Editorial Type:
Article
Article Type:
Research Article

Evaluating and Modeling the Reliability of Continuous No-Rain Forecast from TIGGE Based on the First-Passage Problem and Fuzzy Mathematics

Chenkai Cai aZhejiang Institute of Hydraulics and Estuary (Zhejiang Institute of Marine Planning and Design), Hangzhou, China
bCollege of Hydrology and Water Resources, Hohai University, Nanjing, China

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Jianqun Wang bCollege of Hydrology and Water Resources, Hohai University, Nanjing, China

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Zhijia Li bCollege of Hydrology and Water Resources, Hohai University, Nanjing, China

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Xinyi Shen cSchool of Freshwater Sciences, University of Wisconsin–Milwaukee, Milwaukee, Wisconsin

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Jinhua Wen aZhejiang Institute of Hydraulics and Estuary (Zhejiang Institute of Marine Planning and Design), Hangzhou, China

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Helong Wang aZhejiang Institute of Hydraulics and Estuary (Zhejiang Institute of Marine Planning and Design), Hangzhou, China
dCollege of Civil Engineering and Architecture, Zhejiang University, Hangzhou, China

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Xinyan Zhou aZhejiang Institute of Hydraulics and Estuary (Zhejiang Institute of Marine Planning and Design), Hangzhou, China

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Online Publication:
19 Jun 2023
Print Publication:
01 Jun 2023
DOI:
https://doi.org/10.1175/JHM-D-22-0126.1
Page(s):
1119–1135
Free access

Abstract

As an important reference of reservoir regulation, more and more attention has been paid to the numeric precipitation forecast. Due to the uncertainty of meteorological prediction, reservoir regulation based on precipitation forecasts may lead to flood control risks. Therefore, the reliability of precipitation forecasts is crucial to the formulation of reservoir regulation strategy based on it. In this paper, a reliability assessment model for a continuous precipitation forecast is proposed based on the first-passage problem and fuzzy mathematics. The uncertainty of precipitation forecast is described by the generalized Bayesian model, and the fuzzy reliability of a continuous precipitation forecast can be obtained by the first-passage fuzzy probability model (FFPM). Due to the importance of a no-rain period in flood resource utilization, the no-rain forecasts from four different forecast centers in the Meishan basin are used as an example. The results show that the fuzzy mathematics is helpful in describing the uncertainty of the boundary for the no-rain set, and the fuzzy reliability of the no-rain forecast is affected by the selection of the range for the no-rain forecast, while the influence of the membership function is limited. Furthermore, due to the downward trend of fuzzy reliability as the lead time increases, there is a contradiction between excess water storage of the reservoir and the fuzzy reliability of the no-rain forecast. A longer continuous no-rain period means more excess water storage, but it also faces lower reliability. In actual reservoir regulation, the results of FFPM can be combined with more information to formulate better strategies for reservoir regulation.

© 2023 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Jianqun Wang, wangjq@hhu.edu.cn

Abstract

As an important reference of reservoir regulation, more and more attention has been paid to the numeric precipitation forecast. Due to the uncertainty of meteorological prediction, reservoir regulation based on precipitation forecasts may lead to flood control risks. Therefore, the reliability of precipitation forecasts is crucial to the formulation of reservoir regulation strategy based on it. In this paper, a reliability assessment model for a continuous precipitation forecast is proposed based on the first-passage problem and fuzzy mathematics. The uncertainty of precipitation forecast is described by the generalized Bayesian model, and the fuzzy reliability of a continuous precipitation forecast can be obtained by the first-passage fuzzy probability model (FFPM). Due to the importance of a no-rain period in flood resource utilization, the no-rain forecasts from four different forecast centers in the Meishan basin are used as an example. The results show that the fuzzy mathematics is helpful in describing the uncertainty of the boundary for the no-rain set, and the fuzzy reliability of the no-rain forecast is affected by the selection of the range for the no-rain forecast, while the influence of the membership function is limited. Furthermore, due to the downward trend of fuzzy reliability as the lead time increases, there is a contradiction between excess water storage of the reservoir and the fuzzy reliability of the no-rain forecast. A longer continuous no-rain period means more excess water storage, but it also faces lower reliability. In actual reservoir regulation, the results of FFPM can be combined with more information to formulate better strategies for reservoir regulation.

© 2023 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Jianqun Wang, wangjq@hhu.edu.cn
Keywords: Bayesian methods; Risk assessment; Uncertainty; Numerical weather prediction/forecasting; Water resources

1. Introduction

With the rapid development of social economy and population growth, the global demand for water resources is increasing rapidly (Han and Zhao 2007; Pizarro et al. 2022). At the same time, due to the impact of climate change and human activities, many areas are simultaneously threatened by frequent floods and water shortages. Reservoirs are one of the critical engineering measures in flood control and water resource utilization system, which not only play an important role in intercepting floods during flood seasons to ensure downstream safety, but also needs to guarantee the water supply for downstream in nonflood seasons (Ahmad et al. 2014; Dong et al. 2020; Zhu et al. 2017). However, due to the limited storage of reservoirs, there is usually an inevitable conflict between ensuring the safety of flood control and flood water utilization (Xu et al. 2015). Making full use of the potential of reservoir flood resource utilization under the premise of flood control safety is one of the important issues in hydrological research.

The dynamic control of the flood limited water level (DC-FLWL) is an effective way to make full use of the reservoir capacity for flood resource utilization (Chang et al. 2017; Chou and Wu 2013; Zhou et al. 2013). The FLWL is the upper limit of the water level that is allowed to store water in flood season. The traditional FLWL is a fixed value and determined by the flood regulation for a designed flood, which ignores the real-time precipitation and inflow condition. In contrast, the DC-FLWL means that the FLWL is allowed to fluctuate within specified bounds based on real-time forecast information (Zhou et al. 2014). Currently, due to the continuous improvement of hydrometeorological forecasting and hydrological monitoring systems, the precipitation forecast and flood forecast are gradually being introduced in DC-FLWL for reservoir regulation, and a series of research results has been achieved in China. Zhang et al. (2011) assessed risk of DC-FLWL under different forecast error levels for Dahuofang Reservoir in China. Liu et al. (2015) simulated the risk of forecast-based flood control regulation by the Monte Carlo method. These results show that the forecast information can effectively help reservoir managers formulate reservoir operation strategies in advance to make better use of flood resources, but it may also increase flood control risks. Therefore, it is crucial to use reliable forecast information with a long lead time in DC-FLWL for reservoir regulation and flood resource utilization.

Numerical weather prediction (NWP) is an efficient way to provide future information with a long lead time for DC-FLWL (Chen et al. 2018; Peng et al. 2017; Todini 2017). For example, in the actual reservoir operation, when the precipitation forecast result indicates that there will be a rainstorm in the near future, the reservoir will be threatened by flooding, and the reservoir manager should not store water beyond the FLWL in order to ensure the flood control safety of the reservoir. In contrast, when the reservoir is in the retreat stage of a flood and there is a continuous no-rain period in the next few days, the reservoir can overstore a part of the flood to better utilize the flood resources. The essence of excess water storage is to store a part of flood resources by exceeding the FLWL in the flood retreat stage, and the water level will be reduced to FLWL through the utilization discharge (mainly for power generation and various types of water supply) before the next flood. In this way, the reservoir managers can realize the flood resource utilization and give full play to the benefits of reservoir capacity without bringing additional flood control risk. From the above introduction, it can be found that the length of the no-rain period will largely determine the excess water storage capacity in the reservoir regulation, and the reliability of the no-rain forecast will affect the potential risks for overstoring. If the rainless period ends early, the excess water stored cannot be safely discharged before the flood, which may lead to flood control risks. Unfortunately, although the forecast skill of NWP has been improved in recent years, the error and uncertainty of precipitation forecasts are still inevitable, which means future information with perfect forecast skill cannot be obtained based on NWP (Bauer et al. 2015; Lavers and Villarini 2013; Su et al. 2014). Therefore, it is of great significance to evaluate the reliability of precipitation forecasts to provide a reference for formulation strategies of reservoir operation. To be more specific, it is necessary to know which future information from NWP is reliable and can be used as the basis for DC-FLWL to avoid additional flood risk in reservoir operations.

Most of the current studies are based on flood control risk analysis to evaluate the reliability of forecast information. However, different studies have different definitions of flood control risks for reservoirs. For instance, Hua et al. (2020) assumed that once the no-rain forecast fails, the next flood will occur immediately to simulate the worst-case scenario for risk analysis of precipitation forecast. The different levels of floods will lead to different risks and reliability results. For some studies, a large flood such as the design flood is used for risk analysis, which is a small probability event, especially for large reservoirs, so the results are conservative. On the other hand, if the magnitude of the flood used is too small, the result will be too radical, which is likely to cause flood control risks. Additionally, these studies are based on specific periods or scenarios, and cannot fully assess the reliability of forecast information. Although hydrological and meteorological scholars have proposed plenty of indicators to evaluate the accuracy and reliability of precipitation forecast, such as receiver operating characteristic curve (ROC), Brier score (BS), and continuous ranked probability score (CRPS), these indicators only focus on the reliability of the forecast on specific time scale or single event, and are difficult to evaluate the reliability of continuous precipitation events (Brier 1950; Candille and Talagrand 2005; Hersbach 2000). Specifically, the traditional indicators regard each precipitation forecast from each step of lead time as a single event, rather than a set of continuous events within the whole lead time.

In this paper, a new model based on Bayesian statistics and the first-passage problem is proposed to assess the reliability of continuous daily precipitation forecast from the perspectives of reservoir regulation and flood resource utilization. Fuzzy mathematics is also introduced in the model to describe the fuzziness of the precipitation magnitude classification caused by the uncertain boundary. More detailed information and definitions of the model are introduced in section 2. An example of reliability assessment for the precipitation forecasts from four different global centers are taken over the Meishan Reservoir of the Huaihe River basin in China. Compared to the former studies, the new model provides a general and direct analytical method to assess the reliability of continuous daily precipitation forecasts and can be used as a reference for decision-making in reservoir regulation to improve the utilization of flood resources and avoid flood control risks. Section 3 will introduce an overview of the dataset and study area. Section 4 will detail the model construction and experiment design of this paper. The results and discussion are described in section 5, and finally, the conclusions are provided in section 6.

2. Methodology

The main aim of this study is to develop a model to evaluate the reliability of precipitation forecast for reservoir regulation and flood resource utilization. Due to the highly nonlinearity of the atmospheric system, there are always uncertainty and error in the precipitation forecast from NWP. The probabilistic postprocessing method is one of the most commonly used methods to describe the uncertainty in precipitation forecast. In this paper, the generalized Bayesian model (GBM) developed by Cai et al. (2019) is used to generate the probabilistic precipitation forecast. According to the results from uncertainty analysis, an assessment model is established based on the first-passage problem in reliability theory. The main methods and models are provided below.

a. Generalized Bayesian model

The GBM model is an uncertainty analysis model for precipitation forecast based on Bayesian statistics, which can generate probabilistic precipitation forecasts from single value. Let random variable (r.v.) X indicate the forecasted precipitation, and r.v. Y denote the corresponding true value. The essence of the statistical postprocessing method is to estimate the distribution of Y through statistical methods when we know the value of X = x, i.e., the conditional distribution f(y|X = x). However, the natural rainfall is generally characterized as an intermittent stochastic process, with a mixed marginal distribution function, including both discrete and continuous parts, which makes it difficult to directly apply Bayesian formula. For any real number y, the cumulative probability distribution (CDF) of r.v. Y can be written as
F(Y≤y)={0,y<0P{Y=0}+P{Y>0}P{0<Y<y|Y>0},y≥0.
It can be seen in Eq. (1) that the discrete part is concentrated at zero and means the probability of no rain, and the continuous part can be estimated by a nonnegative distribution. An example of mixed CDF for r.v. Y is shown in Fig. 1.
Fig. 1.
Fig. 1.

An example of mixed CDF for r.v. Y.

Citation: Journal of Hydrometeorology 24, 6; 10.1175/JHM-D-22-0126.1

To solve this problem, the Dirac delta function [Eq. (2)] is employed to transform the distribution law of a discrete random variable into a probability density function (PDF) form called a generalized PDF (GPDF):
δ(x)={+∞,x=00,x≠0,∫−∞+∞δ(x)dx=∫−εεδ(x)dx=1,∀ε>0.
Through Eq. (2), we can get the GPDF of precipitation as
fY(y)=α0δ(y)+(1−α0)fY(y|Y>0),α0=P{Y=0}.
There are two main parts of the GBM model, the prior distribution fY(y) and the likelihood function fX|Y(x|Y = y). For this study, the prior distribution is the marginal distribution of natural precipitation, which can be estimated by the historical observation data [Eq. (3)]. The likelihood function can be drawn by the distribution of error. When the actual precipitation is known, the error of precipitation forecast is
X=y+ε(y).
Through the distribution of ε(y), the conditional distribution fX|Y(x|Y = y) can be drawn by the random variable function:
fX|Y=y(x)={β0,0δ(x)+(1−β0,0)fX|Y=0(x|Y=0),y=0β0,yδ(x)+(1−β0,y)fX|Y=y(x|Y=y>0),y>0,
where β0,0 = P{X = 0, Y = 0} and β0,y = P{X = 0, Y = y > 0}.

Note that it is necessary to divide the likelihood function into two conditions according to the value of y. The error ε is nonnegative when y = 0 due to the sample space of X ≥ 0. Therefore, the continuous part for y = 0 is a nonnegative distribution. If y > 0, the sample space of error ε is [−y, +∞).

Then, we can write the GBM model as
fY|X(y|X=x)=fY(y)fX|Y(x|Y=y)∫−∞+∞fY(t)fX|Y(x|Y=t)dt.
Once the prior distribution [Eq. (3)] and likelihood function [Eq. (5)] are determined, we can use the GBM model [Eq. (6)] to obtain the GPDF of precipitation Y when the precipitation forecast value X = x is known.

It is worth noting that the form of Eq. (6) is the same as the traditional Bayesian formula for continuous random variable, but the functions used in Eq. (6) are GPDF, which means the random variables in the formula can be discrete or mixed.

b. First-passage problem for continuous precipitation forecast

1) Definition of reliability in precipitation forecast

Reliability is a one of the important concepts in the industry field that has been used to estimate the quality of a system or product (Christian et al. 1994; Huang et al. 2021). Reliability is defined as the probability that a product or system will perform its intended function adequately for a specific time or will operate in a defined environment without failure. The time when the product fails for the first time is called the first-passage time (Crandall et al. 1966; Darling and Siegert 1953; Pal and Prasad 2019). Therefore, the definition for failure of a precipitation forecast is the key for the first-passage problem and reliability assessment of a precipitation forecast.

Meteorological researchers usually evaluate the hit rate of a precipitation forecast by the relationship between the magnitude of forecast and observation. Three combinations of the relationship are as follows: hit (the magnitude of the forecast is the same as that of the observation), miss alarm (the magnitude of the forecast is smaller than that of the observation), and false alarm (the magnitude of the forecast is larger than that of the observation) (Cai et al. 2018). Another point that should be taken into consideration for the definition of a precipitation forecast failure is the demand for reservoir regulation. In actual reservoir operations, the reservoir operation strategies can be formulated by magnitudes of flood or precipitation, rather than specific values. However, it is necessary to pay attention to the different impacts of miss alarms and false alarms in reservoir regulation. Generally, if the forecast information indicates a big amount of precipitation or flooding, the reservoir will maintain FLWL without storing excess flood water, so the false alarm will reduce the benefits of flood resources. On the contrary, if there is no rain or only light rain in the future, the reservoir can overstore a part of the water for flood resource utilization. In this case, once a miss alarm occurs, it will cause a great risk of flood control. In this paper, the definition of reliability for a continuous precipitation forecast is the probability that the precipitation will not exceed its corresponding forecast magnitude within the lead time. Therefore, the first passage for a continuous precipitation forecast can be defined as the first occurrence of precipitation exceeding the specified threshold obtained from the forecast information in the specified foreseeable period.

The magnitude of precipitation has been commonly used in accuracy evaluation, such as hit rate and Brier score. Table 1 shows the classification standard of daily precipitation from the meteorological department of China. However, for the first-passage problem of a precipitation forecast, the fuzziness of the magnitude boundary may affect the results of the first-passage time and reliability assessment. For example, the daily precipitation forecast result is 49.8 mm, and the corresponding observation is 50.1 mm. According to Table 1, the magnitude of precipitation forecast is heavy rain, and the threshold is 50 mm. The observation exceeds the threshold, and the precipitation forecast will be considered as a failure, which is undoubtedly too strict. To solve this problem, fuzzy reliability is employed in this paper (Sruthi and Kumar 2021; Wu 2004; Yiping et al. 2014).

Table 1

Classification standard of daily precipitation.
Table 1

2) Fuzzy reliability of precipitation forecast

To solve the fuzziness of the precipitation magnitude boundary in this paper, we are going to introduce the theory of fuzzy reliability in this section. First of all, the basic idea of fuzzy set is shown as follows (Zadeh 1965).

Let X be a universal set and A be a subset of X. We can define a characteristic function χA: X → {0, 1} with respect to A by
χA(x)={1,x∈A0,x∉A.
The fuzzy subset A˜ of X is introduced by extending the characteristic function. A fuzzy set A˜ on the universe set X is mapping A˜:X→[0,1], and A˜(x) represents the degree of membership of each element x ∈ X. The A˜(x) of a fuzzy set A˜ is also called the membership function and can be presented as μA˜(x).

The theory of fuzzy reliability was proposed by Kaiyuan Cai to characterize the fuzziness in system reliability assessment (Cai et al. 1993, 1991). There are three different categories of fuzzy reliability problem: 1) the event is fuzzy, but the probability is clear; 2) the event is clear, but the probability is fuzzy; and 3) both the event and probability are fuzzy. In this paper, only case 1 (i.e., the fuzzy event with clear probability) is discussed. For instance, the probability for heavy rain tomorrow is 80%. Here, the magnitude of precipitation (i.e., heavy rain) may have a fuzzy boundary, but the probability (80%) is a clear value.

As an extension of conventional reliability, the main indicators of fuzzy reliability can be obtained by corresponding conventional reliability form. Compared with the original reliability indicators, these indicators have richer and more complex connotations after adding fuzziness. Here, we provide the main indicators of fuzzy reliability used in this study.

The reliability function R(t) is one of the most important indicators in reliability theory. Define the first-passage time (failure time) T as the time from the start of the system (or product) to the first time the system fails. The failure time T is a random variable with a PDF as f(t). And the CDF of T is
F(t)=P(T≤t)=∫0tf(u)du,t>0.
According to the definition of reliability and Eq. (8), the reliability function R(t) can be calculated as
R(t)=1−F(t)=1−∫0tf(u)du=∫t∞f(u)du.
For fuzzy reliability, let R˜(t) represents the fuzzy reliability function:
R˜(t)=∫t∞μ(A˜)dF(t)=μ(A˜)R(t),
where μ(A˜) is the membership function, which is the bridge connecting R(t) and R˜(t).

It can be seen from Eq. (10) that the fuzzy reliability function R˜(t) is paired with the fuzzy subset A˜. The fuzzy reliability R˜(t) is meaningful only if there is a fuzzy subset A˜. For different fuzzy subsets, the corresponding fuzzy reliability is also different.

Another important indicator of fuzzy reliability assessment for reservoir regulation and flood resource utilization is the mean time to failure (MTTF). In the conventional reliability, the MTTF can be calculated as
MTTF=E(T)=∫0+∞tf(t)dt,
where
f(t)=ddtF(t)=ddt[1−R(t)]=−R′(t).
By integrating the right side of Eq. (12), we can get
MTTF=−[tR(t)]0+∞+∫0+∞R(t)dt.
When MTTF < +∞, [tR(t)]0+∞=0, so
MTTF=∫0+∞R(t)dt.
Since the fuzzy reliability function R˜(t) is closely related to the fuzzy subset A˜, the fuzzy MTTF is also linked to the selected fuzzy subset. Let M˜(A˜) be the fuzzy MTTF for fuzzy subset A˜. Through the definition of mathematical expectation in probability theory, we can get
M˜=∫0+∞t[dF˜(t)dt]dt=∫0+∞td[1−R˜(t)]dtdt=∫0+∞−tdR˜(t).
From Eq. (13), the integral result on the right side of the above equation is
M˜=−[tR˜(t)]0+∞+∫0+∞R˜(t)dt.
Also, if M˜(A˜)<+∞,
M˜=∫0+∞R˜(t)dt.

3) First-passage fuzzy probability model for a continuous precipitation forecast

According to the demand of reservoir regulation, the fuzzy failure of a precipitation forecast defined as the fuzzy magnitude of precipitation is greater than the fuzzy magnitude of a precipitation forecast, and the time when the fuzzy failure of a precipitation forecast occurs during the lead time is the fuzzy first-passage time. Assume that the precipitation forecast results for t (1 < t ≤ d) days are x1, x2, …, xt, respectively, and the corresponding fuzzy probabilities of no fuzzy first passage are P1˜,P2˜,…,Pt˜.

The fuzzy probability of the precipitation forecast can be obtained by the GBM and membership function:
f(A˜|X=x)=∫ΩfY(y)fX(x|Y=y)μA˜(y)∫−∞+∞fY(t)fX(x|Y=t)dtdy.
Equation (18) shows the fuzzy GBM (FGBM) model derived from Eq. (6). Through this formula, we can get the fuzzy probability of different fuzzy subsets.
The main assumption of the fuzzy first-passage model (FFPM) for a precipitation forecast is that the daily precipitation values are independent of each other. Therefore, the distribution of the fuzzy failure time can be simulated by
F˜(t)={1−P˜1,t=1F˜(t−1)+∏i=1t−1P˜i(1−P˜t),t≥2.
With the definition of fuzzy reliability, R˜(t) is
R˜(t)=1−F˜(t)={P˜1,t=11−F˜(t−1)−∏i=1t−1P˜i(1−P˜t),t≥2,
where Pi˜,i=1,2,…,t is the probability of not exceeding the specified fuzzy magnitude.
Since the FFPM for precipitation forecast is a discrete system, the fuzzy MTTF can be calculated by
M˜(A˜)=∑i=1tR˜(i)Δt.
Through Eqs. (19)–(21), the FFPM model makes it possible to assess the reliability of precipitation forecasts with different lead times and different magnitudes. Meanwhile, it also provides an effective way to analyze the lead time selection for reservoir regulation.

c. Excess water storage calculation based on a continuous no-rain forecast

In DC-FLWL management, if there is a continuous no-rain period in the future days, the reservoirs can store excess water storage to improve the flood resource utilization. The volume of excess water storage is related to the length of a continuous no-rain period. Here, we use a simplified example to show the potential excess water storage under the assumption that the precipitation forecast is accurate.

Let d represent the forecasted no-rain period, and there will be rain in d + 1, so we can get Eq. (22) based on the water balance:
Vtd=Vt0+∫t0tdQin(t)dt−∫t0tdQout(t)dt,
where [t0, td] is the consecutive rainless period; Vtd, Vt0 are the corresponding reservoir storage capacities; Qin is the inflow of the reservoir during[t0, td]; and Qout is the outflow of the reservoir during [t0, td], including the discharges of power generation, flood discharge, and various types of water supply.
From Eq. (22), the excess water storage ΔW at t0 can be calculated as
ΔW=Vt0−Vf=∫t0tdQout(t)dt−∫t0tdQin(t)dt+(Vtd−Vf),
where Vf is the water storage capacity corresponding to FLWL of the reservoir.
Considering the requirement of flood control safety, the reservoir should be released to the FLWL at the end of the no-rain period, i.e., Vtd=Vf. Therefore, we can get
ΔW=Vt0−Vf=∫t0tdQout(t)dt−∫t0tdQin(t)dt.
In the actual reservoir regulation, when there is a long consecutive rainless period in the future, the increased demand of water supply, especially for the water demand from irrigation, will lead to a larger outflow of the reservoir (Qout), while the inflow of the reservoir (Qin) will decrease. For this case, it is possible to store more water (ΔW) in the reservoir to obtain greater benefit and reduce the risk of drought. On the other hand, if there is still more precipitation in the future, the reservoir should not store water beyond the FLWL to avoid the flood control risk. Theoretically, this method will not increase the flood control risk and can effectively improve the utilization of flood resources if the no-rain forecast is accurate. However, although some former studies indicate that the precipitation forecast has better performance in no-rain scenario (Cai et al. 2018), it is still impossible to obtain perfect no-rain forecasting. Once the no-rain forecast fails and the excess water storage in the reservoir cannot be discharged in time, this method may bring additional flood control risk. Therefore, it is necessary to fully understand the reliability of no-rain forecast before employing the forecast results as the reference of reservoir regulation and flood resource utilization.

3. Data and study area

a. Study area

The Meishan Reservoir is located in the Dabie Mountains at the junction of Hubei, Henan, and Anhui Provinces and the upper reaches of the Shihe River, a tributary of the Huai River. The Meishan Reservoir is a large-scale water conservancy project with comprehensive utilization functions such as flood control, irrigation, power generation, shipping, and aquaculture. The catchment of the reservoir is a typical monsoon-affected area with a drainage area of 1970 km2, and the primary tasks for the reservoir are to ensure the safety of flood control and water supply. The flood season for this catchment is from May to September. The catchment experienced many major flood events in the past, which severely threatened the safety of the reservoir and downstream cities. Also, as the main water source of the downstream irrigation area, the designed irrigation area for the Meishan Reservoir is over 2500 km2. The contradiction between flood control and water supply is prominent in the management of the Meishan Reservoir, and flood resource utilization based on DC-FLWL is a feasible method to alleviate this contradiction. The location of the Meishan Reservoir is shown in Fig. 2.

Fig. 2.
Fig. 2.

The Meishan basin in the upper Huaihe River basin and the location of the reservoir, hydrological station, and precipitation stations. From the digital elevation model (DEM) data, the elevation of the basin gradually decreases from south to north.

Citation: Journal of Hydrometeorology 24, 6; 10.1175/JHM-D-22-0126.1

The total storage capacity of the Meishan Reservoir is 2.263 billion m3, of which the flood control storage capacity is 1.065 billion m3 and the beneficial reservoir capacity is 0.957 billion m3. The total installed capacity of generators is 4 × 10 MW. The maximum benefit flow is about 170 m3 s−1.

b. Data source

As a major component of The Observing System Research and Predictability Experiment (THORPEX), TIGGE (the THORPEX Interactive Grand Global Ensemble) is a numeric weather forecast dataset consisting of eleven main forecasting centers around the world since 2006 (Bougeault et al. 2010). The main purpose for TIGGE is to improve the ability of high-impact weather forecasting at 2 weeks lead time. In consideration of data integrity, base time, and time step, the control forecasts from four different centers, European Centre for Medium-Range Weather Forecasts (ECMWF), Japan Meteorological Agency (JMA), United Kingdom Meteorological Office (UKMO), and China Meteorological Center (CMA), are selected as the precipitation forecasts in this study. More details of these datasets used in this study are given in Table 2.

Table 2

Configurations of the four TIGGE precipitation forecasts used in this study.
Table 2

The daily precipitation forecasts from the four centers (ECMWF, JMA, CMA, and UKMO) during the flood season in 2015–19 are used as the input for the uncertainty analysis model. Due to the different horizontal resolutions and forecast lengths of the meteorological centers, a series of methods is employed to make the datasets consistent. In this paper, the precipitation forecasts with a base time at 0000 UTC are selected, and the lead time is 7 days. The spatial resolution of the forecast from the centers is converted to 0.50° × 0.50°. Meanwhile, in order to compare with the observed data from ground sites, the precipitation is averaged spatially with gridded area as the weight. The observations measured from the ground precipitation stations with long series data of the basin (as shown in Fig. 2) are used as the reference data, and the areal average observation is calculated by the area-weighted method.

4. Model construction and experimental design

There are two main models used in this study, one is the precipitation forecast uncertainty analysis model, the other is the fuzzy reliability evaluation model. In this section, we discuss the construction and parameter selection of the two models and give the experimental design in the next sections.

a. Model construction

As shown in Eqs. (1)–(6), the GBM is mainly composed of two parts: the prior distribution and the likelihood function. The prior distribution in this study is the marginal distribution of observed precipitation, which can be obtained by the historical observation data and does not vary with the lead time. The forecasted precipitation from different lead times and their corresponding observations are employed to establish the GBM. Accordingly, the ratio of no-rain days and a nonnegative distribution are used to estimate the prior distribution. For the likelihood function, the censored normal distribution is used to describe the distribution of forecast error [Eq. (25)]. All the distributions and their parameters are determined by the Kolmogorov–Smirnov test and maximum likelihood estimation (MLE):
f(ε)=1S(c)σ2πexp⁡[−(ε−μ)22σ2],ε>c,
where S(c)=1−Q[(c−μ)/σ] and Q(t) is the function of standard normal distribution.
The fuzzy first-passage model (FFPM) for the precipitation forecast is developed based on the uncertainty analysis and membership function of the precipitation forecast. The GPDF of the precipitation forecast is obtained from GBM, and the trapezium distribution curve is selected as the membership function for Table 1 in this paper [Eq. (26)]. Note that since the main focus of this study is the reliability of a no-rain forecast, the Eq. (26) only shows the left boundary of the trapezoidal distribution curve. More detailed information about the trapezium distribution curve can be found in Cai et al. (2019):
μA˜(x)={1,x≤ab−xb−a,a<x≤b.0,x>b

b. Experimental design

As there are a large number of different combinations of daily precipitation forecasts within the lead time, the assessment and simulation of the first-passage time or reliability will have certain differences. Meanwhile, the different combinations of daily precipitation have different values in practical application, some specific combinations may be more important for flood resource utilization than others.

From Eqs. (22) to (24), it can be found that the reliability of a no-rain forecast is an important indicator to determine the excess storage of the reservoir during the flood season, which is of great significance to reservoir regulation and flood resource utilization. Therefore, the daily no-rain forecast is employed as an example in this study to show the application potential of the FFPM for precipitation forecast and discuss the key issues in reliability assessment of precipitation forecast. In this experiment, the definition of failure of precipitation forecast is the magnitude of precipitation exceeds the no-rain magnitude, and the time that first failure occurs is defined as the first-passage time. Note that although the analysis and discussion of this paper is based on the no-rain forecast from the study area, the FFPM is general method, and can be used to evaluate the fuzzy reliability of different precipitation magnitude in different areas.

Due to the uncertainty of the precipitation forecast, even if the forecast result is heavy rain, the probability of no rain still exists. Accordingly, the daily “no-rain” forecast in this section does not only refer to the case in which the precipitation forecast is 0 mm, but to a certain interval. When the forecast value falls within this interval, it can be considered as a no-rain forecast. In the following sections, for the convenience of presentation, a forecast selected in this way is referred to as a continuous no-rain forecast. Meanwhile, it can be found that the first-passage problem for a precipitation forecast is also related to the selected interval of a no-rain forecast, so this study will conduct research and discussion on it via two methods. The first method is to use a fixed threshold, that is, to use specific values to represent the boundary of the no-rain forecast interval. The second method is to use a cut set of probability, which means the upper and lower limits of the interval are not fixed, but their corresponding probabilities are fixed. In addition, the influence of fuzzy mathematics used in the reliability assessment is also discussed in this paper. Since the fuzzy set is an extension of traditional set theory, the traditional set (i.e., the set without fuzziness) can be regarded as a special case of fuzzy set, as the membership function shown in Eq. (7). Therefore, the methods in this study can also be used for traditional set, and the fuzzy reliability assessment results from traditional set are used as a comparison to show the influence of fuzziness.

To comprehensively evaluate the reliability of the daily no-rain forecast with different combinations, the method of stochastic simulation is adopted in this paper, the specific steps for reliability assessment of the no-rain forecast are as follows:

  • Step 1: According to the accuracy and uncertainty of the precipitation forecast, determine the reasonable precipitation forecast value interval of the no-rain forecast.

  • Step 2: Based on the GPDF of precipitation, randomly sample within the no-rain interval to generate a daily no-rain sample group during the lead time.

  • Step 3: Calculate the probability of each sample by GBM.

  • Step 4: Calculate the fuzzy reliability, MTTF and other related indicators of no-rain forecast through FFPM to assess the precipitation forecast.

5. Results and discussion

In this section, we assess the fuzzy probability of a no-rain forecast from different centers and no-rain sets, and simulate the fuzzy reliability for a continuous no-rain forecast with different cases. The precipitation forecast and observation data from May to September during 2015–19 are used to estimate the distribution and parameters for the FGBM model.

a. Fuzzy probability of a no-rain forecast

The first step of this study is to generate the fuzzy probability of a no-rain forecast from different centers in TIGGE. For the fuzzy set, the trapezium distribution curve [Eq. (26)] is used to describe the fuzziness of the no-rain boundary. The parameters, a and b, in the membership function are determined by the runoff generation characteristic of the study area. Since precipitation lower than 3 mm has very limited impact on runoff generation of the study area, and precipitation less than 1 mm hardly generates runoff, we adopt the following membership function [Eq. (27)] as the fuzzy set for no rain in this study. As a comparison, the traditional set are also employed, and the membership function can be written as Eq. (28) based on the classification standard in Table 1:
μA˜={1,x≤13−x2,1<x≤30,x>3,
μA˜={1,0≤x<0.10,otherwise.
Through Eqs. (27) and (28), we can get the fuzzy probability of different no-rain sets with the FGBM model when the precipitation forecast is available. Figures 3 and 4 show the fuzzy probability for the two sets from different precipitation forecast values at different lead times. From the two figures, it can be seen that probability peaks for the fuzzy set and traditional set are close to 0–1 mm. Also, the fuzzy probabilities of the two sets show similar trends with the increasing precipitation forecast values. When the precipitation forecast value is lower than 20 mm, both the fuzzy probabilities show a decline trend, while fuzzy probabilities of no rain rise when the forecast value is over 50 mm, and the lead time exceeds 5 days. These results indicate that the large prediction values with long lead times are likely to be overestimated, but different agencies have different degrees of overestimation. The precipitation forecast from CMA has the greatest overestimation in both of the two sets, especially when the lead time is over 5 days, while the results from ECMWF, JMA, and UKMO have less uncertainty for both sets. The uncertainty of the precipitation forecast increases as the lead time grows. The main difference of the results in Figs. 3 and 4 is that the fuzzy probability from the traditional set has a sharper drop when the forecast value is from 0 to 3 mm, while the fuzzy probability for the fuzzy set changes more gently, which is caused by the fuzziness of the set boundary.
Fig. 3.
Fig. 3.

The fuzzy probability of the fuzzy no-rain set from different centers and lead times in the Meishan basin during 2015–19.

Citation: Journal of Hydrometeorology 24, 6; 10.1175/JHM-D-22-0126.1

Fig. 4.
Fig. 4.

The fuzzy probability of the traditional no-rain set from different centers and lead times in the Meishan basin during 2015–19.

Citation: Journal of Hydrometeorology 24, 6; 10.1175/JHM-D-22-0126.1

b. First-passage fuzzy probability model for a continuous no-rain forecast

The combination of continuous precipitation forecasts is infinite. Although we only focus on the no-rain forecast in this section, it is still impossible to fully simulate all scenarios. The historical data are very limited and cannot fully evaluate the fuzzy reliability of no-rain forecasts from the four centers. A viable alternative is to use random samples, which can be easily generated by the Markov chain–Monte Carlo (MCMC) method and the distribution of precipitation forecasts. Meanwhile, in actual application, if the precipitation forecast value is close to 50 mm, despite there still being uncertainty, it is unreasonable to regard it as a no-rain forecast. Therefore, another issue that needs to be considered in random sampling is how to choose a reasonable range of the no-rain forecast. The fixed threshold and cut set of fuzzy probability are selected in this paper to determine the range of a no-rain forecast based on the results from Figs. 3 and 4.

The results of the first-passage fuzzy probability from different lead times and fixed thresholds for the two sets (i.e., the traditional set and fuzzy set for no rain) are shown in Fig. 5. From Fig. 5, it can be found that the fuzzy probabilities of first passage for a no-rain forecast from the traditional set are larger than that from the fuzzy set at all the thresholds. All the fuzzy probabilities of first passage from different sets, centers, and thresholds rise with the lead time increasing. From the comparison of the no-rain forecast from different centers, the performance of CMA is obviously inferior to the other centers for a fixed threshold lower than 1 mm, but with the increase of the fixed threshold, the gap decreases or even disappears. Meanwhile, the no-rain forecast from UKMO has the lowest fuzzy probability of first passage for most lead times and thresholds, especially for the fuzzy set, which indicates the best reliability in the four centers. In addition, it is worth noting that a larger fixed threshold is helpful for flood resource utilization, but it also leads to the increase of failure probability.

Fig. 5.
Fig. 5.

The fuzzy probability of first passage for a continuous no-rain forecast from the four centers with different no-rain sets, lead times, and fixed thresholds.

Citation: Journal of Hydrometeorology 24, 6; 10.1175/JHM-D-22-0126.1

The fuzzy reliability of the no-rain forecast from different centers, lead times, and fixed thresholds for the two sets is illustrated by Fig. 6. From Eq. (20), it can be found that there is a one-to-one linear relationship between the fuzzy reliability and the first-passage fuzzy probability. Therefore, the results in Figs. 6 and 5 are consistent. The reliability of the no-rain forecast can exceed 0.8 for both sets when the threshold and lead time are both small.

Fig. 6.
Fig. 6.

The fuzzy reliability for a continuous no-rain forecast from the four centers with different no-rain sets, lead times, and fixed thresholds.

Citation: Journal of Hydrometeorology 24, 6; 10.1175/JHM-D-22-0126.1

The fuzzy MTTF of the no-rain forecast within 7 days is shown in Fig. 7. The fuzzy MTTF results of both the traditional set and fuzzy set show a decreasing trend with the increase of fixed threshold in Fig. 7. Also, the fuzzy MTTF from fuzzy set is larger than that from traditional set. From the aspect of different centers, the UKMO has the best performance when the fixed threshold is lower than 5 mm. The best fuzzy MTTF is over 5 days, which can be very useful in reservoir regulation and flood resource utilization. Meanwhile, the results from Figs. 5–7 also imply that the fuzziness of the set boundary is worth considering, which can effectively avoid the over-conservative reliability assessment caused by the uncertainty in the set boundary. Also, the selection of the threshold has a great impact on the reliability of the continuous no-rain forecast.

Fig. 7.
Fig. 7.

The fuzzy MTTF for a continuous no-rain forecast from the four centers within 7 days under different no-rain sets and fixed thresholds.

Citation: Journal of Hydrometeorology 24, 6; 10.1175/JHM-D-22-0126.1

Another method to determine the range of a no-rain forecast is to employ the probability cut set. In this study, two different cut sets are used to assess the reliability of no-rain forecasts from the four centers. Four different fuzzy probability cut sets are selected in this study. Meanwhile, note that although Figs. 3 and 4 indicate that the fuzzy probability of no rain increases when the forecast value is over 50 mm, the cut sets with values over 50 mm are not considered in this paper due to the potential flood control risk. Figures 8–10 present the first-passage fuzzy probability, fuzzy reliability, and fuzzy MTTF of the no-rain forecast from different centers and cut sets for the traditional set and fuzzy set. As can be seen in the figures some results are missing because there are no matching cut sets, especially the results from long lead times and cut sets with a large fuzzy probability. Compared with the results of the fixed thresholds, the results of different no-rain sets and centers under the same cut set are very similar, which shows that a suitable no-rain forecast interval can effectively improve the reliability. However, for the results of the cut set, the width of the cut set is another aspect that we should pay attention to. The width of the cut sets from different no-rain sets, centers, and lead times used in the reliability assessment are shown in Fig. 11. In Fig. 11, the width of cut set from the traditional set is much smaller than those from the fuzzy set.

Fig. 8.
Fig. 8.

The fuzzy probability of first passage for a continuous no-rain forecast from the four centers with different no-rain sets, lead times, and cut sets.

Citation: Journal of Hydrometeorology 24, 6; 10.1175/JHM-D-22-0126.1

Fig. 9.
Fig. 9.

The reliability for a continuous no-rain forecast from the four centers with different no-rain sets, lead times, and cut sets.

Citation: Journal of Hydrometeorology 24, 6; 10.1175/JHM-D-22-0126.1

Fig. 10.
Fig. 10.

The fuzzy MTTF for a continuous no-rain forecast from the four centers within 7 days under different no-rain sets and cut sets.

Citation: Journal of Hydrometeorology 24, 6; 10.1175/JHM-D-22-0126.1

Fig. 11.
Fig. 11.

The cut set width of the four centers from different no-rain sets and lead times.

Citation: Journal of Hydrometeorology 24, 6; 10.1175/JHM-D-22-0126.1

According to the results from Figs. 5–11, both the range of the no-rain forecast and fuzziness play an important role in the fuzzy reliability assessment. When the fixed threshold is small and close to the peak of fuzzy probability, the fuzzy reliability of the no-rain forecast shows a significant improvement, which is equivalent to the cut set from a large fuzzy probability value. On the other hand, a narrow range of the no-rain forecast (i.e., small, fixed threshold or cut set of high fuzzy probability) may result in only very limited cases being considered, which is difficult to apply in practice. Therefore, a suitable range for a no-rain forecast is of vital importance in fuzzy reliability assessment of a no-rain forecast. Another factor discussed in this study is the influence of fuzzy mathematics in the reliability assessment of a no-rain forecast. The results from the traditional set are more conservative. The main reason for these results is that the fuzzy boundary of the no-rain set expands the range of the set, which makes the change of fuzzy probability of no rain from different forecast values more subdued, leading to a better performance in fuzzy reliability assessment from fixed thresholds and cut sets. Considering the limited impact of slight precipitation on flood formation, the fuzzy boundary of the no-rain set is acceptable and may be beneficial to flood resource utilization.

c. Sensitivity of membership functions in fuzzy reliability

Another main concern of this study is the sensitivity of the membership function for no-rain forecasts. In this paper, several different boundary values for Eq. (26) are employed as different membership functions for no-rain magnitude, and the values are shown in Table 3. The first case (i.e., a = 0.1 and b = 3) is used to examine the influence of the lower boundary, while the other cases are employed to test the impact from the upper boundary.

Table 3

The boundary values of membership functions for Eq. (26) in this study.
Table 3

First, the fuzzy probability of the no-rain forecast from different membership functions, centers, and lead times are calculated and analyzed. Since the main focus of this study is to assess the fuzzy reliability of the no-rain forecast, more attention should be paid to the forecast value that can be considered as no rain, i.e., the forecast value from 0 to 10 mm. The calculation results are shown in Fig. 12. From Fig. 12, it can be found that all the fuzzy probabilities of no rain from different membership functions are very close and have a similar trend, especially when the forecast values are lower than 1 mm. The results from Case 3 (i.e., the original case) are located between Case 2 and Case 4, which means the value of b can affect the calculation results of fuzzy probability for no rain, and a larger b leads to a greater fuzzy probability of no rain for the same forecast value. The phenomenon is also shown in the comparison between Case 3 and Case 1, indicating that the fuzzy probability of no rain will increase with a larger a under the same input. In addition, the similar performance of Case 1 and Case 2 implies that changing a or b may achieve the same effect in some cases.

Fig. 12.
Fig. 12.

The fuzzy probability of a no-rain set from different membership functions, centers, and lead times.

Citation: Journal of Hydrometeorology 24, 6; 10.1175/JHM-D-22-0126.1

Also, the fuzzy reliability of different membership functions based on thresholds is presented in Fig. 13. The results in Fig. 13 are consistent with the fuzzy probability in Fig. 12. The difference of fuzzy reliability from different cases is very small, which indicates that the membership function is not the dominant factor of fuzzy reliability assessment for the no-rain forecast. Meanwhile, the relationships between fuzzy reliability from different centers and lead times are stable under different cases. To be more specific, the results of fuzzy reliability for the no-rain forecast from CMA have the worst performance with different membership functions, while the results from UKMO are better in most cases and lead times. Therefore, the membership function is an impact factor for fuzzy reliability assessment, but its impact is limited and the comparison between precipitation from different centers is independent of it.

Fig. 13.
Fig. 13.

The fuzzy reliability for a continuous no-rain forecast from the four centers for different membership functions.

Citation: Journal of Hydrometeorology 24, 6; 10.1175/JHM-D-22-0126.1

d. Fuzzy reliability and excess water storage

From Eqs. (22) to (24), the amount of flood resource that can be overstored in the reservoir at the end of a flood event is determined by the length of the no-rain period. However, due to the uncertainty of the precipitation forecast, the future information of the no-rain period is not completely reliable. Therefore, the reservoir managers can only perform the reservoir regulation based on the results of reliability assessment and their own preferences. Here, a simplified example for excess water storage of Meishan Reservoir is shown in Fig. 14 based on the precipitation forecast from UKMO and membership function from Eq. (27). Also, the average inflow during the retreat stage of the 10-yr flood is used as the inflow of reservoir Qin, and the maximum benefit flow is employed to be the average utilization discharge during the no-rain period Qoutxl. As shown in Fig. 14, there is a contradiction between fuzzy reliability of no-rain forecast and excess water storage. To ensure the reliability of flood control, some benefits of flood resource utilization have to be given up. It is vital to strike a balance between the reliability of flood control and the benefits of flood resource utilization. The fuzzy reliability evaluation from Fig. 14 can provide useful information for reservoir managers, but it is still necessary to consider more factors, such as the current situation of the reservoir and the preferences of decision makers in practical applications.

Fig. 14.
Fig. 14.

The relationship between fuzzy reliability and available excess water storage for the Meishan Reservoir.

Citation: Journal of Hydrometeorology 24, 6; 10.1175/JHM-D-22-0126.1

e. Limitation and future work

In this paper, a new general method is proposed to assess the fuzzy reliability of precipitation forecast for reservoir regulation and flood resource utilization. Certainly, as with all scientific research, this study has several obvious limitations. First, only several specific fixed thresholds, cut sets and membership are employed in this study, and their impact on fuzzy reliability assessment is analyzed based on these specific cases. More efforts are still needed to determine the most appropriate model construction for the research in other areas or purposes. Second, the study area used in this paper is not a large basin. Since the spatial variability of precipitation increases with the increase in the basin area, the correlation between the areal average precipitation and the inflow of reservoir [i.e., Qin in Eqs. (22)–(24)] is relatively weak in large basins, which means the reliability assessment for the areal average precipitation may be meaningless. For example, when there is only strong precipitation near the reservoir, the inflow of the reservoir may be quite large, but the areal average precipitation may still be small due to the large basin area. However, it is also worth noting that the fuzzy reliability assessment for the multifunctional reservoirs with relatively small catchment area, which means low spatial variability of precipitation, is still meaningful due to their huge number and prominent contradictions between flood control and water supply, such as the Meishan Reservoir in this paper. In addition, the relationship between fuzzy reliability and available excess water storage for Meishan Reservoir has been briefly discussed in this paper, but a more comprehensive evaluation for the relationship is still necessary in the following studies. Finally, although DC-FLWL is beneficial to flood resource utilization, and some experiments have been taken in different reservoirs, the application of fuzzy reliability assessment for precipitation forecast in DC-FLWL should still be cautious due to the importance of flood control safety. Generally, the FFPM in this paper provides a new method to evaluate the fuzzy reliability of precipitation forecast and can be used as a reference in DC-FLWL and flood resource utilization. However, there is still a lot of work to be done before these methods can be applied to practical reservoir regulation.

6. Conclusions

The development of numeric precipitation forecasts provides a new way to make better use of flood resources and reservoir regulation. However, due to the uncertainty of the meteorological system, the error from the precipitation forecast may lead to flood control risk. Therefore, a reliability assessment of the precipitation forecast is necessary. The traditional indicators of reliability for precipitation forecast focus more on the evaluation of a single event or time step, rather than analyzing a set of continuous time steps. In this study, a reliability assessment model (FFPM) is proposed based on the first-passage problem and fuzzy mathematics. Since the no-rain forecast plays an important role in flood resource utilization, the no-rain forecasts in Meishan Reservoir are employed as an example for fuzzy reliability assessment.

Considering the influence of the range for the no-rain set and fuzzy mathematics, several ranges for the no-rain forecast are used in the fuzzy reliability assessment based on the fuzzy set and traditional set. The results show that the range of the no-rain forecast has a great impact on the fuzzy reliability assessment. A narrow range close to the peak of the fuzzy probability has a better fuzzy reliability, but its practical value is also limited. As a general method, the FFPM can be used for different traditional sets and fuzzy sets. The comparison between the fuzzy set and traditional set indicates that fuzzy mathematics can expand the range of the no-rain set and better describe the uncertainty in the set boundary. The sensitivity of membership functions is also analyzed in this study. The results show that the membership function is not the main influencing factor for fuzzy reliability of the no-rain forecast.

The fuzzy reliability assessment of a continuous no-rain forecast in this study provides a new reference for decision making in reservoir regulation and flood resource utilization, which is conducive to the application of precipitation forecast information, especially the no-rain forecast, in reservoir operation. However, the fuzzy reliability of the continuous no-rain forecast is not the only influencing factor in reservoir regulation; other factors, like reservoir conditions and decision-makers’ preferences should also be taken into consideration. Due to the importance of flood control in reservoir regulation, more efforts and experiments are still needed to find a better balance between the flood control risk and flood resource utilization benefit in reservoir regulation. In addition, the fuzzy reliability can also be used as an indicator to evaluate the performance of precipitation from different centers or magnitudes, and plays an important role in other areas, such as flood forecasts.

Acknowledgments.

This study benefited from the TIGGE dataset provided by European Centre for Medium-Range Weather Forecasts (ECMWF), Reading, United Kingdom. The authors are also grateful to the reviewers of the manuscript for their constructive comments and useful suggestions. The authors declare that they have no competing interests. This study is supported by the National Key Research and Development Program of China (Grant 2016YFC0400909), the China Scholarship Council (CSC), Applied Basic Public Research Program and Natural Science Foundation of Zhejiang Province (LGF22E090007), Soft Science and Technology Plan Project of Zhejiang Province (2022C35022), Science and Technology Plan of Department of Water Resources of Zhejiang Province (RB2020, RC2114), and President’s Science Foundation of Zhejiang Institute of Hydraulics and Estuary (ZIHE22Q014, ZIHE21Q005).

Data availability statement.

The TIGGE datasets can be downloaded at https://www.ecmwf.int/.

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  • Xu, B., P.-A. Zhong, R. C. Zambon, Y. Zhao, and W. W.-G. Yeh, 2015: Scenario tree reduction in stochastic programming with recourse for hydropower operations. Water Resour. Res., 51, 6359–6380, https://doi.org/10.1002/2014WR016828.

    • Search Google Scholar
    • Export Citation

  • Yiping, W., C. Cong, H. Gaofeng, and Z. Qiuxia, 2014: Landslide stability analysis based on random-fuzzy reliability: Taking Liangshuijing landslide as a case. Stochastic Environ. Res. Risk Assess., 28, 1723–1732, https://doi.org/10.1007/s00477-013-0831-x.

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    • Search Google Scholar
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    • Search Google Scholar
    • Export Citation

  • Zhou, Y., S. Guo, P. Liu, and C. Xu, 2014: Joint operation and dynamic control of flood limiting water levels for mixed cascade reservoir systems. J. Hydrol., 519, 248–257, https://doi.org/10.1016/j.jhydrol.2014.07.029.

    • Search Google Scholar
    • Export Citation

  • Zhu, F., P.-A. Zhong, Y. Sun, and W. W.-G. Yeh, 2017: Real-time optimal flood control decision making and risk propagation under multiple uncertainties. Water Resour. Res., 53, 10 635–10 654, https://doi.org/10.1002/2017WR021480.

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    • Export Citation

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    • Export Citation

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    • Search Google Scholar
    • Export Citation

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    • Export Citation

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    • Search Google Scholar
    • Export Citation

  • Zhu, F., P.-A. Zhong, Y. Sun, and W. W.-G. Yeh, 2017: Real-time optimal flood control decision making and risk propagation under multiple uncertainties. Water Resour. Res., 53, 10 635–10 654, https://doi.org/10.1002/2017WR021480.

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    • Export Citation
  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    An example of mixed CDF for r.v. Y.

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    The Meishan basin in the upper Huaihe River basin and the location of the reservoir, hydrological station, and precipitation stations. From the digital elevation model (DEM) data, the elevation of the basin gradually decreases from south to north.

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    The fuzzy probability of the fuzzy no-rain set from different centers and lead times in the Meishan basin during 2015–19.

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    The fuzzy probability of the traditional no-rain set from different centers and lead times in the Meishan basin during 2015–19.

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

    The fuzzy probability of first passage for a continuous no-rain forecast from the four centers with different no-rain sets, lead times, and fixed thresholds.

  • <sc>Fig</sc>. 6.
    View in gallery
    Fig. 6.

    The fuzzy reliability for a continuous no-rain forecast from the four centers with different no-rain sets, lead times, and fixed thresholds.

  • <sc>Fig</sc>. 7.
    View in gallery
    Fig. 7.

    The fuzzy MTTF for a continuous no-rain forecast from the four centers within 7 days under different no-rain sets and fixed thresholds.

  • <sc>Fig</sc>. 8.
    View in gallery
    Fig. 8.

    The fuzzy probability of first passage for a continuous no-rain forecast from the four centers with different no-rain sets, lead times, and cut sets.

  • <sc>Fig</sc>. 9.
    View in gallery
    Fig. 9.

    The reliability for a continuous no-rain forecast from the four centers with different no-rain sets, lead times, and cut sets.

  • <sc>Fig</sc>. 10.
    View in gallery
    Fig. 10.

    The fuzzy MTTF for a continuous no-rain forecast from the four centers within 7 days under different no-rain sets and cut sets.

  • <sc>Fig</sc>. 11.
    View in gallery
    Fig. 11.

    The cut set width of the four centers from different no-rain sets and lead times.

  • <sc>Fig</sc>. 12.
    View in gallery
    Fig. 12.

    The fuzzy probability of a no-rain set from different membership functions, centers, and lead times.

  • <sc>Fig</sc>. 13.
    View in gallery
    Fig. 13.

    The fuzzy reliability for a continuous no-rain forecast from the four centers for different membership functions.

  • <sc>Fig</sc>. 14.
    View in gallery
    Fig. 14.

    The relationship between fuzzy reliability and available excess water storage for the Meishan Reservoir.

Fig. 1.

An example of mixed CDF for r.v. Y.
Fig. 2.

The Meishan basin in the upper Huaihe River basin and the location of the reservoir, hydrological station, and precipitation stations. From the digital elevation model (DEM) data, the elevation of the basin gradually decreases from south to north.
Fig. 3.

The fuzzy probability of the fuzzy no-rain set from different centers and lead times in the Meishan basin during 2015–19.
Fig. 4.

The fuzzy probability of the traditional no-rain set from different centers and lead times in the Meishan basin during 2015–19.
Fig. 5.

The fuzzy probability of first passage for a continuous no-rain forecast from the four centers with different no-rain sets, lead times, and fixed thresholds.
Fig. 6.

The fuzzy reliability for a continuous no-rain forecast from the four centers with different no-rain sets, lead times, and fixed thresholds.
Fig. 7.

The fuzzy MTTF for a continuous no-rain forecast from the four centers within 7 days under different no-rain sets and fixed thresholds.
Fig. 8.

The fuzzy probability of first passage for a continuous no-rain forecast from the four centers with different no-rain sets, lead times, and cut sets.
Fig. 9.

The reliability for a continuous no-rain forecast from the four centers with different no-rain sets, lead times, and cut sets.
Fig. 10.

The fuzzy MTTF for a continuous no-rain forecast from the four centers within 7 days under different no-rain sets and cut sets.
Fig. 11.

The cut set width of the four centers from different no-rain sets and lead times.
Fig. 12.

The fuzzy probability of a no-rain set from different membership functions, centers, and lead times.
Fig. 13.

The fuzzy reliability for a continuous no-rain forecast from the four centers for different membership functions.
Fig. 14.

The relationship between fuzzy reliability and available excess water storage for the Meishan Reservoir.
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