Volume 45, Issue 1
Regular Article
Free Access

Product-error-driven generator of probable rainfall conditioned on WSR-88D precipitation estimates

Gabriele Villarini

Gabriele Villarini

IIHR-Hydroscience and Engineering, University of Iowa, Iowa City, Iowa, USA

Now at Department of Civil and Environmental Engineering, Princeton University, Princeton, New Jersey, USA.

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Witold F. Krajewski

Witold F. Krajewski

IIHR-Hydroscience and Engineering, University of Iowa, Iowa City, Iowa, USA

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Grzegorz J. Ciach

Grzegorz J. Ciach

IIHR-Hydroscience and Engineering, University of Iowa, Iowa City, Iowa, USA

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Dale L. Zimmerman

Dale L. Zimmerman

Department of Statistics and Actuarial Science, University of Iowa, Iowa City, Iowa, USA

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First published: 07 January 2009
Citations: 66

Abstract

[1] The existence of large errors in precipitation products delivered by the network of Weather Surveillance Radar, 1988 Doppler (WSR-88D) radars is broadly recognized. However, their quantitative characteristics remain poorly understood. Recently, the authors developed a functional-statistical model that quantifies the relation between radar rainfall and the corresponding true rainfall in a way that is applicable to the probabilistic quantitative precipitation estimation planned for future use by the U.S. National Weather Service. The model consists of a deterministic distortion function and a random uncertainty factor, both conditioned on given radar rainfall values. It also accounts for the spatiotemporal correlations in the random uncertainty factor. The model components were estimated on the basis of a 6-year-long data sample that considers the effects of seasons, range from radar, and time scales. In this study, the authors present two different applications of the aforementioned uncertainty model: (1) the estimation of rainfall probability maps and (2) the generation of radar rainfall ensembles. In the former, maps of the rainfall exceedance probability for any threshold are produced, given a radar rainfall map. We also present the analytical derivation of the exceedance probability maps at coarser spatial scales. In the latter, the users can generate ensembles of probable true rainfall fields that are consistent with the observed radar rainfall and its error structure. Simulation of the random component is based on the Cholesky decomposition method. Finally, the authors discuss possible uses of these applications in hydrology and hydroclimatology.

1. Introduction

[2] It is commonly agreed that the quality of real-time hydrologic forecasts depends on the adequacy of the rainfall runoff model structure and parameters as well as the accuracy of space-time representation of rainfall input [e.g., Kitanidis and Bras, 1980; Georgakakos and Hudlow, 1984; Georgakakos, 1986; Georgakakos and Smith, 1990; Krajewski et al., 1991; Pessoa et al., 1993; Michaud and Sorooshian, 1994; O'Connell and Todini, 1996; Gupta et al., 1998; Winchell et al., 1998; Bell and Moore, 2000; Ogden et al., 2000; Borga, 2002; National Research Council, 2002; Syed et al., 2003; Smith et al., 2004; Ajami et al., 2007; Gabellani et al., 2007]. While errors due to both factors have received much recent attention in the literature [e.g., Finnerty et al., 1997; Butts et al., 2004; Carpenter and Georgakakos, 2004; Wagener and Gupta, 2005; Borga et al., 2006; Huard and Mailhot, 2006; Kavetski et al., 2006a, 2006b; Kuczera et al., 2006; Oudin et al., 2006], data assimilation and ensemble forecasting frameworks are considered effective means of improving forecasts in the face of uncertainty [e.g., Kavetski et al., 2002; Vrugt et al., 2005; Carpenter and Georgakakos, 2006a; Kavetski et al., 2006a; Moradkhani et al., 2005a, 2005b, 2006; Oudin et al., 2006; Russo et al., 2006; Vrugt et al., 2006; Ajami et al., 2007; Gabellani et al., 2007; Smith et al., 2008]. Both approaches require specification of realistic uncertainty descriptions for the hydrologic models and their rainfall input.

[3] In this paper, we focus on uncertainty of rainfall input that is provided by weather radar observations. Specifically, we formulate a simulation tool that can provide hydrologic models with rainfall input that has an error structure consistent with observational evidence. We illustrate the simulation model with two applications: (1) generation of probabilistic maps of rainfall conditioned on radar observations, and (2) generation of an ensemble of rainfall input. Before we describe the simulation model, we briefly acknowledge some key earlier developments.

[4] The benefits of synthetic radar rainfall data were recognized long ago by Grayman and Eagleson [1971], Krajewski and Georgakakos [1985], Krajewski [1987], and Seo et al. [1990], who conducted studies of flood and rainfall observation systems in the spirit of today's Observing System Simulation Experiments, or OSSE [e.g., Arnold and Dey, 1986; Lord et al., 1997; Tong and Xue, 2005; Xue et al., 2007]. In pursuit of more realistic synthetic radar rainfall data, models evolved from being purely statistical, or on the basis of a conceptual understating of the uncertainty involved [e.g., Krajewski and Georgakakos, 1985; Carpenter et al., 2001; Georgakakos and Carpenter, 2003; Russo et al., 2006], to a hybrid approach that combined statistical descriptions of space-time rainfall with a physics-based mechanism of radar observations [e.g., Krajewski et al., 1993, 1996; Anagnostou and Krajewski, 1997; Borga et al., 1997; Jordan et al., 2003]. Further realism was attained by replacing statistical models of rainfall with high-resolution mesoscale numerical weather prediction models [e.g., Sharif et al., 2002, 2004]. Other authors used even simpler approaches to investigate the effects of radar rainfall uncertainties on the inferred properties of hydrologic processes [e.g., Carpenter and Georgakakos, 2006a, 2006b; Villarini et al., 2007].

[5] Despite these efforts, one could argue that the existing approaches do not mimic the actual radar rainfall uncertainty because they are not based on rigorous empirical evidence. Our work addresses this shortcoming by combining an empirically based comprehensive model of radar rainfall uncertainty developed by Ciach et al. [2007] with actual radar rainfall products. Their product-error-driven uncertainty modeling approach is based on the functional-statistical representation of the relationships between radar rainfall estimates and the corresponding area-averaged rainfall based on a 6-year-long sample of WSR-88D (Weather Surveillance Radar, 1988 Doppler) weather radar and rain gauge data. A brief description of this model is presented in section 2.

[6] After describing the model, we present two applications. We first derive the rainfall probability of exceedance maps conditioned on a given observed radar rainfall map and information on its uncertainty. The maps provide answers to the typical hydrological question: “What is the probability that, for a given (observed) radar rainfall value, the corresponding actual areal rainfall exceeds a specified threshold?” We also describe the extension of this procedure for areas that are larger than the single pixel considered in the original WSR-88D precipitation products by Ciach et al. [2007]. In the second application, we develop a generator of ensembles of probable rainfall fields conditioned on the observed radar rainfall fields and consistent with the known error structure (notice that the emphasis will be on the spatial dependencies of the radar rainfall errors and we will neglect the temporal correlation of the errors for the reasons discussed in section 4). Such ensembles can be used directly as the probabilistic input in the ensemble hydrological forecasts. We conclude with a summary and discussion of the limitations of the proposed approach.

2. Modeling Radar Rainfall Errors and Their Spatiotemporal Correlations

[7] To characterize the relation between radar rainfall (RR) and the corresponding areally averaged pixel-scale rainfall (RA), Ciach et al. [2007] proposed a model in which RA values, conditioned on given radar rainfall values, are expressed as the product of a deterministic component and a random component:
equation image
where the deterministic distortion function, h(RR), accounts for the systematic conditional biases, whereas the stochastic factor, ɛ(RR), describes the remaining random errors. Both of the two components in (1) depend on the observed radar rainfall values. In developing the model, the authors computed the overall bias (defined as the ratio of the rain gauge and radar rainfall sample totals) and removed it from the radar rainfall estimates. Then, the deterministic component and the conditional distributions of the random component were estimated using nonparametric functional estimation schemes derived from kernel regression methods [see also Villarini et al., 2008]. In these procedures, the true areal rainfall values, RA, corresponding to the given radar rainfall estimates are approximated using the available single rain gauge measurements. As discussed by Ciach et al. [2007], the relatively small spatial variability of hourly rainfall in Oklahoma [Krajewski et al., 2003; Ciach and Krajewski, 2006] justifies the use of this straightforward approximation without significant adverse effects. Note that Ciach et al. [2007] used a multiplicative error model. However, the discrepancy between true rainfall and radar rainfall could be defined in other ways (e.g., as a difference; or applying a transformation, such as logarithmic, on both RA and RR), leading to different results and perhaps conclusions. Discussing models different from (1) is outside the scope of this paper and should be addressed in future studies.

[8] Ciach et al. [2007] used a 6-year-long sample of level 2 data from the Oklahoma City WSR-88D radar site (KTLX) to obtain the abovementioned nonparametric estimates of the radar rainfall uncertainty model components. The data sample covers the period from January 1998 to December 2003. The radar rainfall products represent hourly accumulations (in mm) averaged over Hydrologic Rainfall Analysis Project (HRAP) grids [e.g., Reed and Maidment, 1999] that are approximately 4 km by 4 km pixels. The sample was complemented with the corresponding rain gauge records from the Oklahoma Mesonet [e.g., Brock et al., 1995] and the Agricultural Research Service (ARS) Micronet [e.g., Allen and Naney, 1991]. The rainfall measurements from these two networks are treated directly as the values of areally averaged pixel-scale rainfall RA that correspond to the radar rainfall values estimated over the HRAP grids containing the rain gauges.

[9] To account for different synoptic conditions, Ciach et al. [2007] divided the data set into three seasons: cold (January, February, March, November, and December), warm (April, May, and October), and hot (June, July, August, and September). In addition, they investigated the dependence of the estimated radar rainfall error components on the distance from the radar site by dividing the radar umbrella into five zones, from zone 1, the closest to the radar site, to zone 5, at about 200 km away from it (see Ciach et al. [2007] for details).

[10] To obtain a convenient model, Ciach et al. [2007] analytically approximated the nonparametric estimates of the two components in the radar rainfall error model using simple parametric models. The deterministic distortion function and the standard deviation of the random component are parameterized using power law functions:
equation image
equation image
where B0 is the overall bias (ratio of rain gauge and radar rainfall sample totals), ah and bh are the coefficients of the parametric model of the deterministic distortion function, and σ0e, ae, and be are the coefficients of the parametric model of the random component. According to Ciach et al. [2007], the conditional frequency distributions of the stochastic factor, ɛ(RR), can be described by the Gaussian distribution model that, for each radar rainfall value, has expectation equal to 1 and standard deviation estimated from the data. As discussed by Ciach et al. [2007], the use of the Gaussian distribution is supported by the fact that we consider only conditional errors, mitigating the effects of high skewness displayed by rainfall itself.
[11] The estimates of the spatial and temporal correlations in the random component are presented in the form of scattergrams by Ciach et al. [2007]. Below, we complete the results by parameterizing these correlation functions. We fitted the following two-parameter functions from the modified exponential family to the points in the scattergrams of the spatial and temporal correlation coefficients in the random radar rainfall error component using a standard least squares approach:
equation image
equation image

[12] In (4) and (5) above, as is the spatial scale and bs is the shape parameter of the fitted spatial correlation function; at is the time scale and bt is the shape parameter of the fitted temporal correlation function; Δs and Δt are the separation lags in space and time respectively; and the expression subscripts S and T denote the season and time scale, respectively.

[13] Although Ciach et al. [2007] show the scattergrams of spatial and temporal correlation coefficient of ɛ(RR) at all five distance zones, here we limited the fitting of the parametric models (4) and (5) to zone 2 only. The presence of the Micronet network in this zone provides much more information about the radar rainfall error correlations than in all the other zones. Furthermore, this is the only zone where the information about ρs at relatively small intergauge distances (below 10 km) is available. Finally, the scattergrams do not show any regular differences between the spatial and temporal correlations in different zones. Consequently, we decided to use the parametric correlation function estimates based on zone 2 data as representative of the whole radar field considered here.

[14] In Figure 1, we show both the intergauge correlation coefficients of the random error component, ɛ(RR) and the parametric function (4) fitted to the points at an hourly scale for different seasons. In each case, the modified exponential function effectively captures the general pattern of the empirical results. In Table 1, we present the fitted coefficients as and bs in the correlogram model (4) for three time scales and three seasons. The correlation distance, as, of the random factor in the cold season is much greater than in the other two seasons. This seems to capture the properties of precipitation in central Oklahoma, with the prevalence of stratiform systems in the cold season and a strong contribution of convection in the warm and hot seasons. On the other hand, the shape parameter, bs, is practically the same for each season, at least at the hourly time scale. In Figure 2, we show analogous results for the temporal correlation of the random factor at the hourly scale for different seasons. As was true for the spatial correlation functions, the fitted shape coefficients are similar for the different seasons, whereas the temporal correlation scale in the cold season is much larger than in the other two.

Details are in the caption following the image
Empirical spatial correlation of the random component with a two-parameter exponential model approximation for zone 2 at the hourly scale for the three seasons and the entire data set.
Details are in the caption following the image
Empirical temporal correlation of the random component with a two-parameter exponential model approximation for zone 2 at the hourly scale for the three seasons and the entire data set.
Table 1. Parameterization Results for the Spatial Correlation Function, ρss), of the Random Component, equation image(RR), in the Radar Rainfall Uncertainty Modela
Hourly 3 Hourly Daily
Cold Warm Hot Cold Warm Hot Cold Warm Hot
Zone 2 236.7 (0.37) 37.0 (0.39) 41.9 (0.37) 603.1 (0.34) 79.4 (0.42) 57.0 (0.40) 478.0 (0.36) 138.4 (0.48) 73.3 (0.49)
  • a The fitted coefficient values are the correlation distance as and the shape factor bs (shown in parentheses).

[15] In summary, the radar rainfall uncertainty model developed by Ciach et al. [2007] has the flexibility to account for different synoptic conditions approximated by the three seasons, different time averaging scales, and radar range effects [e.g., Fabry and Zawadzki, 1995; Smith et al., 1996]. The deterministic and random radar rainfall error components, as well as the spatial and temporal correlation functions of the random factor, can be approximated using simple parametric models. We use these analytical parameterizations in the two applications presented below.

3. Probability of Exceedance Maps

[16] The first application of the model is the generation of maps with the probability of exceedance of some predetermined threshold Rthres by the true rainfall. This scenario is facilitated by the fact that the random component can be described by a Gaussian distribution. In fact, for a certain pixel, we can write
equation image
where RA is a random variable.
[17] In particular, from Ciach et al. [2007] we obtain
equation image
that can be standardized as
equation image
Now we can write
equation image
where Φ( ) is the standard normal cumulative distribution function (CDF).
[18] It is possible to express the normal CDF in terms of the erf function [e.g., Johnson et al., 1994] as follows:
equation image
From (8) and (10), we have P(RARthres) = 0.5 when h( ) is equal to Rthres. If the values of the deterministic distortion function are larger than the threshold, we have P(RARthres) > 0.5. On the other hand, if h < Rthres, then P(RARthres) < 0.5.

[19] As an example, consider the hourly rainfall map in Figure 3a and a threshold value of 20 mm. Figure 3b shows the corresponding values of the deterministic distortion function, while Figure 3c plots the probabilities of exceedance by the true rainfall. Figures 3b and 3c compare the areas with higher rainfall map to areas with higher probability of exceedance. This feature is even more evident if we create maps in which we highlight only the pixels with a probability of exceedance larger than a certain probability value. In Figure 3d, we plot the case for P(RA ≥ 20 mm) ≥ 0.50. From Figure 3d, it is evident that the areas of higher rainfall are in agreement with the pixels exceeding 0.5-probability of exceedance.

Details are in the caption following the image
Images of (a) the original DPA map, (b) the deterministic distortion function, (c) the probability of exceedance for a threshold Rthres of 20 mm, and (d) the map of P(RA ≥ 20 mm) ≥ 0.50.
[20] Using the information about the spatial correlation of the random component, it is possible to extend this formulation to a group of n pixels, which in reality could represent a basin of interest. In this case, exploiting the properties of a linear combination of Gaussian random variables [e.g., Kottegoda and Rosso, 1997], we can rewrite equations (6) and (7) as
equation image
equation image
where ρs is the spatial correlation of the random component as defined in equation (4).
[21] Standardizing the Gaussian variable in (12) as
equation image
we obtain
equation image
Note that it is possible to select different values of Rthres for each pixel or group of pixels.

4. Ensemble Generator

[22] Consider a map (or an array) representing the spatially distributed rainfall field. Then, taking into account the season and time scale as in the radar rainfall error correlation functions (4) and (5), the conditional radar rainfall uncertainty model (1) can be simply expressed as
equation image
where x and y are the field coordinates (or array indexes), RA(x, y) is the stochastic map of probable true rainfall conditioned on estimated radar rainfall values, and RR(x, y) is the given radar rainfall map. The radar distance zone in this expression is fully determined by coordinates x and y. In the application discussed in this study, the radar maps, RR(x, y), are given as the original Digital Precipitation Array (DPA) products generated by the Precipitation Processing System module of the NEXRAD (Next Generation Weather Radar) system [e.g., Fulton et al., 1998]. By using the power law form in (2), the areas with zero radar rainfall values are preserved by the model (15) because of its multiplicative structure. Moreover, according to the parameterization in (3), the standard deviation tends to +∞ for RR→0. In this case, for very small radar rainfall estimates (RR ≤ 0.5 mm), we have set the value of the standard deviation of the random component equal to σequation image[RR = 0.5].
[23] To generate ensembles of the probable rainfall fields, it is sufficient to focus on the random error component, ɛ(x, y), in the radar rainfall uncertainty model (1). The remaining deterministic factor, h(x, y), is determined by the radar rainfall values. The random component, ɛ(x, y), is described by a Gaussian distribution with mean equal to 1, and its standard deviation, σequation image[RR(x, y)], is a function of radar rainfall values and spatial and temporal correlation functions, ρss) and ρtt), as described in section 2. In general, the simulation of Gaussian random variables correlated in space or time can be performed according to well-established techniques (see Gneiting et al. [2005] for a review). However, the simulation of spatially and temporally correlated Gaussian random fields presents a much higher degree of complexity [e.g., Gneiting et al., 2007]. To build on the results of Ciach et al. [2007], who estimated the spatial and temporal correlations of the random component separately, we would need to assume a separable spatiotemporal covariance model, which can be written as
equation image
where D ⊆ ℜ2 and T ⊆ ℜ1 are, respectively, the spatial (in our case a regular grid with side n) and temporal domains over which we want to generate space-time Gaussian-distributed random fields.

[24] However, as we subsequently show in this section, on the basis of the results by Ciach et al. [2007], we are not able to generate spatiotemporally correlated Gaussian random fields, and we will have to simplify our error model by neglecting the temporal dependencies. In the literature, several methods have been proposed to generate correlated Gaussian random fields: matrix factorization, spectral turning bands, circulant embedding, and cutoff embedding [e.g., Mantoglou and Wilson, 1982; Davis, 1987; Dietrich and Newsam, 1993; Gneiting et al., 2005]. For our purpose, we selected the Cholesky decomposition method [e.g., Cressie, 1993], whose advantages include exactness (asymptotically, an ensemble of simulated field series has the correlation structure exactly as required) and an ability to account for the dependence of the variance on the given radar rainfall values to reproduce the estimated properties of the radar rainfall uncertainty model. Its major drawback is that it is computationally intensive, which may limit the size of the simulation domain. With the number of time lags, r, the size of the space-time variance-covariance matrix is equal to n2r, which corresponds to floating point operations of the order of O(n6r3) [e.g., Golub and Van Loan, 1996]. Consequently, the method may not be computationally feasible.

[25] The selected simulation methodology is based on the Cholesky decomposition [e.g., Kreyszig, 1999] of the variance-covariance matrix V. To use the Cholesky decomposition method, the matrix must be positive definite and symmetric. In this case, these conditions are satisfied because the variance-covariance matrix from the modified exponential correlation functions is positive definite and symmetric [e.g., Chilès and Delfiner, 1999].

[26] By assuming a separable spatiotemporal covariance model as in (16), we can write V as a tensor product as follows:
equation image
where
equation image
equation image
and si,j = sisj, t0,k = t0tk; the subscript r represents the number of time lags (Figure 4); and image is set equal to 1 because of the identifiability problem. As far as the temporal component is concerned, we decided to use negative time steps to underline the fact that we are not trying to perform temporal forecasting.
Details are in the caption following the image
Representation of the n × n grid considered for the simulation of random Gaussian fields correlated in space and time.
[27] In (16), we assumed that the variances at certain time lags are obtained by multiplying the variances at lag zero by a constant. In fact, if we expand the tensor product in (17), we have
equation image
where σnn+12, σnn+22, …, are equivalent to σ12, σ22, …, at lag t−1 (Figure 4).

[28] However, we already know σ12, σ22, …, σnn+12, σnn+22, …, since we compute them from the radar scans at lag t0, t−1, …, on the basis of equation (3). In general, there is no guarantee that precomputed image (with k ∈ [−1; −r]) satisfy the conditions (20). Because of this limitation, we decided to neglect the time component of the random errors and to consider only their spatial correlation.

[29] Applying the Cholesky decomposition to the variance-covariance matrix in (18), we can rewrite Σs as the product of a lower (L) and an upper triangular (LT) matrices and obtain Σs = LLT. We simulate ɛ, with the mean μɛ equal to 1 and the desired spatial correlation as [e.g., Cressie, 1993]
equation image
where η is a vector of uncorrelated Gaussian random variables with zero mean and unit variance:
equation image
Therefore, given a certain DPA map, to generate an ensemble of m radar rainfall fields, we have to simulate m realizations of ɛ and multiply them by the corresponding h.

[30] To assess the performance of the Cholesky decomposition method for this study, we compared the input spatial correlation function (based on the spatial correlation function at hourly scale, from zone 2, and for the warm season) with the one estimated from 1000 realizations of ɛ( ) (from a real DPA field) obtained from the abovementioned approach. As shown in Figure 5, the Cholesky decomposition method accurately simulates fields with the desired spatial correlation function and dependence of the variance on the radar rainfall estimates. If any ɛ(x, y) values were negative, we would set them to zero to avoid unrealistic negative rainfall. Additionally, we have set an upper threshold of 305 mm/h which, according to the National Weather Service, corresponds to the maximum observed rainfall value for the United States.

Details are in the caption following the image
Comparison between the input correlation (hourly scale, zone 2, and warm season) and the one estimated from 1000 realizations of ɛ( ) generated using the Cholesky decomposition method.

[31] To visually illustrate the work of the simulation scheme, we selected a DPA field (Figure 6a) and generated three probable true rainfall maps conditioned on the original radar rainfall fields (Figures 6b, 6c, and 6d). Looking at the patterns of the three simulated maps, we can see that they are consistent with the one in the original DPA field.

Details are in the caption following the image
(a) Original DPA map and (b, c, and d) three probable true rainfall fields conditioned on the original radar rainfall map.

[32] One can verify the above numerical simulation procedure using the analytical approach described in section 3. The probability of exceedance map can be estimated by analyzing the results of the ensemble generator. Given a DPA map, we generated 10,000 realizations and computed the map with the probability of exceedance of a threshold of 20 mm (Figure 7b). The comparison between this map and the one obtained from the methodology described in the previous section (Figure 7a) reveals an excellent agreement.

Details are in the caption following the image
Comparison of the probability of exceedance map for a threshold Rthres of 20 mm (a) obtained using the analytical formulation and (b) based on 10,000 realizations from the ensemble generator.

5. Discussion, Conclusions, and Directions

[33] Radar rainfall products based on WSR-88D radars are increasingly popular in hydrology because of their broad availability and ability to provide continuous coverage of large areas with relatively good spatial and temporal resolution. However, radar rainfall estimates are corrupted with large uncertainties that originate from many sources. Even though the currently available rain gauge data are mostly based on sparse networks of point measurements, we cannot ignore the fact that they can be quite accurate in comparison with radar rainfall estimates. The results of Ciach et al. [2007] show that, for strong rainfall, the standard error in the hourly radar rainfall products is about 50%. At the same time, the standard deviation of the local random errors in tipping bucket measurements (local means within the range of about 50–200 m, depending on spatial rainfall variability) is only about 1–2% [e.g., Ciach, 2003; Ciach and Krajewski, 2006]. Of course, even if partially corrected, the effects of wind under catch, out splash, and wetting evaporation losses significantly increase these local random errors. Nevertheless, the accuracy of local rainfall data from a correctly located, well-manufactured, and well-maintained rain gauge can still be about an order of magnitude better than the accuracy of radar rainfall data. Consequently, good quality data from dense rain gauge networks are indispensable as an affordable ground reference for all systematic and comprehensive analyses that attempt a realistic mathematical characterization of the radar rainfall error structure.

[34] In this study, we developed two methods of radar rainfall uncertainty description that can be directly applied to hydrological forecasts using radar-based rainfall estimates as input. Both methods are based on the empirically based radar rainfall uncertainty model presented by Ciach et al. [2007]. The first development concerns constructing the probability of exceedance maps of true rainfall conditioned on the given radar rainfall map. These maps also account for the estimated radar rainfall error properties. The second development presented above is the generation of synthetic probable rainfall fields conditioned on given radar rainfall estimates. The generator can reproduce the conditional and unconditional biases in radar rainfall products, the conditional statistical distributions of the remaining random errors, and the spatial correlations in the random errors. All of these properties of the radar rainfall error structure have been identified using a large data sample and nonparametric functional estimation methods. In the course of developing these two procedures, we made several assumptions and simplifications that require more discussion. In most cases, further efforts, including extensive data analyses and new ideas that emerge from these analyses, are necessary to achieve any real progress regarding these issues.

[35] The first simplification concerns the estimation of the spatial correlation functions of the random error factor. As discussed in section 2, we used the correlation estimates from one distance zone containing the ARS Micronet network as representative for the entire radar domain. It is unclear to us whether this is an acceptable assumption. However, we do not have enough data to obtain meaningful estimates in the other zones. The availability of dense rain gauge networks at different distances from the WSR-88D radars would provide invaluable information about the possible range dependences in the spatial and temporal correlation functions of radar rainfall errors. Another simplification of our model is that we estimate the spatial and temporal correlations of the random component even though its standard deviation is nonhomogeneous, possibly affecting the estimation of the random error space-time dependencies. However, at this stage it is not clear how to address this limitation. In the future, a model in which the weak second-order stationarity assumptions are fulfilled would overcome this limitation and simplify the structure of the ensemble generator.

[36] Another assumption concerns the model of conditional probability distributions of the random error factor. On the basis of the exploratory data analyses by Ciach et al. [2007] and considering its well known mathematical tractability, we selected the Gaussian model. However, since the Gaussian distribution is defined over the infinite support of (−∞; +∞), and given the multiplicative nature of the radar rainfall error model, some of the rainfall values simulated in this manner can be negative and require correction. Currently, we address this issue by setting the negative values to zero, although such a solution results in a departure from the Gaussian distribution. This problem pertains mostly to the weak radar rainfall values and has a small impact on the strong rainfall values that are the most important in hydrologic applications. Nevertheless, more accurate modeling of the error distribution in the region of weak and moderate rainfall remains an open issue. Furthermore, the adequacy of the Gaussian model requires further research, as proper treatment of the upper quantiles of the random factor up to the 99% level, or possibly even higher, is important in hydrologic applications. Another element that may have to be investigated in more detail in future studies is the modeling of radar rainfall uncertainties for extreme rainfall events, which would be particularly important when generating probability maps with high threshold values. We have already discussed simplifications concerning the temporal structure of the radar rainfall errors. Even though we decided to neglect it in this study for the aforementioned reasons and focus on the spatial dependencies, we are currently exploring other approaches to eliminate this limitation and generate joint spatiotemporal replicates.

[37] Thus, we are fully aware of the weaknesses and simplifications in the results presented here. Nevertheless, these results constitute considerable progress in characterizing the radar rainfall uncertainties in an empirically supported way. The generator we present provides a meaningful tool for use within the increasingly popular probabilistic and ensemble forecasting and data assimilation frameworks.

[38] The first venue developed here, the generation of exceedance probability maps, can directly apply to the flash flood forecasting problem [e.g., Georgakakos, 2006]. For example, it can help the transition of the current Flash Flood Guidance System (FFGS) from a deterministic to a fully stochastic [Ntelekos et al., 2006] framework. Another possible application of the probability maps is the development of new validation methods for space-based rainfall estimates, methods that account for the uncertainties in the radar-based rainfall estimates. The second development presented here, the technique for generating the ensembles of probable rainfall fields consistent with the observed radar rainfall maps and its error structure, can have many applications. For example, such ensembles can be directly used in the ensemble streamflow predictions based on radar rainfall input. They can also be applied to investigate the possible effects of radar rainfall uncertainties on the analyses of spatial scaling in rainfall [Villarini et al., 2007]. Moreover, they can be used in the probabilistic quantitative precipitation forecasts (PQPF) that use the observed radar rainfall data both as part of the initial conditions and in the process of updating the forecasting results. All these potential applications of our current results are beyond the scope of this paper.

Acknowledgments

[39] This work was supported by NSF grant EAR-0309644 and the contract under the PQPE initiative by the Office of Hydrologic Development of the U.S. National Weather Service. The opinions expressed in this work are those of the authors and do not necessarily reflect those of NSF, NOAA, or their subagencies. The second author also acknowledges partial support of the Rose and Joseph Summers endowment.