This study presents a probabilistic radar rainfall estimation (PRRE) model to quantify the reliability and accuracy of the resulting radar rainfall estimates at ungauged locations from a radar-based quantitative precipitation estimation (QPE) model. This model primarily estimates the quantiles of the radar rainfall errors at ungauged locations by incorporating seven spatiotemporal variogram models with a nonparametric sample quantile estimate method based on the radar rainfall errors at rain gauges. Then, by adding the resulting error quantiles to the radar rainfall estimates, the corresponding radar rainfall quantiles can be obtained. The QPE system Quantitative Precipitation Estimation Using Multiple Sensors (QPESUMS) provides hourly observed and radar precipitation for three typhoons in the Shinmen reservoir watershed in Northern Taiwan, which are used in the model development and validation. The results indicate that the proposed PRRE model can quantify the spatial and temporal variations of radar rainfall estimates at ungauged locations provided by the QPESUMS system. Also, its reliability and accuracy could be evaluated based on a 95% confidence interval and occurrence probability resulting from the cumulative probability distribution established by the proposed PRRE model.

## INTRODUCTION

Rainfall is an essential and important input in many climatological, hydrological, and hydraulic applications. Traditionally, rain gauges are the major source for measuring surface rainfall, but many related hydrological applications probably are limited due to no sufficient or appropriately located rain gauges. In other words, the resulting estimation from gauged data are imprecise due to the density of most rain gauge networks being insufficient to adequately describe spatial variation in storm patterns (Clark & Slater 2006). Recently, remote sensing precipitation products, such as radar rainfall estimates, have been widely used to provide information on rainfall's spatiotemporal structure, because of their large areal coverage and high resolution as compared to traditional rain gauge measurements (Sharif et al. 2002; Mandapaka et al. 2010). Therefore, the radar rainfall estimates are widely used in hydrometeorological and water resource applications, such as land-surface hydrological and hydraulic models. They are also used in the validation of numerical weather prediction and climate models, numerous hydrological forecasting, and for the minimization of flood and flash flood-related hazards (e.g., Peters & Easton 1996; Hossain et al. 2004; Jorgeson & Julien 2005; Bowler et al. 2006; Smith et al. 2007; Chiang & Chang 2009; Aghakouchak et al. 2010; Liguori et al. 2011).

The remainder of this paper is organized as follows: the model concept and framework are addressed in the next section, Methodology, and the results from the model development and demonstration are provided in the Results and discussion section. The paper ends with a Conclusions section, which summarizes a discussion of the results and suggestions for future study.

## METHODOLOGY

Recent weather radar systems can provide rainfall characteristics, such as the rainfall amount, duration, and storm pattern, with higher spatial and temporal resolution than was previously possible from gauged data (Aghakouchak et al. 2010). However, the radar rainfall estimate might be influenced by uncertainties in relevant parameters of quantitative rainfall estimation models and gauged data, so that some errors could occur which may have the estimates deviate from the recorded observations at the rain gauges. In general, a complete statistical characterization of radar rainfall uncertainties should involves the biases, error variances, conditional error distributions, and a description of the error's dependence on time and space (Mandapaka et al. 2010). Therefore, based on the spatial and temporal statistical properties of the insufficient radar rainfall errors calculated at the rain gauges, this study develops the PRRE model to be used in a reliability assessment for the radar rainfall estimates at ungauged locations. Details of the model's development process are presented below.

### Model concept

This study develops the PRRE model based mainly on the idea from Kirstetter et al. (2010). A geostatistical technique is used for calculating the difference in the point-rainfalls resulting from the precipitation estimation method and the rain gauges, but without the areal-average rainfall used by the Kirstetter model. This difference in the point-rainfall is defined as the spatiotemporal radar rainfall error
1
where Rgauge(x, t) and Rradar(x, t) serve as the observed rainfall and radar estimate at time t and location x, respectively. Since the radar rainfall estimate is the spatiotemporal variable, the associated error can be regarded as the spatiotemporal variate. This study first estimates the error at the ungauged locations by means of the spatiotemporal analysis with the radar rainfall error calculated at the rain gauges. Then, by using the estimated error at the ungauged locations, the corresponding error quantiles are obtained through a statistical analysis. Finally, the cumulative probability distribution, which is composed of quantiles, for the radar rainfall estimate at the ungauged location can be established by combining the original estimates with the resulting error quantiles.

According to the aforementioned concept, the proposed PRRE model consists of four steps: (1) calculation of the radar rainfall errors using gauged and radar rainfall data at the rain gauges; (2) spatiotemporal analysis for the radar rainfall error at the ungauged location; (3) statistical analysis for the radar rainfall error at the ungauged location; and (4) reliability analysis for the radar rainfall estimate at the ungauged location. The details of these steps are given in the following sections.

### Spatiotemporal analysis for the radar rainfall error at the ungauged location

In this study, the uncertainty found in the radar rainfall estimate is a result of associated errors. Because the radar rainfall error can only be measured at rain gauges, it is necessary to estimate its error at ungauged locations in accordance with known errors at the rain gauges. This section describes the algorithm used for estimating radar rainfall errors in the proposed PRRE model.

#### Estimation of the radar rainfall error using the kriging method

The first step in the proposed PRRE model is to calculate the difference between the radar rainfalls estimated by the precipitation estimation model and observed data. This difference is defined as the radar rainfall error from Equation (1) at the rain gauges. Note that a negative error indicates that the precipitation estimation model overestimates the rainfall, while a positive error means that the radar rainfall could be underestimated.

Since the radar rainfall error serves as the spatiotemporal variable, the spatiotemporal analysis can be applied in the estimation of the radar rainfall error at the ungauged locations. In general, the spatiotemporal analysis basically establishes the relationship between the variation in the variables with distances in time and space, which can be carried out using the kriging method associated with the variogram models. Therefore, the parameters of the spatiotemporal variogram model can be calibrated using the calculated at the rain gauges, and the corresponding can be estimated for the time step at ungauged location x* using the following equation:
2
in which serves as the kriging weights estimated by the kriging method with the mth spatiotemporal variogram model; NX and NT denote the duration of a rainstorm event and the number of rain gauges; and is the radar rainfall error for time step it at rain gauge ix. Note that this study assumes the radar rainfall error to be isotropic, so that the spatial variogram model is determined without any consideration of direction.

#### Description of spatiotemporal variogram model

The spatiotemporal variogram model is widely applied to deal with the spatially and temporally correlated variable. The spatiotemporal variogram which measures the degree of spatial and temporal of a random variable over certain distances in time and space is described as follows:
3
where hs and ht represent the distance and time lag; Z(x,t) denotes the spatiotemporal variable at time t and position x. A number of variogram models have been published to describe the behavior of spatiotemporal variograms, such as the product of variograms (Rodriguez-Iturbe & Mejia 1974; De Cesare et al. 1997), the integrated product of variograms (Dimitrakopoulos & Luo 1993), and the product-sum model (De Cesare et al. 2001, 2002). This last model has three advantages over the others: (1) it can provide a large class of flexible models that require less constraint symmetry between the spatial and temporal correlation components; (2) it does not need an arbitrary space-time metric; and (3) it can be fitted to data using relatively straightforward techniques, similar to those developed for spatial-based variograms (Gneiting et al. 2007). Therefore, this study uses the product-sum method to calculate the spatiotemporal variogram of Z(x,t). The concept of the spatiotemporal variogram model is briefly introduced below.
The product-sum spatiotemporal variogram is defined in terms of separate spatial variogram and temporal variogram as (De Cesare et al. 2002):
4
where and are the respective spatial and temporal covariances in which hs and ht are the lag distance in time and space; Cs(0) and Ct(0) serve as the sills, which are defined as the variograms' limits, and which are generally used as the parameters of theoretical variogram models. Table 1 shows commonly used theoretical variogram models. The coefficients are k1, k2, and k3 and they can be computed by using the following equations (De Cesare et al. 2001, 2002):
5
where Cst(0,0) denotes the sill of the spatiotemporal covariance and can be calculated from a plot of the spatiotemporal variogram plot.
Table 1

Definition of variogram models and their associated parameters

Model γ(hRange of h
1. Spherical
model
0 < ha
h > a
2. Exponential model  h > 0
3. Gaussian model  h > 0
4. Nugget model h = 0
h > 0
5. Cubic model  0 < ha
h > a
6. Circular model  0 < ha
h > a
7. Pentaspherical
model
0 < ha
h > a
Model γ(hRange of h
1. Spherical
model
0 < ha
h > a
2. Exponential model  h > 0
3. Gaussian model  h > 0
4. Nugget model h = 0
h > 0
5. Cubic model  0 < ha
h > a
6. Circular model  0 < ha
h > a
7. Pentaspherical
model
0 < ha
h > a

c and a denote the sill and influence ranges, and h the distance (Davis 1973).

#### Estimation of radar rainfall error with various variogram models

Numerous variogram models can be selected for the spatiotemporal analysis as shown in Table 1, which describe various variables' behavior in time and space. The selection of the best-fit variogram model can be done using the standardized average error and the standardized kriging variance (Kumar & Remadevi 2006; Evrendilek & Ertekin 2007; Wu et al. 2011). However, Barancourt et al. (1992) indicated that the accuracy and reliability of results from the spatiotemporal analysis depend on the selection of the best-fit variogram model. In fact, the variogram plays an important role in realizing the spatiotemporal characteristics and estimating spatiotemporal variates in ungauged locations. Therefore, the uncertainties in the selection of variogram models and the associated parameters are taken into account when this study develops the proposed PRRE model. This study adopts results from the kriging method associated with seven theoretical variogram models to estimate the radar rainfall error values. That is, there are seven estimated values at a particular time step for a specific ungauged location. Note that the Nugget model is also selected as a variogram model in this study, because the other six variogram models do not take the nugget effect into account.

### Statistical analysis for radar rainfall error at ungauged locations

The aim of this section is to estimate the quantile of the radar rainfall error at ungauged locations using the results from the spatiotemporal analysis. The reliability analysis for the radar rainfall estimate would be carried out based on the resulting error quantile. The process for estimating the rainfall error quantile in the proposed PRRE model is addressed below.

#### Estimation of radar rainfall error quantile

Using the seven estimated errors from the section Spatiotemporal analysis for the radar rainfall error at the ungauged location, the statistical properties can be calculated accordingly, including statistical moments and quantiles associated with various theoretical probability distribution functions which are implemented using the statistical analysis. Generally speaking, in the statistical analysis, the best-fit probability functions should be first identified by means of the Kolmogorov–Smirnov or Chi-square test to establish the variable's quantile relationship, i.e., the cumulative probability distribution. However, it is complicated due to the difficulty in determining the best-fit probability function from a large number of candidate distributions and parameter estimation procedures (Haddad & Rahman 2011). To avoid this problem, this study employs a nonparametric method to calculate the quantiles of the radar rainfall error , i.e., the weighted likelihood sample quantile estimator method (Yang & Tung 1996). This method can establish the quantile relationship from the finite sample data, but without the probability functions. Then, the corresponding quantile for the radar rainfall estimate at the ungauged location can also be estimated.

#### Description of the weighted likelihood sample quantile estimator

The weighted likelihood sample quantile estimator method is addressed below. Consider a set of n independent observations, (), of a random variable from a distribution function, F(x). They can be arranged in ascending order to form a set of n order statistics as with X(1) being the smallest observation and X(n) the largest. The maximum likelihood estimator of the unknown distribution function is an empirical one, Fn(x), which is defined as:
6
where is the indicator variable having a value of 1 if or 0 otherwise. Hence, when x = X(r), then the value of the empirical distribution function is , for r = 1,2, …, n.
Although the sample quantile estimator has the advantage of being computationally simple and intuitively understandable, it does not satisfy the symmetrical property of the population quantile and experiences a substantial lack of efficiency due to the variability of the individual order statistics (Sheather & Marron 1990). To obtain a more stable quantile estimator and smoother distribution, this study employs a weighted likelihood estimator to estimate the quantiles. The weighted likelihood estimate is presented here where the estimator for the pth quantile is defined using the weighted average of the order statistics (Yang & Tung 1996) as:
7
where with ω being the band width that contains a set of order statistics that are deemed significant in contributing to the estimation of Xp. Yang & Tung (1996) suggested that the band width covers the order range whose log-likelihood function values are no smaller than 2.0 below the log-likelihood value at the optimal rank. Figure 1 shows the determined bandwidth.
Figure 1

Graphical illustrating determination of band width ω for the weighted likelihood quantile estimator (Yang & Tung 1996).

Figure 1

Graphical illustrating determination of band width ω for the weighted likelihood quantile estimator (Yang & Tung 1996).

### Reliability analysis for radar rainfall estimate at ungauged location

Since the quantiles are points taken at regular intervals from the cumulative distribution function (CDF) of a random variable, the cumulative probability distribution for the radar rainfall estimate resulting from a QPE model can be obtained by combining the radar rainfall estimate and the resulting error quantiles in the section Statistical analysis for radar rainfall error at ungauged locations. Since the reliability of a random variable can be presented in terms of the associated occurrence probability under consistent conditions, the reliability analysis can be carried out by assessing the variations of the cumulative probability and associated statistical properties of the radar rainfall estimate. The process for evaluating the reliability of the radar rainfall estimate is shown below in the proposed PRRE model.

#### Estimation of quantiles of radar rainfall estimate

In the proposed PRRE model, the quantiles of the radar rainfall estimated for time step at ungauged location can also be calculated by summing up the original radar rainfall estimate and the estimated error quantile by the weighted likelihood sample quantile estimator method as:
8
And the corresponding cumulative probability can be obtained by using the following equation:
9

#### Modification of radar rainfall quantile using a logistic regression method

Referring to Equation (8), the radar rainfall estimate's negative quantile is possibly obtained where the absolute value of the error quantile is greater than the radar rainfall estimate. To prevent this problem, this study first calculates the conditional probability of the estimate's non-negative quantile by using the following equation with the cumulative probability function derived in Equation (9):
10
where r* stands for a specific value of the radar rainfall estimate. Then, the logistic regression equation is used to establish the relationship of the conditional probability with the corresponding non-negative quantile of the radar rainfall estimate. This non-negative quantile can then be calculated in accordance with a particular cumulative probability. The logistic regression equation is addressed below.
The logistic regression equation is helpful in establishing the relationship between a set of independent variables and a dependent variable that takes only two dichotomous values (Dai & Lee 2003). It can be expressed as:
11
where P and Xi are the ith independent variables and the corresponding occurrence probability. βi(i = 1,2, …,) are the coefficients with α denoting the intercept that is calibrated from the sample. For this study, P is defined as the conditional probability of the non-negative radar rainfall and the independent variable X serves as the log-value of the associated non-negative radar rainfall . Therefore, Equation (11) can be rewritten as:
12
Intercept αlge and coefficient βlge are determined using the calculated positive quantiles of the radar rainfall estimates in Equation (8). Since can be regarded as the spatiotemporal variable for a specific rain gauge, the associated logistic regression equation, Equation (12), should be derived for each time step during the rainstorm event at ungauged location. That is, the intercept αlge and coefficient βlge vary with time. Finally, the modified quantiles of radar rainfall estimation for a specific cumulative probability p* can be obtained from Equation (12) as:
13

Eventually, by using the proposed PRRE model, the cumulative probability distribution for the radar rainfall estimate at ungauged locations can be established with varying cumulative probabilities from 0.00001 to 1.0.

## RESULTS AND DISCUSSION

The PRRE model is proposed to quantify stochastic information on the radar rainfall estimate at ungauged locations based on its associated error at gauged locations. This study adopts a cross-validation method to clearly address and demonstrate the capability of the proposed model in the quantification of the statistical properties, including the mean, variance, and occurrence probability of the radar rainfall estimate. Note that each rain gauge involves both the gauged and radar rainfall data recorded during a rainstorm event which are provided by the QPE model. The detailed framework for the model validation is shown below.

• Step [1]: Select a rain gauge as the validation point, assuming it is a non-gauged location, and the remaining gauges are defined as the calibration points.

• Step [2]: Calculate the radar rainfall error for each time step during a rainstorm event at calibration points using Equation (1) and compute the associated statistical properties.

• Step [3]: Calibrate the parameters of the proposed PRRE model, including the parameters of seven variogram models in time and space, the kriging weights, the sample quantile estimator parameters, and the coefficients of the logistic regression equation.

• Step [4]: Establish the quantile relationship (cumulative probability distribution) of the radar rainfall estimate at the validation point.

• Step [5]: Calculate the 95% confidence interval and the corresponding occurrence probability of the radar rainfall estimate at the validation point. According to the estimated occurrence probability and the existing radar rainfall error, evaluate the reliability of the radar rainfall estimations.

• Step [6]: Return to Step [1] to select another gauge as the validation point and carry out results for the model validation.

The graphical framework of the proposed PRRE model is shown in Figure 2.

Figure 2

Framework of cumulative probability of radar rainfall established using the proposed PRRE model.

Figure 2

Framework of cumulative probability of radar rainfall established using the proposed PRRE model.

### Study area and data

This study chose the Shinmen reservoir catchment area (see Figure 3) for the development and validation of the proposed PRRE model. The Shinmen reservoir which is located upstream from the Dahan River basin in northern Taiwan provides irrigation, hydroelectric power, a fresh water supply, flood prevention, and sightseeing. The precipitation estimation system QPESUMS provides the hourly radar rainfall and observations for three typhoons, Morakot (2008/08/06 21:00 to 2008/08/10 09:00), Megi (2010/10/16 21:00 to 2010/10/19 20:00), and Fanapi (2010/09/18 09:00 to 2010/09/19 21:00). Moreover, the observed rainfall data for the three typhoons at the ten rain gauges in the Shinmen reservoir catchment, as shown in Table 2, are used as the study data.

Table 2

Information on rain gauges in the Shinmen reservoir catchment

Gauge Name Location
R1 Ga-la-he 290,495.4 2,725,234
R2 Ba-ling 288,792.6 2,730,767
R3 Chi-duan 297,238.4 2,727,102
R4 Yu-feng 280,363.4 2,728,900
R5 Bai-shi 271,949.4 2,715,963
R6 Shinmen 273,867.9 2,745,779
R7 Xi-yue-si-shan 285,447.5 2,719,683
R8 Zhen-xi-bao 280,387.5 2,717,825
R9 Xia-yun 285,386.1 2,743,680
R10 Gao-yi 285,409.9 2,734,450
Gauge Name Location
R1 Ga-la-he 290,495.4 2,725,234
R2 Ba-ling 288,792.6 2,730,767
R3 Chi-duan 297,238.4 2,727,102
R4 Yu-feng 280,363.4 2,728,900
R5 Bai-shi 271,949.4 2,715,963
R6 Shinmen 273,867.9 2,745,779
R7 Xi-yue-si-shan 285,447.5 2,719,683
R8 Zhen-xi-bao 280,387.5 2,717,825
R9 Xia-yun 285,386.1 2,743,680
R10 Gao-yi 285,409.9 2,734,450
Figure 3

Locations of 14 rain gauges in the Shinmen reservoir catchment.

Figure 3

Locations of 14 rain gauges in the Shinmen reservoir catchment.

QPESUMS (Gourley et al. 2001, 2004) provides accumulated precipitation estimates on a 1.3 km × 1.3 km grid for any period of time using algorithms that automatically remove radar artifacts, employ differential Z-R relationships, and integrate data from multiple sensors. In the QPESUMS, the precipitation estimates are automatically updated every 10 min. In detail, QPESUMS has several sub-processes that are used to provide the most optimal QPE. These sub-processes include bright band identification, segregation of convective versus stratiform areas, delineation of precipitation phase, vertical profile of reflectivity determination, satellite integration and precipitation estimation, and rain gauge bias corrections. In Taiwan, the QPESUMS system is developed by Taiwan's National Weather Bureau (CWB) in cooperation with the National Severe Storm Laboratory (NSSL). It is a real-time operating system using multiple sensors. For the most part, these include surface rain gauges and Doppler radars. They provide high-resolution QPE by means of a reflectivity–precipitation (power-law Z-R) relationship and the quantitative precipitation forecast (QPF) with auto-warning capability. In the QPESUMS system, rainfall information is collected from a network of 406 automatic rain gauges and 45 ground stations for a real-time bias adjustment and is currently updated every 10 min for radar rainfall estimates (Chen et al. 2007). In recent years, the QPESUMS system has been used to monitor rainfall from typhoons (e.g., Lee et al. 2006) and for flood forecasting (Vieux et al. 2003).

### Statistical analysis for hourly radar rainfall error

Before developing the proposed PRRE model to evaluate the reliability and accuracy of the hourly radar rainfall estimates provided by QPESUMS, it is necessary to calculate the hourly radar rainfall error and its associated statistical properties at rain gauges. Figure 4 shows the difference between the hourly radar rainfall estimates and observed data, i.e., the radar rainfall error, at ten validation points in the Shinmen reservoir watershed for Typhoon Morakot and its statistical properties are listed in Table 3. It can be seen that the hourly radar rainfall error varies with time and location. In detail, on average, the radar rainfall estimates at the Chiduan, Yufeng, Zhen-Xi-Bao, and Gaoyi gauges are overestimated due to negative mean values of the radar rainfall errors. The associated standard deviations approximately range from 2.0 and 5.0 mm, except at the Gaoyi gauge where the standard deviation reaches 8.0 mm. It is also seen that the radar rainfall estimates at the Chiduan, Shinmen, and Gaoyi gauges deviate more significantly from the observations than those at other rain gauges due to the large variation bounds of the radar rainfall errors, i.e., 38.3 mm (Chiduan gauge), 32.4 mm (Shinmen gauge), and 33.5 mm (Gaoyi gauge). These results indicate that there are different spatial and temporal variations in the hourly radar rainfall errors, which can then be regarded as the spatiotemporal variables. The results for Typhoons Megi and Fanapi are the same as Morakot's.

Table 3

Mean and standard deviation of the radar rainfall error at ten rain gauges for Typhoon Morakot

Variation bound
Gauge Mean Standard deviation Upper Lower
Galahe 0.190 2.526 10 − 10.5
Baling 0.204 2.712 12.1 − 7.7
Chiduan − 0.209 4.030 25.9 − 12.4
Yufeng − 0.238 2.534 − 6
Baishi 0.241 2.994 9.2 − 10.1
Shinmen 0.419 3.923 24.5 − 7.9
Xi-yue-si-shan 0.195 3.323 13.4 − 7.4
Zhen-Xi-Bao − 0.330 2.573 6.7 − 9.9
Xiayun 0.456 3.162 15.9 − 6
Gaoyi − 4.163 8.386 9.9 − 33.5
Variation bound
Gauge Mean Standard deviation Upper Lower
Galahe 0.190 2.526 10 − 10.5
Baling 0.204 2.712 12.1 − 7.7
Chiduan − 0.209 4.030 25.9 − 12.4
Yufeng − 0.238 2.534 − 6
Baishi 0.241 2.994 9.2 − 10.1
Shinmen 0.419 3.923 24.5 − 7.9
Xi-yue-si-shan 0.195 3.323 13.4 − 7.4
Zhen-Xi-Bao − 0.330 2.573 6.7 − 9.9
Xiayun 0.456 3.162 15.9 − 6
Gaoyi − 4.163 8.386 9.9 − 33.5
Figure 4

Comparison of radar and observed hourly rainfalls at ten validation points for Typhoon Morakot.

Figure 4

Comparison of radar and observed hourly rainfalls at ten validation points for Typhoon Morakot.

In summary, the hourly radar rainfall errors, which are attributed to the uncertainties in reflectivity characteristics and sampling data, vary with time and space, so that their spatial and temporal variations probably influence the accuracy and reliability of hourly radar rainfall estimates. In general, the spatial and temporal variation can be described in terms of a spatiotemporal variogram. This is commonly quantified by the theoretical variogram models. Consequently, it is fair to say that this study develops a PRRE model by taking into account the uncertainties in the radar rainfall errors in time and space. In this study, these uncertainties would be quantified by the spatiotemporal variogram models.

### Model development

According to the validation procedure, the parameters of the proposed PRRE model should be first determined using a cross-validation method with the hourly radar rainfall errors at the calibration points (rain gauges) for a specific rain gauge, which is then taken as the validation point. The calibrated parameters for the proposed PRRE model include the parameters of the spatiotemporal variogram models and a logistic regression. The relevant results are expressed below.

#### Parameter calibration of spatial and temporal variogram model

Referring to Table 1, parameters (a and c) of the variogram models for time and space should be calibrated in advance, which this study accomplished by using a modified genetic algorithm (Wu et al. 2011). Table 4 shows the mean and standard deviation of the parameters a and c as calculated from seven spatiotemporal variogram models at ten validation points for Typhoon Morakot. It can be observed that the average parameter a for the various spatial variogram models ranges from 6 to 20 m, and its standard deviation is located between 0.8 and 9.45 m. As for the parameter c in the spatial variogram model, its mean varies from 11 to 16 mm, and the standard deviation ranges, approximately, from 1.7 to 3.6 mm. Since the parameters a and c denote the influence distance that represents the maximum range in which the variogram varies with distance, and the scale of variogram, respectively, the different spatial variations of the radar rainfall should exist for the ten validation points. The same conclusion can be reached from the temporal variogram model's parameters. Consequently, it is necessary to consider the spatial and temporal correlations for the radar rainfall error, which is the basic modeling concept for the probabilistic radar rainfall estimate in this study.

Table 4

Statistics of parameters for spatial and temporal variogram models in space and time for Typhoon Morakot

Parameter
Space Time
a c a C
Model Mean Standard deviation Mean Standard deviation Mean Standard deviation Mean Standard deviation
Spherical 7.364 2.872 11.088 1.735 3.765 1.154 15.914 4.084
Exponential 6.937 1.904 15.487 3.570 4.314 1.446 22.336 11.471
Gaussian 5.749 0.802 13.073 2.864 2.955 0.926 14.396 2.873
Nugget 20.978 9.450 11.548 1.826 11.270 4.218 14.400 0.849
Cubic 8.123 2.231 10.261 1.880 3.238 0.771 13.068 1.899
Circular 6.981 1.830 12.161 2.612 4.212 1.417 17.808 4.263
Pentaspherical 6.028 2.223 11.290 3.777 5.838 1.216 28.195 5.823
Parameter
Space Time
a c a C
Model Mean Standard deviation Mean Standard deviation Mean Standard deviation Mean Standard deviation
Spherical 7.364 2.872 11.088 1.735 3.765 1.154 15.914 4.084
Exponential 6.937 1.904 15.487 3.570 4.314 1.446 22.336 11.471
Gaussian 5.749 0.802 13.073 2.864 2.955 0.926 14.396 2.873
Nugget 20.978 9.450 11.548 1.826 11.270 4.218 14.400 0.849
Cubic 8.123 2.231 10.261 1.880 3.238 0.771 13.068 1.899
Circular 6.981 1.830 12.161 2.612 4.212 1.417 17.808 4.263
Pentaspherical 6.028 2.223 11.290 3.777 5.838 1.216 28.195 5.823

#### Parameter calibration of logistic regression equation

Referring to Equation (8), the negative quantile of the radar rainfall would be obtained in a case of the radar rainfall estimate being less than the absolute value of its negative estimated quantile error. To avoid negative quantiles for the hourly radar rainfall estimate obtained by the proposed PRRE model, this study employs a logistic regression method to improve these negative quantiles for this particular cumulative probability. Since there is a unique logistic regression equation for each time step (hour) at the ten validation points, this study takes the average parameters of the logistic equations for each hour to discuss the applicability of the logistic regression to be applied in the proposed PRRE model. Figure 5 indicates the average values of the parameters for the logistic regression equations and their corresponding coefficient of determination (R2). This is a measure of how well the regression line represents the data (very well if approaching 1) at validation points for each hour during Typhoon Morakot. The average coefficient of the determination for R2 is located between 0.7 and 0.9, and its average is approximately 0.8. In addition, the average coefficients of αlge and βlge vary with the time steps. It can be concluded that the logistic regression equation for the hourly radar rainfall can well describe the changes in the occurrence cumulative probability of the radar rainfall estimations along with the corresponding quantiles. In addition, Figure 6 shows that the average R2 values of the logistic regression equations for each hour during typhoons Megi and Fanapi are about 0.7 and 0.9, respectively. That is to say, they approach 1. These results have the same implications for typhoons Megi and Fanapi as they did for Typhoon Morakot.

Figure 5

Average coefficients (αlge, βlge) and the corresponding coefficient of determination (R2) for the logistic regression equations at various hours for Typhoon Morakot.

Figure 5

Average coefficients (αlge, βlge) and the corresponding coefficient of determination (R2) for the logistic regression equations at various hours for Typhoon Morakot.

Figure 6

Average coefficient of determination (R2) for the logistic regression equations at various hours for Typhoons Megi and Fanapi.

Figure 6

Average coefficient of determination (R2) for the logistic regression equations at various hours for Typhoons Megi and Fanapi.

### Establishment of a cumulative probability distribution for hourly radar rainfall estimates

Using the proposed PRRE model incorporated with the previously calibrated parameters, the cumulative probability distribution, i.e., the relationship of quantiles with various occurrence probabilities, can be established. Figure 7 shows the cumulative probability distribution for the hourly radar rainfall quantiles associated with Morakot's beginning, middle, and end time, i.e., the 1st, 42nd, and 72nd hour, at the ten validation points. It can be seen that the temporal trend of the cumulative probability distribution at the various validation points are similar, except at the Shinmen gauge where the probability distribution for the 1st hour significantly departs from those at the 42nd and 72nd hour due to the different temporal changes in the observations and radar rainfall estimates (see Figure 4). This is because the Shinmen gauge locates downstream from the Shinmen reservoir watershed and is away from other rain gauges, so that its rainfall characteristics are probably different.

Figure 7

Cumulative probability distribution of radar rainfall for the 1st, 42nd, and 72nd hour for ten validation points during Typhoon Morakot.

Figure 7

Cumulative probability distribution of radar rainfall for the 1st, 42nd, and 72nd hour for ten validation points during Typhoon Morakot.

From Figure 7, it is known that the cumulative probability distribution varies with time and space, and the probability associated with an inherently larger radar rainfall error is less than 1 with a smaller error. For example, in a case of the radar rainfall estimate quantile for the 42nd hour being 3.9 mm at the Baishi gauge, the corresponding probability, the error of which is equal to 5.1 mm, approximates 0.2. However, the radar rainfall estimates and its associated errors at the 72nd hour are about 2 mm and 1 mm, respectively, and the corresponding probability reaches 0.99. This implies that a high occurrence probability for the radar rainfall estimate corresponds to less radar rainfall error. That is to say, the occurrence probability is inversely related to the existing hourly radar rainfall error. As a result, the cumulative probability distribution of the hourly radar rainfall estimate derived by the proposed PRRE model can be applied in the reliability assessment for results from the QPE model.

### Reliability and accuracy assessment for the radar rainfall estimate

The 95% confidence interval is defined as the range of quantiles at cumulative probabilities between 2.5 and 97.5%. Since the confidence interval accounts for an estimate's reliability, its width gives some idea about the possible range of uncertainties in the unknown variables. For example, a wider confidence interval in relation to the estimate indicates instability. Moreover, the occurrence probability of a variable can account for the degree of likelihood of the variable being equal to or less than a specific value. Therefore, this study calculates the 95% confidence interval and the occurrence probability corresponding to the hourly radar rainfall estimate using the proposed PRRE model. Therefore, the applicability of the proposed PRRE model can be evaluated by discussing the relationship between the radar rainfall error and its resulting stochastic information.

#### 95% confidence interval

Figure 8 shows the hourly radar rainfall estimates at the ten validation points for Typhoon Morakot and the estimated 95% confidence intervals by the proposed PRRE model. It can be seen that the 95% confidence interval width depends on time step and location. The largest width occurs at the time associated with the high hourly radar rainfall estimates (from the 1st to the 25th hour). On average, the width of the 95% confidence interval for Typhoon Morakot approximately stays at a constant 3.5, except at the Shinmen gauge, which has an approximate average of 4.8. By comparing Figures 3 and 7, the hourly radar rainfall estimates and the observed data mostly are located in the 95% confidence interval, except at the Gaoyi gauge where the upper and lower bounds of the 95% confidence interval are close to the hourly radar rainfall estimates, but the observed rainfalls are significantly less than the estimates. Although a narrow width in the confidence interval represents a stable estimate, the hourly radar rainfall estimates at the Gaoyi gauge are overestimated as compared to the observed rainfalls. Consequently, the proposed PRRE model shows that there is a high probability that Morakot's hourly rainfalls as provided by QPESUMS could match real-time observations everywhere in the Shinmen reservoir watershed, except the Gaoyi gauge.

Figure 8

Radar hourly rainfall and its associated 95% confidence interval at ten validation points for Typhoon Morakot.

Figure 8

Radar hourly rainfall and its associated 95% confidence interval at ten validation points for Typhoon Morakot.

Figure 9 shows the comparison between the average width of the 95% confidence interval and the existing absolute value of the radar rainfall error for three typhoons. It is evident that the temporal change in the absolute value of the radar rainfall error is similar to the 95% confidence interval width. Namely, the deviation in the radar rainfall error increases with the width of the 95% confidence interval. This implies that the deviation of the radar rainfall estimate is positively related to the absolute value of the radar rainfall error. In other words, on average, the reliability of the radar rainfall is correlated with the width of the 95% confidence interval. Therefore, it is known that the reliability assessment of the hourly radar rainfall estimate can be carried out by investigating the change in the width of the 95% confidence interval estimated by the proposed PRRE model.

Figure 9

Comparison of average radar rainfall error and its corresponding average width of 95% confidence interval (95% CI) for three typhoons.

Figure 9

Comparison of average radar rainfall error and its corresponding average width of 95% confidence interval (95% CI) for three typhoons.

#### Occurrence probability

In addition to the 95% confidence interval, the occurrence probability, which corresponds to the hourly radar rainfall as a result of the cumulative probability distribution established by the proposed PRRE model, can be used for assessing the accuracy of the hourly radar rainfall estimate as compared to the actual observations. Figure 10 presents the hourly radar rainfall estimates and the corresponding occurrence probability at the ten validation points for Typhoon Morakot. It can be seen that a lesser hourly radar rainfall error (on average −0.79) has a greater occurrence probability between the 1st and 25th hour (on average 0.7) at the validation points, except for the Gaoyi gauge. At the Gaoyi gauge, the mean value of the hourly radar rainfall error reaches −11.58 during the 1st and 25th hour, but the corresponding probability approximates 0.45. This indicates that the QPESUMS significantly overestimates the hourly radar rainfall estimate; however, the corresponding width of 95% confidence interval is, on average, over 6.1. Therefore, using the proposed PRRE model, it can be known that the QPESUMS can produce an unstable, and less accurate, hourly radar rainfall estimate at the Gaoyi gauge for Typhoon Morakot.

Figure 10

Radar hourly rainfall and the corresponding occurrence probability at ten validation points for Typhoon Morakot.

Figure 10

Radar hourly rainfall and the corresponding occurrence probability at ten validation points for Typhoon Morakot.

Figure 11 indicates the average occurrence probability of the hourly radar rainfall and the absolute inherent error at the ten validation points for all three typhoons. It is known that, on average, a high occurrence probability for the radar rainfall estimate is associated with the low absolute value of the rainfall error. That is, the deviation of the radar rainfall estimate is negatively related to the occurrence probability. As a result, the accuracy of the hourly radar rainfall estimate can be quantified by means of the occurrence probability calculated by the proposed PRRE model. This conclusion is the same argument as discussed in the results from the 95% confidence interval.

Figure 11

Comparison of averaged absolute radar rainfall error and its corresponding average occurrence for three typhoons.

Figure 11

Comparison of averaged absolute radar rainfall error and its corresponding average occurrence for three typhoons.

## CONCLUSIONS

The radar rainfall may be influenced by several uncertainties, such as the reflectivity–rainfall (Z-R) relationships, the reflectivity's vertical profile, and the spatial and temporal sampling. The differences existing between the radar rainfall estimates and actual observations are known as the radar rainfall error. It is necessary to evaluate the effect of the uncertainties attributed to radar rainfall error on the reliability and accuracy of the radar rainfall estimates. Therefore, this study develops a PRRE model based on the radar rainfall error to quantify the reliability and accuracy of the radar rainfall estimates at ungauged locations, which are originally provided by a radar-based QPE model.

The hourly radar and observed rainfalls from three typhoons provided the QPE system QPESUMS and are used as data in the study of the Shinmen reservoir watershed of Northern Taiwan. The proposed PRRE model is validated by discussing the relationship between the radar rainfall estimate and the corresponding cumulative probability distributions derived from the model. The validation results show that the occurrence probability of the hourly radar rainfall estimate calculated by the proposed PRRE model is inversely proportional to the existing error, so that the larger probability accounts for the more accurate radar rainfall estimate. In addition, the narrow width of the resulting 95% confidence interval of the radar rainfall indicates less uncertainty in the radar rainfall estimates.

The isotropic variable in the spatiotemporal analysis is assumed to be the radar rainfall error; however, the error is actually anisotropic (Kirstetter et al. 2010). Therefore, a spatial variogram model for the radar rainfall error is obtained from various sources in order to improve the reliability of the estimated rainfall error at the ungauged locations. Since the proposed PRRE model utilizes identical variogram models in both time and space for a particular time step, future work could adopt different types of spatial and temporal variogram models. No doubt more estimates of rainfall errors can be obtained and used in the proposed PRRE model so as to improve its reliability. Moreover, since the stochastic information on the radar rainfall estimate can be obtained from the proposed PRRE model, it could be applied to the deterministic hydrological model and hydraulic routing to produce a probabilistic runoff and corresponding water level, which would be helpful in the watershed's management and flood forecasting and prevention operations.

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