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 ac-curate reference data on area-averaged rainfall. Radar±raingauge (R±G) com-parisons are commonly used to assess and to validate the radar algorithms, but large dierences of the spatial resolution between raingauge and radar measure-ments prevent any straightforward interpretation of the results. We assume that the R±G dierence variance can be partitioned into the error of the radar area-averaged 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 rig-orous statistical terms. An extensive data set is used to illustrate the ESM ap-plication. 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 as-sumed. 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. Ó1999 Elsevier science Limited. All rights reserved
1 INTRODUCTION
A common goal of radar rainfall (RR) estimation techniques is to produce grid-averaged rainfall accu-mulations 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 as-pects 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 prob-lem 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 ra-tional quantitative utilization of the radar remote sens-ing 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-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 (representative-ness) 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 un-known 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 in-formation 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
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PII: S 0 3 0 9 - 1 7 0 8 ( 9 8 ) 0 0 0 4 3 - 8
*
Corresponding author.
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 Schert-zer,19 Crane,7Over and Gupta,20 and references there-in), it becomes clear that direct comparisons of data from the two sensors are problematic. The large reso-lution 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 re-sulting in the fact that raingauge data cannot be directly treated as aground truthreference for the area-averaged rainfall. Although comparisons of radar data with the corresponding raingauge accumulations are often car-ried 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 Wic-hern12). Second, the RR estimation error variance seems to be the most important error feature, after the esti-mation 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 rain-gauge rainfall averages, it can be fairly reliably deter-mined and removed from the RR products as shown, for example, by Steineret al.25or Ciachet al.6The 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 extrac-tion 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 esti-mates 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:
VarfRrÿRgg Varf RrÿRa ÿ RgÿRag
VarfRrÿRag ÿ2 CovfRrÿRa; RgÿRag
VarfRgÿRag; 1
whereRais the true area-averaged rainfall accumulation for the speci®ed time and area,Rrits 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 be-tween the truth and the radar, and bebe-tween the truth and the raingauge, are uncorrelated, then the above R± G variance can be partitioned as follows:
VarfRrÿRgg VarfRrÿRag VarfRgÿRag: 2
Recognizing the importance of the area-point dier-ence, 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:
VarfRrÿRag VarfRrÿRgg ÿVarfRgÿRag: 3
Formula (3) enables the assessment of one of the important RR estimation error attributes through sub-tracting 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 deter-mined 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:
VarfRrÿRgg 1
N
XN
i1
Rr i ÿRg i2; 4
second term in eqn (3). Not much work has been done so far on the problems of the area-point dierence sta-tistics, 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 sec-tion.
In eqn (4) and further in this study, it is assumed that the RR estimation algorithm and the raingauge mea-surements are free of an overall bias. This means that the unconditional ensemble means of the true area-av-eraged accumulations, the raingauge accumulations and the radar accumulations are the same:
EfRgg EfRrg EfRag; 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 av-eraged 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 dis-cussed 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 es-timates 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 (Krajewskiet al.17). We believe that to achieve reason-able 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 ra-dar 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
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 tech-nique 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 rain-gauge 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 area-point dierences to the R±G comparison problems was discussed by Zawadzki28 who, based on Taylor hy-pothesis considerations, suggested raingauge accumula-tion times, which in a sense correspond to the rainrate spatial averages. The eect of spatial averaging on the rainfall variability was extensively studied by Rodri-guez-Iturbe and Mejia23 and applied to hydrological design (Rodriguez-Iturbe and Mejia,22 Bras and Ro-driguez-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 es-tablish 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 area-point dierences of the rainfall accumulations were also investigated by Bras and Rodriguez-Iturbe3and Silver-man et al.24Those studies were concerned with spatial scales of the order of tens and hundreds of kilometers, typical for the operational hydrological networks. Sta-tistics of the area-point dierences at the scales of modern weather radar resolution, and their conse-quences for the RR estimation/validation questions, have not been addressed in the literature yet.
near-point raingauge measurement process and the true rainfall averages over some areaAis assumed to be:
Ra1
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 measure-ment process as a high-resolution sampling of the rain-fall accumulation ®elds. Accepting this link allows us omit the notion of the strictly point rainfall process and the conceptual diculties associated with its mathe-matical 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 consid-ered, the spatial rainfall process is second order homo-geneous, so that the expected values and variances of the point rainfall ®eld are equal:
EfRg xg lg; 7a
VarfRg xg r2g; 7b
for each pointxwithin the areaA, then also the mean of the area-averaged rainfall is the same:>
EfRag E 1
From the practical point of view, it is dicult to say what is the upper limit of the area size for the homo-geneity 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:
VarfRgÿRag VarfRgg ÿ2 CovfRg;Rag
VarfRag; 9
where the ®rst term is given by eqn (7b). The second term can be written as:
CovfRg;Rag Ef Rg xg ÿlg Raÿlgg; 10 wherexgis the raingauge location within the radar grid. SubstitutingRa with eqn (6) and changing the order of expectation and integration, one obtains the explicit form of the covariance term for the locally homoge-neous ®eld:
where q(á, á) is the ®eld correlation function. The third
term in eqn (9), the variance of the area-averaged rain-fall, can be expressed the same way, through the point ®eld variance and the correlation function:
VarfRag Ef Raÿlg
Substituting eqns (7b), (11) and (12) into eqn (9), the ®nal form of the area-point dierence variance is ob-tained:
which will be used in further applications to derive es-timates of the area-point part for the error separation eqn (2) and RR error variance estimation eqn (3). Ac-cording 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 ex-ample 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 WSR-88D 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.
3.1 Data sample and z±r conversion
and carefully scrutinized by NASA, and was a part of the NASA TRMM Ground Validation experiment (Ciachet 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 reso-lution, although more information from the scales of hundreds of meters would de®nitely improve the reli-ability of the error estimates. The data sample covers a
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 tem-poral spacing. The raw radar re¯ectivities from the base scan are converted to rainfall rates using a simple power-law Z±R relationship ZAá Rbg. The value of parameter b1.26 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 uncondi-tional 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 accumu-lations 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 mea-surements, which is the fundamental problem in this study.
3.2 Estimation of the small scale correlation structure
For each accumulation time considered here, the sta-tistical dependencies between the gauges have to be
de-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
re-gion is 40 km East and 14 km South from the radar.
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-callednugget eect:
q d q0exp ÿ d d0
; 14
whereq0 <1 is the immediate correlation jump (nugget)
parameter, d0 the long range correlation distance, and the argument d is the separation distance. The expres-sion 1 )q0 gives a value of the ®eld correlation drop at very small distances and will be called hereafter a de-correlation parameter. The nugget eect model is often used in applied spatial statistics to describe very small scale variability of a stochastic ®eld (Cressie8). The ex-ponential 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 area-point error part eqn (13), and also that, due to very limited information on the small scale rain®eld correla-tion 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 pre-sented 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 scalesd0 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 ofq0that 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:
VarfRavg Var 1 n
Xn
i1 Rg i
( )
1
n2 nVarfRgg n nÿ1CovfRg i;Rg jg
1
n VarfRgg nÿ1VarfRggq0; 15
wherenis the number of gauges in the group (n5) and Rav the rainfall average over those gauges. From that, after substituting the variances with their sample esti-mates, one can obtain the following expression for the q0 estimator:
q0
^
nVar^ fRavg ÿVar^ fRgg
nÿ1Var^ fRgg
; 16
which is used for the analyses presented in this section. The estimates of 1ÿq0
^
are shown in Fig. 3 as function of the accumulation time. Regular drop of the decor-relation 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 power-law (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 distanced0 in the exponential part of eqn (14) is much bigger than the distances considered, the integrals in eqn (13) can be also simpli®ed:
VarfRgÿRag r2g 1ÿ 2 A
Z
A q0exp
ÿjxgÿxj
d0
dx2
Fig. 3. Decorrelation parameter 1ÿq0
^
1
A2 Z
A Z
A
q0exp ÿ
jxÿyj
d0
dx2dy2
r2g 1ÿ2q0q0 r2g 1ÿq0: 17
This leads to the following estimate of the area-point part in eqn (2):
Var^ fRgÿRag Var^ fRgg 1ÿq0
^
; 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 dif-ference 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 accumu-lation time. In part (a) of this ®gure, the solid line pre-sents the R±G sample variance eqn (4) in nondimensional terms ± its square root normalized by the rainfall accumulation sample mean (averaged un-conditionally 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
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 ac-cumulation time. The ratio of the raingauge to radar error variances is in Fig. 5(b). One can see that the rel-ative 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 instan-taneous 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
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 re-mains 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 three-day accumulation periods the value of the decorrelation parameter is about 0.01, and thus, the correlation co-ecient of the point and area-averaged rainfall accu-mulations 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 com-bines with the decorrelation in eqn (18). This also demonstrates a more general fact that, once extremely variable processes are concerned, even very high corre-lation levels still leave room for large dierences between the random variables, and that in such situations, con-sidering the correlations only can be misleading.
4 DISCUSSION OF THE ESM ASSUMPTIONS
This section elaborates on the assumptions and re-quirements 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 parti-tioning as in eqn (2) is that the RR estimation error and the area-point rainfall dierence are uncorrelated ran-dom variables:
CovfRgÿRa; RrÿRag 0; 19
or that at least the covariance term is negligible:
CovfRgÿRa; RrÿRag VarfRrÿRgg: 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 in-volved). 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 anaverage 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
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
rain®eld as a stochastic process, onlyRg in the covari-ance term depends on the raingauge location. Both area-averaged 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:
1
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 pur-pose 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 posi-tions 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 situ-ations, 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 re-duces the impact of the covariance term on the RR error variance estimates. This further suggests that the cor-relation 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 con-clusion would be a special experiment which will be discussed later.
4.2 Other assumptions and application questions
It is also assumed that the random variablesRr andRg 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 ac-cumulation ®elds which, in general, does not have to be true. The raingauges are subject to serious
undercatch-ment errors that have to be investigated and corrected (Robinson and Rodda,21Lindroth18). 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 in-cludes the point mean, variance and correlation func-tion, 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, ex-cept of the areas with strong orographic eects. The homogeneity assumption might become more demand-ing, 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 nonho-mogeneity. The latter is not trivial and is beyond the scope of this paper. For the example presented in Sec-tion 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 ab-sence of local orographic eects, and/or statistical tests for consistency of the rainfall sample means and vari-ances should be applied. Elaboration on such tests is beyond the scope of this paper.
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 ques-tions 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 es-timation error variance, an important characteristic of the radar product uncertainties. The method is based on partitioning of the R±G dierence variance into the er-ror of the radar grid-averaged estimate, and the area-point background originating from the large discrep-ancy 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 devel-oped here is quite general. For practical application, the method requires information on the rain ®eld correla-tion structure at the scales below the resolucorrela-tion of the remote sensing rainfall products. To assure satisfactory precision of the results, it has to be applied to a su-ciently 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 res-olution 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 compar-isons which are contaminated with the raingauge rep-resentativeness noise.
The question is how to verify the ESM assumptions and its range of applicability. Simulation models of space-time rainfall and RR measurement (Foufoula-Georgiou and Krajewski,9 Krajewski et al.,17 Ana-gnostou and Krajewski1) can be the most eective to study some problems like the method sampling prop-erties or its extensions to nonhomogeneous and non-stationary situations. For testing some other questions, like the zero-covariance assumption eqn (19), the mod-els 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 re-mote sensing techniques is through analysis of su-ciently 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 accu-mulations with adequate spatial resolution. Most likely, if this ground truth were to be based on a dense rain-gauge network, its spatial resolution should be at least one order of magnitude better than the radar product resolution. Thisbrute forcesolution 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 infor-mation 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 de-signs of the operational raingauge networks are required to initiate substantial progress in this area. We want to conclude by suggesting a concept of ahierarchical clus-ter networkdesigned 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
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