauge within the
radar grid. It is obvious that, for each realization of the
On the estimation of radar rainfall error variance
rain®eld as a stochastic process, only Rg in the covariance term depends on the raingauge location. Both areaaveraged rainfall and the radar estimate are associated
with the grid area as a whole and do not change when
dierent possible gauge positions are considered. Thus,
one can area-average the covariance term as follows:
Z
1
CovfRg x ÿ Ra ; Rr ÿ Ra gdx2
A A
Z
1
Cov
Rg xdx2 ÿ Ra ; Rr ÿ Ra
A A
CovfRa ÿ Ra ; Rr ÿ Ra g
Covf0; Rr ÿ Ra g 0:
21
This result indicates a feasible practical way out, if for
a ®xed raingauge position the covariance term happens
to be signi®cant. Estimating the R±G dierence variance
eqn (4), one needs to construct a number of radar grids
from the raw radar data, so that the gauge positions
within them cover the grid area uniformly. Derivation
eqn (21) implies that the covariance term cancels out
from the variance averaged over all the positions. Of
course, to use eqn (3) for the RR error estimation, the
same averaging has to be applied to the area-point term
eqn (13). This technique is computationally demanding
and has not been implemented is this study, as its purpose is the presentation and general discussion of the
ESM.
In many practical circumstances, one does not have
the ¯exibility to arbitrarily change the radar grid positions around the raingauge. This happens when RR
estimate products in a ®xed grid are only available, for
example, if the ESM is to be applied to evaluate the
WSR-88D (Klazura and Imy14) products. In such situations, the averaging eqn (21) of the covariance term
cannot be used, but one can use a network of many
raingauges under the radar umbrella instead. Typically,
the raingauges are located randomly within each radar
grid and this natural gauge position randomization reduces the impact of the covariance term on the RR error
variance estimates. This further suggests that the correlation term of the partitioned random variables in
eqn (1) is most likely negligible in practice. However, it
seems that the most convincing validation of this conclusion would be a special experiment which will be
discussed later.
4.2 Other assumptions and application questions
It is also assumed that the random variables Rr and Rg
are free of an overall bias, which is equivalent to eqn (5)
being ful®lled. This implies an appropriate calibration of
the RR estimates as mentioned in previous sections.
However, it also requires that raingauge measurements
are bias-free, ®ne resolution samples of the rainfall accumulation ®elds which, in general, does not have to be
true. The raingauges are subject to serious undercatch-
593
ment errors that have to be investigated and corrected
(Robinson and Rodda,21 Lindroth18). Here, we assume
good quality of the raingauge data and our concern is
with the random component of RR estimation error.
An important assumption that is necessary for the
derivation of the area-point dierence variance eqn (13)
is the local homogeneity of the rainfall accumulation
®eld. It requires the second order stationarity of the
stochastic ®eld within the area of a radar grid. This includes the point mean, variance and correlation function, which have to be constant within the grid, as
demonstrated by the derivation eqn (8), for example. If
a single radar grid of the size of a few kilometers is
considered, this assumption is not very demanding, except of the areas with strong orographic eects. The
homogeneity assumption might become more demanding, if several raingauges scattered under the radar
umbrella have to be combined for the sample to be
suciently big. In this situation, the large-scale rain®eld
homogeneity can be simply assumed, if realistic, or the
ESM has to be extended to cope with the ®eld nonhomogeneity. The latter is not trivial and is beyond the
scope of this paper. For the example presented in Section 4, homogeneity in the whole domain is assumed
because only a group of close gauges is considered. In a
more general situation, the grids that are candidates for
the ESM application should be scrutinized for the absence of local orographic eects, and/or statistical tests
for consistency of the rainfall sample means and variances should be applied. Elaboration on such tests is
beyond the scope of this paper.
Our discussion has to be completed with consideration of several statistical problems, if obtaining realistic
estimates is the goal. The question of sample size, which
has to be suciently large for computation of second
order ®eld statistics, is a standard problem of applied
spatial statistics (see e.g. Cressie8) and will not be analyzed here. An appropriate simulation model can be an
excellent tool to investigate sampling properties of the
EMS and to determine its data demands for dierent
rainfall ®eld characteristics. The spatiotemporal ®eld
ergodicity that allows the estimation of the ensemble
statistics through their spatiotemporal sample averages
is a deep and dicult question and scienti®c literature
on it is abundant. Of course, the rainfall process ergodicity is implicitly assumed here by attributing to the
sample statistics a meaningful interpretation. There is,
however, another point of practical importance that is
hardly mentioned in the literature. It concerns the fact
that, if one uses a rainfall data sample that includes
raingauges from a large area of hundreds of kilometers
and several months of observations that include many
types of precipitation systems, then treating it as a
statistically homogeneous sample might be a crude
simpli®cation. For example, one might ask what is the
meaning and the value of a spatial correlation function
which is evaluated from a mixture of rainfall ®elds that
594
G. J. Ciach, W. F. Krajewski
have dierent correlation distances (most likely a quite
common case). Can it still be used reliably to estimate
the errors using the ESM described here? These questions of applied statistics certainly should be thoroughly
investigated and will be included in our future studies on
the method.
5 SUMMARY AND CONCLUSIONS
In this study, we have outlined a statistical method that
allows a realistic quantitative assessment of the RR estimation error variance, an important characteristic of
the radar product uncertainties. The method is based on
partitioning of the R±G dierence variance into the error of the radar grid-averaged estimate, and the areapoint background originating from the large discrepancy between the raingauge measurement resolution and
the grid size of the RR products. The ESM can also be
applied to other remote sensing rainfall estimation
techniques (e.g. satellites) because the apparatus developed here is quite general. For practical application, the
method requires information on the rain ®eld correlation structure at the scales below the resolution of the
remote sensing rainfall products. To assure satisfactory
precision of the results, it has to be applied to a suciently large data sample.
An extensive data set is used to illustrate the ESM
application. The proportions of the error components in
the R±G dierence variance are studied as functions of
rainfall accumulation time. The times from 5 min
through 4 days are considered, and the radar grid resolution of 4 ´ 4 km, which is close to the WSR-88D
product grid, is assumed. The results suggest that the
area-point component is a dominant part of the R±G
dierence at short time scales, and remains signi®cant
even for the 4-day accumulations. The time behavior of
the RR estimation errors obtained in Section 3 is the
major objective of the ESM developed in this study. It
can be regarded as a statistically supported uncertainty
characteristic of hydrological radar products, and is
de®nitely more informative than the raw R±G comparisons which are contaminated with the raingauge representativeness noise.
The question is how to verify the ESM assumptions
and its range of applicability. Simulation models of
space-time rainfall and RR measurement (FoufoulaGeorgiou and Krajewski,9 Krajewski et al.,17 Anagnostou and Krajewski1) can be the most eective to
study some problems like the method sampling properties or its extensions to nonhomogeneous and nonstationary situations. For testing some other questions,
like the zero-covariance assumption eqn (19), the models might be less useful because they are based on other
assumptions that might be in direct relation to the ones
that are to be tested. This is aggravated by the fact that
there are practically no data on the rainfall structure at
the scales of the order of tens and hundreds of meters to
calibrate such models.
It seems that the only way to get more insight into the
impact of the small scale rainfall variability on the remote sensing techniques is through analysis of suciently detailed experimental evidence. Carefully
designed, very high resolution networks covering several
single radar grids could probably tell us more on the
subject than hundreds of raingauges scattered within the
whole radar observation ®eld. To approach directly the
radar estimation/validation problems, one would like to
be able to accurately measure the actual rainfall accumulations with adequate spatial resolution. Most likely,
if this ground truth were to be based on a dense raingauge network, its spatial resolution should be at least
one order of magnitude better than the radar product
resolution. This brute force solution is unfeasible on the
operational basis, however, a few experiments would
give information necessary to parameterize the problems
involved and to develop more ecient methods. Such
experiments based on small scale network designs are
currently being implemented at Iowa City Municipal
Airport in Iowa, at the Washita Basin in Oklahoma, and
near Warsaw in Poland. Thus, repeating our analysis
with another data and obtaining more accurate information on the radar and raingauge uncertainties will be
possible in the near future.
Regarding operational exploitation of the ESM and
other RR estimation and validation methods, new designs of the operational raingauge networks are required
to initiate substantial progress in this area. We want to
conclude by suggesting a concept of a hierarchical cluster network designed to collect the information on small
scale rain®eld variability, together with the usual rainfall
accumulations. Its essence is in installation of clusters of
two or more raingauges at a site. Such small clusters,
with the gauges separated by a 1±2 m, would improve
the reliability of the data collection. Separating these
elementary clusters with distances ranging from tens to
hundreds of meters would provide information on the
small scale variability on a routine basis. Such designs
could be quite inexpensive nowadays and could also
help in solving other numerous dilemmas of the remote
sensing rainfall estimation.
ACKNOWLEDGEMENTS
This work was supported by NASA grant NAG 52084. We would like to thank the NASA TRMM oce
for the preparation of a good quality data sample. We
also appreciate helpful comments provided by C.G.
Collier on an earlier version of this paper.
On the estimation of radar rainfall error variance
REFERENCES
1. Anagnostou, E. N. and Krajewski, W. F., Simulation of
radar re¯ectivity ®elds: algorithm formulation and evaluation. Water Resour. Res., 1997, 33(6), 1419±1429.
2. Barnston, A. G., An empirical method of estimating
raingauge and radar rainfall measurement bias and resolution. J. Appl. Meteor., 1991, 30, 282±296.
3. Bras, R. L. and Rodriguez-Iturbe, I., Evaluation of mean
square error involved in approximating the areal average
of a rainfall event by a discrete summation. Water Resour.
Res., 1976, 12, 181±184.
4. Bras, R. L. and Rodriguez-Iturbe, I., Network design for
the estimation of area mean of rainfall events. Water
Resour. Res., 1976, 12, 1185±1195.
5. Bras, R. L. and Rodriguez-Iturbe, I., Random Functions
and Hydrology. Dover, New York, 1993.
6. Ciach, G. J., Krajewski, W. F., Anagnostou, E. N., Smith,
J. A., Baeck, M. L., McCollum, J. R. and Kruger, A.,
Radar rainfall estimation for ground validation studies of
the Tropical Rainfall Measuring Mission. J. Appl. Meteor., 1997, 36(6), 735±747.
7. Crane, R. K., Space-time structure of rain rate ®eld. J.
Geophys. Res., 1990, 95, 2011±2020.
8. Cressie, N. A. C., Statistics for Spatial Data. Wiley, New
York, 1993.
9. Foufoula-Georgiou, E. and Krajewski, W. F., Recent
advances in rainfall modeling, estimation, and forecasting.
In Reviews of Geophysics, US National Report to International Union of Geodesy and Geophysics 1991±1994,
1995, pp. 1125±1137.
10. Harrold, T. W., English, E. J. and Nicholass, C. A., The
accuracy of radar derived rainfall measurements in hilly
terrain. Q. J. R. Met. Soc., 1974, 100, 331±350.
11. Hendrick, R. L. and Comer, G. H., Space variations of
precipitation and its implications for rain gauge network
design. J. Hydrology, 1970, 10, 151±163.
12. Johnson, R.A. and Wichern, D.W., Applied Multivariate
Statistical Analysis. Prentice-Hall, Englewood Clis, NJ,
1992.
13. Kitchen, M. and Blackall, R. M., Representativeness
errors in comparisons between radar and gauge measurements of rainfall. J. Hydrology, 1992, 134, 13±33.
14. Klazura, G. E. and Imy, D. A., A description of the initial set
of analysis products available from the NEXRAD WSR88D system. Bull. Am. Meteor. Soc., 1993, 74, 1293±1311.
595
15. Krajewski, W. F., Co-kriging of radar-rainfall and rain
rage data, J. Geophys. Res., 1987, 92(D8), 9571±9580.
16. Krajewski, W. F. and Duy, C.J., Estimation of homogeneous isotropic random ®elds structure: a simulation
study. Computers and Geosciences, 1988, 14, 113±122.
17. Krajewski, W. F., Anagnostou, E. N. and Ciach, G. J.,
Eects of the radar observation process on inferred rainfall
statistics. J. Geophys. Res., 1996, 101, 26, 493±502.
18. Lindroth, A., Reduced loss in precipitation measurements
using a new wind shield for raingauges. J. Atmos. Oceanic
Technol., 1991, 8, 444±451.
19. Lovejoy, S. and Schertzer, D., Fractals, raindrops, and
resolution dependence of rain measurements. J. Appl.
Meteor., 1990, 29, 1167±1170.
20. Over, T. M. and Gupta, V. K., Statistical analysis of
mesoscale rainfall: dependence of a random cascade
generator on large-scale forcing. J. Appl. Meteor., 1994,
33, 1526±1542.
21. Robinson, A. C. and Rodda, J. C., Rain, wind and the
aerodynamic characteristics of rain gauges. Meteor. Mag.,
1969, 98, 113±120.
22. Rodriguez-Iturbe, I. and Mejia, J. M., The design of
rainfall network in time and space. Water Resour. Res.,
1974, 10, 713±728.
23. Rodriguez-Iturbe, I. and Mejia, J. M., On the transformation of point rainfall to areal rainfall. Water Resour.
Res., 1974, 10, 729±736.
24. Silverman, B. A., Rogers, L. K. and Dahl, D., On the
sampling variance of raingauge networks. J. Appl. Meteor., 1981, 20, 1468±1478.
25. Steiner, M., Houze, Jr., R. A. and Yuter, S. E., Climatological characterization of three-dimensional storm structure from operational radar and rain gauge data. J. Appl.
Meteor., 1995, 34, 1978±2007.
26. Zawadzki, I., Errors and ¯uctuations of raingauge
estimates of areal rainfall. J. Hydrology., 1973, 18,