Flood risk analysis for flood control and sediment transportation in sandy regions: A case study in the Loess Plateau, China

Accepted Manuscript Research papers Flood risk analysis for flood control and sediment transportation in sandy regions: a case study in the Loess Plateau, China Aijun Guo, Jianxia Chang, Yimin Wang, Qiang Huang, Shuai Zhou PII: DOI: Reference:

S0022-1694(18)30157-4 https://doi.org/10.1016/j.jhydrol.2018.02.076 HYDROL 22625

To appear in:

Journal of Hydrology

Received Date: Revised Date: Accepted Date:

25 September 2017 24 February 2018 27 February 2018

Please cite this article as: Guo, A., Chang, J., Wang, Y., Huang, Q., Zhou, S., Flood risk analysis for flood control and sediment transportation in sandy regions: a case study in the Loess Plateau, China, Journal of Hydrology (2018), doi: https://doi.org/10.1016/j.jhydrol.2018.02.076

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Flood risk analysis for flood control and sediment transportation in sandy regions: a case study in the Loess Plateau, China Aijun Guo1, Jianxia Chang*1, Yimin Wang1, Qiang Huang1, Shuai Zhou1 1

State Key Laboratory of Eco-hydraulics in Northwest Arid Region of China (Xi’an University of

Technology), Xi’an 710048, China *Corresponding author: E-mail: [email protected]

ABSTRACT Traditional flood risk analysis focuses on the probability of flood events exceeding the design flood of downstream hydraulic structures while neglecting the influence of sedimentation in river channels on regional flood control systems. This work advances traditional flood risk analysis by proposing a univariate and copula-based bivariate hydrological risk framework which incorporates both flood control and sediment transport. In developing the framework, the conditional probabilities of different flood events under various extreme precipitation scenarios are estimated by exploiting the copula-based model. Moreover, a Monte Carlo-based algorithm is designed to quantify the sampling uncertainty associated with univariate and bivariate hydrological risk analyses. Two catchments located on the Loess plateau are selected as study regions: the upper catchments of the Xianyang and Huaxian stations (denoted as UCX and UCH, respectively). The univariate and bivariate return periods, risk and reliability in the context of uncertainty for the purposes of flood control and sediment transport are assessed for the study regions. The results indicate that sedimentation 64

triggers higher risks of damaging the safety of local flood control systems compared with the event that AMF exceeds the design flood of downstream hydraulic structures in the UCX and UCH. Moreover, there is considerable sampling uncertainty affecting the univariate and bivariate hydrologic risk evaluation, which greatly challenges measures of future flood mitigation. In addition, results also confirm that the developed framework can estimate conditional probabilities associated with different flood events under various extreme precipitation scenarios aiming for flood control and sediment transport. The proposed hydrological risk framework offers a promising technical reference for flood risk analysis in sandy regions worldwide.

Keywords: Flood risk analysis; Flood control and sediment transport; Copula; Uncertainty; Loess Plateau

1. Introduction Examining flood risk is extremely beneficial and indispensable for hydraulic engineering design and flood mitigation. This topic has received enormous worldwide attention to date, whether from a univariate or multivariate perspective (Rossi et al., 1984; Burn,1990; Webb and Betancourt, 1992; Brath et al., 2006; Grimaldi and Serinaldi, 2006; Shankman et al., 2006; Wood and Lettenmaier, 2006; Zhang and Singh, 2006; Kay et al., 2009; Reddy and Ganguli, 2012; Viglione et al., 2013; Fu and Butler, 2014; Sraj et al., 2015; Ozga-Zielinski et al., 2016). However, previous studies 65

focused predominantly on the hazards associated with floods, i.e., the phenomenon that higher flood magnitude results in larger flood losses, e.g., economic loss, environmental loss (Barredo, 2009; Zaman et al., 2012; López and Francés, 2013; Jongman et al., 2014; McAneney et al., 2017). Therefore, the probability/risk of flood discharge/volume exceeding a threshold (i.e., the design flood of downstream flood control works) is the provided information for flood control departments (Volpi and Fiori, 2014; Vittal et al., 2015; Li et al., 2016). However, it is worthy of note that there are many non-negligible practical benefits of floods, such as enhancing ecosystem health, replenishing reservoirs, recharging ground water, or transporting sediment in river channels (Eaton and Lapointe, 2001; Ward et al., 2001; Thomaz et al.,2007; Richter and Thomas,2007; Dahan et al., 2008; Rosenberg et al., 2011). In this context, higher flood discharge/volume signifies more benefits, and vice versa (VanRheenen et al., 2004; Payne et al., 2004; Di Baldassarre et al., 2017; Harden and O’Connor, 2017). Therefore, knowledge of the probability of floods non-exceeding certain thresholds is equally crucial for water resource managers and policy-makers, particularly on sediment transport issues. However, research on this issue has not been conducted to date, and thus performing univariate and bivariate flood risk analyses of the hazards and benefits associated with floods has motivated our work. In this study, our interest is the impact of floodwater on sediment transport.

66

Water discharge and sediment transport are two primary elements that maintain the healthy development of river channels, particularly in sandy regions (Song et al., 2010; Tian et al., 2016). During low-to-mean discharge stages, sediment accumulates in the watercourses; during floods, these accumulated sediments are resuspended and transported downstream (Ogston et al., 2000; Ciszewski, 2001; Lenzi et al., 2006). Therefore, effective flow discharge during floods is crucial for transporting these sediments and maintaining the balance between sediment load and water volume. Otherwise, low flow discharge during floods induces sediment accumulation, which triggers channel deposition and subsequently results in the decline in channel gradient and the shrinkage of the river channel. Consequently, the sediment transport capacity of the river channel reduces, and thus seriously impairs the flood regulation and affects regional flood control security (Xu, 2002; Wang et al., 2005; He et al., 2015). In light of the above, investigating the probability of floods less than certain thresholds (i.e., the appropriate instream flow requirements for sediment transport) is of great importance to maintain the vigor of river ecosystems and the safety of regional flood control systems.

Additionally, another noteworthy point in flood risk analysis in practical applications is the uncertainty caused by the small size of hydrological samples (i.e. the limited size of hydrological records) (Saad et al., 2014; Fan et al., 2016; Qi et al., 2016; Serinaldi, 2016). This uncertainty source is frequently overlooked in the univariate and 67

multivariate frequency analyses due to the difficulties of estimation and interpretation. Salvadori et al. (2011), Serinaldi (2013) and Serinaldi and Kilsby (2015) highlighted that this uncertainty should be seriously considered to avoid misinterpretation of bivariate return periods and incoherent conclusions when assessing the risk of extreme hydrometeorological events. Michailidi and Bacchi (2017) stated that flood risk evaluation without accounting for this uncertainty is deceiving. In spite of this, however, this issue has not yet received significant attention (Serinaldi, 2013, 2015; Dung et al., 2015; Serinaldi and Kilsby, 2015; Zhang et al., 2015). Therefore, in this study, the sampling uncertainty in bivariate risk evaluation is assessed by using a Monte Carlo-based algorithm.

Given the above concerns, a framework for univariate and bivariate flood risk estimation under uncertainty is developed for flood control and sediment transport in sandy regions. First, the univariate risk of floods is calculated by considering both the damages and benefits of floods. In rain-dominant watersheds, floods are generally caused by extreme precipitation events (Milley et al., 2008; Blöschl et al., 2005; Machado et al., 2015; Huang et al., 2016). Additionally, in practical terms, reducing the flood risk also requires information on extreme precipitation events (Kiem and Verdon-Kidd, 2013; Madsen et al., 2014; Liu et al., 2017). Therefore, the second part of this study examines the bivariate risk of floods and extreme precipitation events for local flood control and the sediment transport of river channels. Here, the commonly 68

used copula functions are employed (Salvadori 2004; Salvadori et al., 2010; Zhang et al., 2011; Zhang et al., 2013; Fu and Butler, 2014; Ozga-Zielinski et al., 2016; Fan, et al., 2016). Finally, the conditional probabilities of occurring different flood discharge magnitudes under given various extreme precipitation scenarios are estimated according to the joint dependence structure between floods and extreme precipitation events, which is beneficial for practical flood control and sediment transport.

The Loess Plateau (LP), located in the middle Yellow River basin of China, is one of the most severely eroded areas in the world (Zhang and Liu, 2005) and is known as the “cradle of Chinese civilization”. To date, the LP has received much attention from the government and international organizations (Wei et al., 2006; Cong et al., 2009; Zheng et al., 2011; Zhao et al., 2013; Liang et al., 2015). Annual average soil erosion on the LP reaches approximately 2000-2500 t/km2, and more than 60% of the area suffers severe soil loss (Zhang et al., 2008; Zhao et al., 2013). Severe soil erosion results in the transport of vast amounts of sediment, up to approximately 1.6 billion t/year, into the Yellow River, which gives this river the highest sediment concentration in the world (Li et al., 2009). Most of the sediment is deposited on the bottom of the river channel when the river flows downstream (He et al., 2015). The severe sedimentation in the lower Yellow River makes the bed continuously rise and poses a great challenge to flood control (Xu et al., 2002; Peng et al., 2010; Wang et al., 2016). Moreover, the sediment deposition is much higher during the flood season (June to September) than 69

that in other months (Fang et al., 2008). Therefore, to decrease the stagnated and accumulated sediment in the channel, a certain amount of instream flow during the flood season is extremely crucial. Investigating the probability of floods that do not exceed certain flood discharge magnitudes can provide valuable information for sediment transport in river channels. However, most previous studies have primarily focused on the frequency of low flows and drought events because of the arid and semi-arid continental monsoon climate on the LP (Ma et al., 2013; Du et al., 2015; Huang et al., 2014). Studies investigating flood risk remain few, and even fewer take sediment transport into account.

Therefore, the main objective of this work is to examine the univariate and bivariate risk of flood/and extreme precipitation events in catchments of the LP. The primary novelty of this study is (1) the design of a flood risk analysis framework for flood control and sedimentation management and the derivation of the conditional probability of different flood discharge magnitudes under various extreme precipitation scenarios and (2) the identification of the associated uncertainty in univariate and bivariate flood risk evaluation by a Monte Carlo-based algorithm to provide more robust insights for practical flood control and sedimentation management. Although catchments of the LP are selected as study regions, methods adopted in this study have no restrictions that would prevent them from being applied to other sandy regions.

70

The remainder of the paper is constructed as follows. Section 2 states the problem in the catchments of the LP. Section 3 describes the data collection. Section 4 introduces the methods adopted in this study. The results and discussion are presented in Section 5. Section 6 summarizes the main conclusions drawn from the study.

2. Statement of the problem Our study area is the Weihe River basin, situated between 104-107°E and 33–34°N (Figure 1). The basin has a typical continental climate, and lies in the semi-humid and semi-arid transitional zone in the southern part of the LP (Peng et al., 2015). The Weihe River (hereafter WR) provides the water supply for 9300 km2 of fertile fields in the Guanzhong Plain and more than 61% of the Shaanxi Province’s population (Guo et al., 2017). Additionally, the start-point of the well-known Silk Road Economic Belt, Xi’an City, is situated in this basin. -------------------------------------

Insert Figure 1 here -------------------------------------

The WR flows into the Yellow River with hyper-concentrated sediment at Tongguan station (Figure 1), where the Yellow River makes an approximately 90° turn and flows east. The WR basin is one of the most serious soil loss areas on the LP. Areas suffering from severe soil loss cover approximately 65% of the total land area of this basin (Song et al., 2010). Serious soil loss has caused severe sediment deposition in the lower reach 71

of the WR, which poses great challenges for local flood control (Li et al., 2006; Li et al., 2013).

Sedimentation in the Sanmenxia Reservoir (shown in Figure 1), 113.5 km away from Tongguan station, has enhanced the bed elevation at Tongguan (Wu et al., 2004; Wang et al., 2007). During 1960-1969, the Tongguan elevation abruptly rose from 323 m to 328.5 m (Long and Chien, 1986; Long, 1996). The rising Tongguan elevation triggered retrogressive siltation waves in the WR and prevented sediment transport from the WR into the Yellow River. Continuous sediment deposition thus occurred in the lower WR because of the rising Tongguan elevation.

The coupled effects of serious soil loss in the WR basin and increasing Tongguan elevation has caused shrinkage of the WR channel and a decline in its flood discharge capability. As a result, the flood risk has increased, and the capacity of levees to prevent floods in the lower WR has been reduced from a 50-year flood frequency to the current 10-year frequency (Song et al. 2010). Regional industrial and agricultural economic productivity and, more importantly, Xi’an, the capital city of Shanxi province in the lower reach of the WR, are gravely threatened.

72

In light of the above changes, flood control problems in the WR basin are twofold: one is that flood peak discharge exceeds the design flood of the levees downstream, and the other is sedimentation from severe soil loss in the upper WR and retrogressive siltation waves induced by the increasing Tongguan elevation that prevent sediment transport, which increases flood risks. Given the above information, the aims of flood control operations in the WR basin are to ensure the safety of levees downstream and people living near the WR, and promote sediment transport of the WR.

3. Dataset The backwater region of the Sanmenxia Reservoir extends to the Xianyang hydrological station (Wang and Hu, 2009). Moreover, the upper catchment of Xianyang station is a flood-prone region in the WR basin. Huaxian station is the control station of the WR. Therefore, the upper catchments of Xianyang and Huaxian stations are selected as study regions in the current work (shown in Figure 1); they are abbreviated as UCX and UCH, respectively.

Annual maximum flood records (denoted as AMF) for 1960-2014 from the Xianyang and Huaxian stations are used. The locations of the two stations are displayed in Figure 1. The data were provided by the hydrology bureau of the Yellow River Conservancy Commission, which takes the main responsibility of monitoring, collecting and disposing hydrological information in the Yellow River basin. The data quality was 73

strictly controlled before being released according to the Specification for complication and publication of hydrological yearbook (SL 460- 2009) and Measures of the Yellow River

for

the

hydrological

management

(http://www.yellowriver.gov.cn/zwzc/zcfg/gz/200612/t20061222_75846.html).

Daily precipitation data for 1960-2014 are provided by the China Meteorological Data Sharing Service System (http://cdc.cma.gov.cn). The flood discharge is closely linked to the accumulated rainfall amounts before the occurrence of annual peaks (Teegavarapu, 2012), and the extreme precipitation event Pr used in this work is thus defined as: l n

Pr   Raini

(1)

i l

where Pr denotes the accumulated rainfall from the 1 st to the i-th day, l is the occurrence time of peak discharge Q , n  n  0,1, 2,3, 4  indicates the lag time (i.e., the time from peak discharge to the beginning of rainfall), and Raini means the i-th day of rainfall. Pr1, Pr2, Pr3, Pr4 and Pr5 represent the accumulated 1-, 2-, 3-, 4- and 5-day consecutive rainfall amounts (i.e., n=0, n=1, n=2, n=3, and n=4), respectively. The Thiessen polygon method is applied to compute the areal accumulated rainfall.

To select the extreme precipitation events most closely correlated to AMF, the correlation coefficients Spearman’s rho and Kendall’s tau are computed (

74

Table 1). The Spearman’s rho and Kendall’s tau are rank-based coefficients, which are

robust to departures from normality. From Table 1, it can be found that Pr2 and Pr3 are most closely correlated with AMF gauged at Xianyang and Huaxian stations, respectively. -------------------------------------

Insert Table 1 here -------------------------------------

4. Methodologies As mentioned in the Introduction, the aim of this paper is to develop a univariate and copula-based bivariate hydrological risk framework focusing on flood control and sediment transport for sandy regions. Figure 2 presents the schematic diagram of this developed framework. Following Figure 2, Sections 4.1 and 4.2 illustrate the univariate and bivariate flood risk analyses. Section 4.3 exhibits the determination of conditional probability of flood peak discharge under given extreme precipitation scenarios. Approach on uncertainty estimation for the flood risk evaluation and conditional probability determination is described in Section 4.4. 75

-------------------------------------

Insert Figure 2 here -------------------------------------

4.1. Univariate flood risk analysis 4.1.1. Marginal distribution selection The generalized extreme value distribution (Gev), Pearson type III distribution (P3), lognormal distribution (Logn), normal distribution (Norm) and gamma distribution (Gam) are adopted to fit the AMF and Pr series. These parametric distributions are popular in modeling the marginal distributions of extreme hydrological events due to their better performance (Zhang et al., 2007; Benkhaled et al., 2014; Fan et al., 2016; Kamal, et al., 2017). The maximum likelihood (ML) method is used to estimate the parameters

of

marginal

distributions.

The

goodness-of-fit

test

(the

Kolmogorov-Smirnov (K-S) approach) is employed to evaluate the validity of these distribution models (Zhang and Singh, 2012). The K-S statistic aims to quantify the largest vertical difference between the empirical and estimated distributions (Massey, 1951), which is defined as: K -S  sup F  x   Fe  x 

(2)

x

where sup denotes the supremum, and F  x  and Fe  x  indicate the estimated and empirical distributions, respectively. Here, the empirical distribution is estimated by the Gringorten plotting-position formula (Gringorten, 1963). The p-value of the K-S statistic was estimated by the Monte Carlo simulation, for details see (Lilliefors, 1969).

76

The R packages available from (Viglione, et al., 2013; Novack-Gottshall and Wang, 2016) were used for the p-value estimation.

Then, the corrected Akaike information criterion (AICc) is used to select the most appropriate model from among the candidate distribution models, which is much stricter than the classical AIC, particularly when the sample size is limited (Burnham and Anderson, 2004). The AICc is defined by: AICc  2 ln  maximized loglikelihood   2Pa 

2 Pa  Pa  1 n  Pa  1

(3)

where Pa is the number of estimated parameters and n is the length of observations.

The Bayes Information Criterion (BIC) (Schwarz, 1978) is also used to select the appropriate model, which penalizes the log-likelihood function by adding the number of parameters multiplied by the logarithm of the sample size n. The specific formula is expressed as: BIC  2 ln  maximized loglikelihood   ln  n  Pa

(4)

The best fitting model is selected as having the minimum AICc and BIC values.

4.1.2. Univariate return periods and risk In the current work, the univariate flood risk is first derived based on the return period (also called recurrence intervals) of the annual maximum flood discharge. Based on the 77

flood control problems in the WR basin described in Section 2, the return period is defined as (1) the average interval in years between AMF events that exceeds a certain magnitude amf1 and (2) the average interval in years between AMF events that does not exceed a certain magnitude amf2. amf1 and amf2 denote the design flood (hereafter, DF) of levee downstream and instream flow requirements for sediment transport (hereafter, IFRST) in the lower WR, respectively. Generally, scouring the sediment in the river channel is implemented under the premise of ensuring the safety of downstream levees. Therefore, IFRST should be less than DF.

The two types of return periods are associated with the probabilities of AMF exceeding DF and AMF not exceeding IFRST (shown in Figure 3 (a-b)). AMF  DF and AMF  IFRST events are defined as dangerous events for the safety of a local flood

control system; otherwise, AMF  DF and AMF  IFRST events are conductive. From the perspective of a local flood control system, the IFRST
Insert Figure 3 here -------------------------------------

For the sake of simplicity, the conditions of AMF exceeding the DF event, AMF not exceeding the IFRST event, and

 AMF  IFRST or AMF  DF

1, 2 and 3, respectively. 78

are denoted as Case

The random variable X represents the year of the first occurrence of the previously-mentioned events. Under the assumption of independence and stationarity, probabilities of occurrence of the Case 1, Case 2 and Case 3 events for the first time in year X  x (x=1, 2, …, ∞) are expressed as follows (Mood et al. 1974; Salas and Obeysekera, 2014; Du et al., 2015): Case 1:

f  x   P( X  x)  1  F  AMF  DF

x 1

F  AMF  DF = 1  p1 

x 1

p1

(5)

where f  x  is the geometric probability law; p1  F  AMF  DF and F (▪) is the cumulative distribution function (denoted as CDF) of the annual flood discharge. Case 2:

f  x   P( X  x)  1  F  AMF  IFRST 

x 1

F  AMF  IFRST   1  p2 

x 1

p2 (6)

where p2  F  AMF  IFRST  . Case 3:

f  x   P( X  x)  1  F  AMF  IFRST or AMF  DF    1  p1  p2 

x 1

x 1

 F  AMF  IFRST or AMF  DF 

 p1  p2 

Therefore, the corresponding return period, i.e., the expected value X, is:

79

(7)

1   p1  1 T  E ( X )   xf  x    x 1  p2  1   p1  p2

Case 1

8-1

Case 2

8-2 

Case 3

8-3 

Similarly, the variance of X, the expected value of the squared deviation from E(X), is expressed as:

2 2 Var ( X )  E  X  E  X     E  X 2   E  X   

1  p 1  2 p  1   1  p2  2   p2  1  p  p 1 2  2   p1  p2 

Case 1 Case 2

(9)

Case 3

The coefficient of variation of X, a standardized measure of the relative change of E(X), is defined as:

Cv( X ) 

Var  X  EX 

 1  p1    1  p2   1  p1  p2

Case 1 Case 2

(10)

Case 3

In practical flood management, risk is the probability of occurrence of an exceedance flood event, dangerous or undesirable event over a given project life of n years (Salas 80

and Obeysekera, 2014; Read and Vogel, 2015; Du et al., 2015; Fan et al., 2016). Therefore, the hydrological risk can also be expressed as:

n

Risk   x 1

 n x 1 n Case 1  p1  1  p1   1  1  p1  x 1   n x 1 n f  x    p2  1  p2   1  1  p2  Case 2  x 1 n  x 1 n  p1  p2   1  p1  p2   1  1  p1  p2  Case 3 x 1 

(11)

Note that the risk of failure can be computed from the return period T, i.e., n

 1 Risk  1  1   .  T

The reliability, which signifies the probability that a dangerous event will not occur within a project life of n years, i.e., that a system will remain in a satisfactory case within its lifetime, is defined as:

1  p1 n  n  Reliability  1  Risk  1  p2   n 1  p1  p2 

Case 1 Case 2

(12)

Case 3

Risk (Eq. 11) and Reliability (Eq. 12) are derived for Case 1-Case 3 and can be applied to other regions worldwide with similar flood control problems.

4.2. Bivariate flood risk analysis

81

4.2.1. Copula function construction The copula, as introduced by Sklar (1959) is a powerful tool for modeling the dependence structures of individual variables. The main advantage of copulas is that they allow modelling the joint dependence structure without any restrictions on marginal distributions. Therefore, the copula function has been extensively used in the hydrology field (Zhang and Singh, 2006; Karmakar and Simonovic, 2009; Reddy and Ganguli, 2012; Zhang et al., 2012; Fan et al., 2016; Fan et al., 2017; Xu et al., 2017). A d-dimensional copula is defined as a multivariate distribution function F [0,1]d → [0,1], linking standard uniform marginal distributions. Formally, the copula can be divided into two components: individual univariate distributions, and a copula function describing dependence structures between variables based on the copula and its parameter(s).

According to Sklar's theorem, one d-dimensional multivariate distribution function F for random variables X1, X2, …, Xd with marginal distribution of F1, F2, …, Fd can be expressed as F  X1 ,

, X d   C  F1  X1  ,

,Fd  X d   

(13)

where  is the copula parameter vector and Fd  X d   F  X  x  is the marginal distribution of X d .

82

Tail dependence is another important property of a copula (Embrechts, 2003), which can be used to measure the dependence among extremal random events. For the sake of simplicity, the theoretical details of the coefficient of tail dependence are referred to Aas (2004).

In the current work, the ML method is used to estimate the copula parameter(s). The Cramér–von Mises statistic (Cramér, 1928; von Mises, 1931) is used for goodness-of-fit testing.

In the field of hydrology, the elliptical copulas (Student t copula, Gaussian copula, etc.) and Archimedean copulas (Clayton copula, Frank copula, Gumbel copula, etc.) are two most frequently used copula families (Genest et al., 2007; Kao and Govindaraju, 2008; Ma et al., 2013; Sraj, et al., 2015; Vernieuwe, et al., 2015; Xiong et al., 2015; Chang et al., 2016). In this study, we select the Student t copula (Appendix A) to model the joint dependence structure between AMF and Pr. There are several reasons behind this selection. First, compared to the bivariate Student distribution, marginals of Student t copula are not restricted to the Student t distribution with the same degrees of freedom (Rodriguez, 2007). Additionally, the Student t copula can capture the dependence between variables in the left and right tails without giving up flexibility to describe dependence in the center; by contrast, other popular copulas capture one pattern of tail dependence (such as Clayton copula, Gumbel copula) or zero 83

tail dependence (such as Gaussian copula, Frank copula). However, present study focuses on the dependence structure between pairs of AMF and Pr values in the left tail (i.e. the lower-left quadrant) and right tail (i.e. upper-right quadrant), as shown in Figure 4. Given the above, the Student t copula is employed to characterize the joint behavior of AMF and Pr values in this study. Readers interested in more theoretical details of the Student t copula are referred to the papers by Cambanis et al. (1981) and Fang et al. (2002).

4.2.2. Joint return periods and risk To conduct a bivariate risk analysis, the commonly-used joint probability behaviors focus on the so-called “OR”, “AND” and “COND” events (Sraj, et al., 2015; Zhang et al., 2015; Serinaldi, 2016). Figure 4 displays the joint probabilities of “OR”, “AND” and “COND” operators for Case 1 and Case 2. The corresponding joint probability behaviors for Case 1 (shown in the upper panel of Figure 4) are expressed as: (1): (AMF>DF) OR (Pr> pr), (2): (AMF>DF) AND (Pr> pr), and (3) (AMF>DF) |(Pr> pr).

The joint probability behaviors for Case 2 (shown in the lower panel of Figure 4) are: (1): (AMF≤IFRST) OR (Pr≤ pr), (2): (AMF≤IFRST) AND (Pr≤ pr), and (3) (AMF≤IFRST) |(Pr≤ pr) ------------------------------------84

Insert Figure 4 here -------------------------------------

From the perspective of a local flood control system, there are various combinations of flood and precipitation events for the bivariate case. Additionally, once the probabilities of the above joint probability behaviors for Case 1 and Case 2 are calculated, that for Case 3 can be derived by the addition and subtraction formula of probability and the generalized total probability formula. Therefore, we restrict our attention to the bivariate joint probability behaviors for Case 1 and Case 2 in the current work for the sake of simplicity.

The joint return periods for Case 1, also called primary return periods, can be expressed as: TOR 

1 1  P  AMF  DF  OR  Pr  pr  1  C  FAMF  DF  ,FPr  pr  

TAND 

1 P  AMF  DF  AND  Pr  pr 

1 = 1  FAMF  DF   FPr  pr   C  FAMF  DF  ,FPr  pr   TCOND 

1

P

 AMF  DF  Pr  pr 



(14)

(15)

1 P  AMF  DF  , Pr  pr  P  Pr  pr 

1 = 1  FAMF  DF   FPr  pr   C  FAMF  DF  ,FPr  pr   1  FPr  pr 

85

(16)

Substituting Eq. (15) into Eq. (16), it can be found that Eq. (16) can also be written as

TCOND  1  FPr  pr    TAND   TAND .

The primary return periods for the Case 2 are defined as: TOR 

1 P  AMF  IFRST  OR  Pr  pr 

1  FAMF  IFRST   FPr  pr   C  FAMF  IFRST  ,FPr  pr   TAND 

1 1  P  AMF  IFRST  AND  Pr  pr  C  FAMF  IFRST  ,FPr  pr  

TCOND 

1

P

 AMF  IFRST   Pr  pr 



(17)

(18)

1 P  AMF  IFRST  , Pr  pr  P  Pr  pr 

1 = C  FAMF  IFRST  ,FPr  pr  

(19)

FPr  pr 

Substituting Eq. (18) into Eq. (19), it can be found that Eq. (19) can be expressed as

TCOND   FPr  pr   TAND   TAND .

Considering the above, several inequalities between TOR , TAND and TCOND exist for any given copula:

TOR  TAND

(20)

TCOND  TAND

(21)

86

The corresponding Risk and Reliability associated with the return period of joint probability behaviors can be expressed as:  1 Risk  1  1    T

n

 1 Reliability  1    T

(22) n

(23)

4.3. Conditional statistics of AMF under given precipitation scenarios In practical terms, water resources managers want to identify how possible it is that AMF exceeds or does not exceed certain thresholds under given extreme precipitation events. Therefore, we examine the conditional probability of AMF under various extreme precipitation scenarios.

The conditional exceedance probability of AMF > DF given Pr=pr is defined as: FAMF DF Pr  pr  1 

C  FAMF  DF  ,FPr  pr   FPr  pr 

 1  FAMF DF Pr  pr

(24)

The conditional unexceedance probability of AMF ≤ IFRST given Pr=pr is expressed as: FAMF IFRST Pr  pr 

C  FAMF  IFRST  ,FPr  pr   FPr  pr 

87

 FAMF  IFRST Pr  pr

(25)

According to the formula of holohedral probability and Bayes formula, the conditional probability that IFRST  AMF  DF given Pr=pr is denoted as: FIFRST AMF DF Pr  pr  FAMF DF Pr  pr  FAMF IFRST Pr  pr

(26)

Given Pr=pr, Fmin DF,IFRST AMF  max DF, IFRST Pr pr denotes the probability of a flood peak discharge less than the design flood of levee downstream and higher than the instream flow requirement for sediment transport.

4.4. Uncertainty estimation for joint probability behaviors To determine the impact of sampling uncertainty on the evaluation of risk of bivariate extreme hydrometerological events and conditional probabilities of AMF given precipitation scenarios, the following Monte Carlo-based procedures (motivated by Serinaldi (2016)) are designed: 1. Estimate the vector of parameters  (    x , y ,  ) of the joint distribution



H  x, y;   C F  x; x  ,F  y; y  ;



simultaneously using the ML method

(given in Appendix B); 2.

Simulate

B

Z *   X * ,Y *    xij , yij  ,

bivariate

 i  1,

samples

,n; j  1,

of

size

n

as

,B  from the fitted bivariate model

H  x, y;  . n is equal to the length of the observed sample. B is set to 9999 in this

study;

88

3. Estimate the vector of parameters  j for the simulated sample Z *j using the maximum likelihood method; 4. Compute the primary ( TOR , TAND and TCOND ) return periods for different

 x, y 

pairs. Moreover, compute the conditional probabilities of x under given different y scenarios; 5. Estimate confidence intervals for TOR , TAND , TCOND and conditional probabilities at the 95% confidence level.

5. Results and discussion 5.1. Marginal distribution selection To test the randomness of variables, the autocorrelations for AMF and Pr series are computed and shown in Figure S1 (supplemental material). It can be inferred from Figure S1 that these autocorrelations cannot pass the significance test at the 5% significance level. Therefore, we accept the hypothesis that AMF and Pr in the two study regions are random variables.

Figures S2 and S3 illustrate the fitted marginal distributions by the Gev, P3, Logn, Norm and Gam distributions for the AMF and Pr series in the UCX and UCH, respectively. The two figures both display relatively high degree of similarity among the five curves in the two regions. The estimated parameters for these candidate distributions are displayed in Table S1 (supplemental material). Given the similarity 89

among the five distributions, we performed the K–S goodness-of-fit test and estimated the AICc and BIC to select the most appropriate distributions to model the AMF and Pr series. The corresponding results exhibited in Table S2 demonstrate that these candidate distributions are suitable for fitting the distribution of AMF and Pr series. With the assist of AICc and BIC indicators, the Gam distribution is selected as the most appropriate one to model the AMF and Pr series in the two regions.

5.2. Copula function construction Once the marginal distribution is chosen, the next step is to estimate the copula parameters. The estimated parameters of the Student t copula are listed in Table . The goodness-of-fit statistics Sn of the Cramér–von Mises criterion and its associated p-value based on N=9999 parametric bootstrap samples are also given in Table . Large value of Sn indicates the large distance between the estimated and the empirical copulas. p-value>0.05 means that the estimated copula can be accepted at the 5% significance level. Results displayed in Table illustrate that the Student t copula can be applied to model the dependence structure between AMF and Pr series in each region at 5% significance level. -------------------------------------

Insert Table 2 here -------------------------------------

5.3. Flood risk analysis for flood control and sediment transport 90

5.3.1. Univariate flood risk analysis Figure 5 displays the univariate flood return period and its associated confidence interval (denoted as CI) for Case 1 and Case 2. In the current work, the sampling uncertainty related to the limited size of hydrological records is discovered by the Monte Carlo sampling method (i.e., the parametric bootstrapping method). The AMF observations are resampled 9, 999 times, and for each resample, the Gamma distribution is fitted (thin gray lines in Figure 5). In this study, each simulation is generated from a random seed, and the figure thus displays the effects of sampling variability on the simulated AMF series. To select the percentiles, the 9,999 quantile estimates are ranked. The 250th and 9750th values are used to depict the 2.5 and 97.5 points of the distribution, i.e., the 95% CI (red dotted lines in Figure 5). One can assess the risk of occurrence of different flood magnitudes for both flood control and sediment transport purposes from Figure 5. -------------------------------------

Insert Figure 5 here -------------------------------------

The width of CI for flood return periods at the 95% confidence level is used to measure the uncertainty (shown in Figure 5). Wider CI denotes larger uncertainty. The results highlight that large uncertainties in univariate flood return periods exist for both Case 1 and Case 2. For a given return period, the 95% CI of DF (involved in Case 1) and

91

IFRST (involved in Case 2) is wide. Specifically, CI is wide for high flows and narrow for low flows in Case 1, whereas it is the opposite in Case 2.

For Case 1, for a univariate return period of 15 years in the UCX, the DF can span from 3873 to 5941 m3/s (34.81% difference); for 20 years, it ranges from 4165 to 6450 m3/s (35.43% difference). As for Case 2 in the UCX, the IFRST for a univariate return period of 15 years at the 95% confidence interval spans from 287 to 695 m3/s (58.71% difference), and for 20 years, it spans from 234 to 613 m3/s (61.83% difference). From the above, it can be found that univariate flood frequency analyses for flood control and sediment transport both are plagued with considerable uncertainty. The large uncertainty makes it difficult for the management departments to determine the scientific design flood for hydraulic engineering and flood control operation of reservoirs. Therefore, reducing the uncertainty of flood return period estimation is essential for flood control in practice, particularly the sampling uncertainty. However, it is worthy of note that, although the length of hydrological records used in this study is comparatively long (55 years) for a hydrological context, it is still not sufficient to constrain the uncertainty to tight limits. Therefore, as Merz and Blöschl (2008) and Dung et al (2015) suggested, increasing the information content by temporal, spatial or causal data expansion, such as the historical and paleo-flood data, is necessary and beneficial for responding such high uncertainty and obtain a more reliable estimation of flood return period. As for selecting the suitable DF for the hydraulic engineering 92

(e.g. levee, dam) and determining appropriate IFRST for sediment transport in the context of high uncertainty, as Hu et al. (2013) suggested, management departments can employ the point estimate of flood design value and add one modification value for it. The modification value should be determined with the assist of CI of flood return periods. Apparently, the larger the modification value is, the lower risk of flood levee overflowing and the higher construction cost of the lee is, otherwise, the reverse. Specifically, the importance of flood control objects is another one important reference factor when determining the modification value.

Zhang et al. (2014) discovered that the Tongguan elevation decreases when flood peak discharge observed at Huaxian station is more than 2000 m3/s and that the decrease degree is dramatic when flood peak discharge is more than 3000 m3/s. Li et al. (2010) highlighted that flood peak discharge more than 2500 m3/s observed at Huaxian station is significantly negatively correlated with the Tongguan elevation; flood peak discharge less than 2500 m3/s is insufficient to decrease the Tongguan elevation. Therefore, 2500 m3/s is treated in the current work as an effective IFRST in the WR basin. Local flood control departments have rechecked the flood control ability of hydraulic structures in the lower WR and suggested that the gauged flood discharge at Huaxian station should be less than 5760 m3/s for the safety of the levee in the WR basin.

93

Table shows the risk and reliability of occurrence of different flood events for flood control and sediment transport in the WR basin. The hydraulic project life is given as 10 years, i.e., n=10 for the Case 1 (Eq. (11)). With respect to sediment transport, water-sediment regulation by reservoirs is generally considered an effective measure to solve the sedimentation problem in the Yellow River (Wang et al., 2006; Li and Sheng, 2011; Miao et al., 2016). Wu et al. (2016) suggested that the water-sediment regulation can be performed every 3 years, at least with the premise of meeting the water demands of agriculture, industry and daily life. Therefore, we define risk as the probability of occurrence of flood peak discharge less than 2,500 m3/s over 3 years, i.e., n=3 for the Case 2 (Eq. (11)).

Take the UCH for example, it can be seen from Table

that the probability of

occurrence of the (AMF>5760 m3/s) event (i.e., Case 1) is 0.05 [95% CI: 0.01-0.09] and that the corresponding return period reaches 22.14 years [95% CI: 11.22-76.49 years]. During the design life of 10 years, the risk that an AMF>5,760 m3/s event will occur before or at the 10th year is 0.37 [95% CI: 0.12-0.61]. On the other hand, within the 10 years, the reliability that an AMF>5,760 m3/s event will not occur is 0.71 [95% CI: 0.46-0.92]. In fact, it can be deduced from the formulas of Risk and Reliability (i.e. Eqs. 11-12) that the risk varies inversely to the return period, while the reliability is proportional to the return period. Therefore, if management departments tend to

94

decrease the so-called risk or increase the reliability in practice, increasing the design flood level for levees is compulsory.

For the Case 2 in the UCH, the probability that an AMF≤2,500 m3/s event will occur is huge, 0.66 [95% CI: 0.56-0.77], which means that this event will occur once every 1.51 years [95% CI: 1.30-1.77]. The corresponding risk reaches 0.96 [95% CI: 0.92-0.99], whereas the reliability is only 0.15 [95% CI: 0.08-0.26]. The high risk implies that scouring sediment in the river channel depending merely on the natural flood event is insufficient. Therefore, floodwater released by the reservoir in the upstream region of Huaxian station is crucially important for sediment transport in the lower WR. The Dongzhuang reservoir located in the Jinghe River (the largest tributary of WR) is under construction, with completion scheduled by 2022. A primary goal of this reservoir is to conduct water-sediment regulation to reduce the sedimentation in the lower WR, and thus to decrease the flood risk. Then, the Tongguan elevation will be decreased. Consequently, the safety of the flood control system in the lower WR will be further enhanced.

Case 3 (i.e., AMF>5,760 m3/s OR AMF≤2,500 m3/s) reflects the events impairing the safety of the local flood control system. Therefore, according to the addition formula of probability, the probability of a Case 3 event occurring is the sum of F (AMF>5,760) and F (AMF≤2,500 m3/s), which is 0.69 [95% CI: 0.62-0.78]. The probability is close to that 95

of Case 2 (0.66 [95% CI: 0.56-0.77]). It signifies that sedimentation will be the primary factor impairing the safety of the local flood control system. Therefore, appropriate water-sediment regulation scheme using the Dongzhuang reservoir is of great significance for sediment transport in the WR basin, and the sum of discharge from the Dongzhuagn reservoir and that gauged at Xianyang station can’t be less than 2,500 m3/s. In doing so, probability of floods due to channel sedimentation will be reduced greatly.

5.3.2. Bivariate return period and risk analysis Building the joint distribution of AMF and Pr and evaluating their bivariate risk are beneficial for better understanding risks between extreme hydro-climatic events (Candela, et al., 2014). Figure S4 and Figure S5 (supplemental material) exhibit the primary return periods for flood control and sediment transport in the WR. Decision-makers can identify the corresponding flood risk information for different flood and extreme precipitation events from Figure 6-7, which is extremely beneficial for practical flood control and mitigation (Fan et al., 2016). -------------------------------------

Insert Figures 6-7 here -------------------------------------

In the lower WR, 5,760 and 2,500 m3/s are critical thresholds for flood control and sediment transport. Therefore, we calculated the return periods for different flood-related events and their 95% CI in Figure 6-7 and Table to provide more 96

complete information. Note that in the UCX, F(AMF<5,760) and F(AMF<2,500) calculated using the fitted Gamma distribution are 0.9633 and 0.6635, respectively. In the UCH, F(AMF<5,760) and F(AMF<2,500) are 0.9511 and 0.4708, respectively. Using the fitted Gamma distribution of the Pr2 (Pr3) series, we calculated the values of Pr2 (Pr3) with equal probabilities of F(AMF<5,760) and F(AMF<2,500) events in the UCX (UCH) (listed in Table ). Values of Pr2 with equal probabilities of F(AMF<5,760) and F(AMF<2,500) events in the UCX are 58.12 mm and 28.89 mm, respectively, while values of Pr3 in the UCH are 52.96 mm and 26.95 mm, respectively. -------------------------------------

Insert Table 4 here -------------------------------------

From Figure 6-7 and Table , it can be inferred that the uncertainties of bivariate return periods induced by the limited sample size are very large and should generate critical concern for future flood mitigation. Taking the UCH as an example, the TOR for an (AMF>5760 OR Pr3>48.32) event is 9.35 years [95% CI: 5.65-21.52 years], which overlaps that for (AMF>5760 OR Pr3>67.37), i.e., 20.14 years [95% CI: 10.10-66.87 years]. Similarly, the TOR for an (AMF<2,500 OR Pr3<15.19) event is 2.06 years [95% CI: 1.68-2.66 years], which overlaps that for (AMF<2,500 OR Pr3<29.27), i.e., 1.5 years [95% CI: 1.30-1.79 years]. This phenomenon, i.e., the return periods overlap, is also found in the works of (Serinaldi, 2013; Dung et al., 2015; Zhang et al., 2015; Serinaldi, 2016). Due to the considerable uncertainty of the flood risk assessment, Michailidi and Bacchi (2017) stated that, any attempt to obscure this uncertainty could 97

create a false notion about its existence in a multivariate problem. Therefore, taking the uncertainty into account when developing flood control policies is of critical importance . Specifically, as suggested in Section 5.3.1, management departments can improve the levee design standard and increase water release from reservoirs (although it would increase the cost and decrease economic profit) with the assist of the above uncertainty evaluation results.

For the sake of simplicity, we don’t display the Risk and Reliability results in Figure 6-7 and Table 4. If one is interested in the Risk and Reliability of different flood-related events, the derived Eq. (22) and Eq. (23) can be used to compute Risk and Reliability on the basis of the return periods of different flood events.

These large uncertainties pose significant challenges for the design of hydraulic engineering, and the development of flood control and mitigation scheme. Particularly, the Guanzhong Plain (an important Chinese agricultural production region) and Xi’an City (known as one of the four major ancient civilization capitals) both lie in the floodplain of the WR (Huang, et al., 2014). Therefore, the uncertainty of copula-based risk analyses for this region should generate critical concern, particularly when developing policies for flood control and hazard mitigation.

98

5.4. Conditional probabilities of AMF under given extreme precipitation events Based on the joint distribution of AMF and Pr, the conditional probabilities of AMF under given Pr scenarios can be examined. In present study, according to the specific thresholds of flood peak discharge for flood control and sediment transport, the conditional probabilities FAMF DF Pr  pr , FAMF IFRST Pr  pr and FIFRST AMF DF Pr  pr are explored. These conditional probabilities help to identify the possibilities of the occurrence

of

 AMF  DF

event,

 AMF  IFRST 

event,

and

IFRST  AMF  DF event under different extreme precipitation conditions. In

practice, with the assist of rainfall forecast information, management departments can deduce the conditional probabilities of different flood events by constructing the joint dependence between AMF and Pr, and subsequently develop scientific flood control policies for flood control and sediment transport.

In practical terms, given a potentially extreme future precipitation scenario, the probabilities of the occurrence of specific flood peak discharge exceeding DF, not exceeding IFRST and varying in the domain [IFRST, DF], can be discovered from Figure 8. This information would be important for performing effective flood control measures to ensure the safety of local flood control systems. Moreover, we also discovered the uncertainties of these conditional probabilities. The results are also shown in Figure 8, from which a flood control department can identify the conditional

99

probabilities and their 95% confidence intervals for any extreme precipitation scenarios of interest. -------------------------------------

Insert Figure 8 here -------------------------------------

Taking UCH as an example, conditioned on Pr3=52.96 mm in the future, the conditional probabilities FAMF DF Pr  pr , FAMF IFRST Pr pr and FIFRST
6. Summary and conclusions In this study, a univariate and bivariate hydrological risk framework is developed to perform flood risk analysis for flood control and sediment transport. This framework considers both the design flood of downstream hydraulic structures and instream flow requirements for sediment transport. Consequently, the univariate and bivariate return period, risk and reliability are derived for the purposes of flood control and sediment transport. The bivariate hydrological risk is examined by exploiting the copula-based joint distribution model of annual maximum flood discharge (AMF) and extreme precipitation (Pr). Moreover, the conditional probabilities of different flood events 100

under various extreme precipitation scenarios are estimated by the joint distribution of AMF and Pr. This information is valuable for maintaining the safety of local flood control systems. Moreover, one Monte Carlo-based algorithm is proposed to disclose the impact of sampling uncertainty on hydrological risk evaluation in order to provide more robust and practical information for local flood control and hydraulic engineering design.

Two sub-catchments of the Loess Plateau (UCX and UCH, the upper catchments of Xianyang and Huaxian stations) are selected as the study regions. Floods and sedimentation in river channels in the two regions greatly impair the safety of local flood control systems. Our framework is thus used to analyze the hydrological risk in the study regions. Primary conclusions are shown as below.

Sedimentation in river channels poses greater threat to the safety of flood control systems, compared with the flood peak discharge exceeding the design flood of downstream hydraulic structures in the two sub-catchments of the LP. Given this, management departments should pay more attention on reservoir operation scheme for scouring sediment in river channel under the premise of ensuring the safe operation of hydraulic structures in the regions.

101

The uncertainties of univariate and bivariate flood risk in the UCH and UCX are considerably large. For example, in the UCH, return periods of the (AMF>5760 m3/s OR Pr >58.12 mm) event can reach between 9.53 and 59.99 years. The considerable uncertainty should generate critical attention, particularly when developing policies for flood control and mitigation. Specifically, confronted with the great uncertainty of flood risk assessment, to ensure the safety of human life and property during floods, management departments can improve the levee design standard and increase water release from reservoirs for sediment delivery. Additionally, by applying the conditional probability distributions of AMF upon various extreme precipitation scenarios (Eqs 24-26), management departments can deduce the conditional probabilities of different flood events with the assist of rainfall forecast information, and subsequently develop scientific flood-relevant policies for flood control and sedimentation management.

The developed flood risk assessment framework can potentially help decision-makers in flood control and mitigation by providing valuable insights toward assessing flood risk focusing on flood control and sediment transport in sandy regions. This framework is also promising for use in other sandy basins worldwide.

Acknowledgments

102

This work was supported by the National Key Research and Development Program of China (2016YFC0400906, 91647112), National Natural Science Foundation of China (51679187, 51679189), Doctorial Innovation Fund in Xi’an University of Technology (310-252071606, 310-252071605), and the China Scholarship Council (CSC). Sincere gratitude is extended to the editor and the anonymous reviewers for their professional comments and corrections.

Appendix A Student t copula The cumulative distribution function of the Student t copula for the variables X1 and X2 is given by

C  F1  X1   u,F2  X 2   v ,    

1

t



u 



1

t



 s 2  2  st+t 2  v 1 1    1   2   2 1   2  

 1



2

dsdt

(A1) where  is the degrees-of-freedom parameter, t is the univariate Student’s t distribution function, and  is the correlation between t1  u  and t1  v  .

Appendix B Maximum Likelihood method. Procedures of the maximum likelihood method are given as follows. The joint distribution of (x, y) is defined as: 103



H  x, y;   C F  x; x  ,F  y; y  ;



(B1)

where    x , y ,  . Therefore, h  x, y;  can be written as (Cherubini et al., 2004, p.154):





h  x, y;   c F  x; x  ,F  y; y  ; f  x; x  f  y; y 





where c F  x; x  ,F  y; y  ; 



(B2)

 F  x; x  ,F  y; y  ; F  x; x  F  y; y 



denotes the density

function of the copula. Let

 x1 , y1  , , xT , yT 

be the random sample of size T from copula C. The

log-likelihood function is expressed as:







l     log  h  x, y;     log c F  x; x  ,F  y; y  ; f  x; x  f  y; y  (B3) T

T

t 1

t 1

Then, maximizing the log-likelihood function l   generates:



MLE   x MLE  , y MLE  , MLE 



(B4)

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Figure 1. Sketch map of the study area on the Loess Plateau. Weiyu 37 and Weiyu 10 are two sediment load measuring stations, which are close to the Xianyang and Huaxian stations, respectively. Figure 2. Flowchart of flood risk evaluation for flood control and sediment transport in the sandy regions. Figure 3. Schematic depicting the probabilities of AMF exceeding/not-exceeding a certain magnitude. Figure 4. Illustration of joint probabilities for the “OR”, “AND” and “COND” events. Figure 5. Return periods for AMF series obtained by the fitted Gamma distribution in the UCX and UCH. The thin gray lines denote the return periods using the Monte Carlo simulated AMF data. The red solid lines represent the return periods of observed AMF data. The red dotted lines represent the 5% percentile and 95% percentile return periods. 121

Figure 6. Primary return periods of (AMF>5760) OR (Pr>pr), (AMF>5760) AND (Pr>pr) and (AMF>5760) | (Pr>pr) events for flood control in the UCX and UCH. The thin gray lines denote the primary return periods using the Monte Carlo simulated AMF and Pr data. The red solid lines represent the return periods of observed AMF and Pr data. The red dotted lines represent the 5% percentile and 95% percentile return periods. Figure 7. Primary return periods of (AMF<5760) OR (Pr
122

Flood control problems in

Flood control for safety of downstream levee

Flood control for promotion of sediment transport

Flood control for safety of local flood control system

(Case 1)

(Case 2)

(Case 3)

sandy regions

Case 1

Case 2

AMF

Case 3

AMF

AMF

Dangerous event

p1

Dangerous event

p1

DF

Safe

q=1-p2

q=1-p1

DF

event

q=1-p1-p2

IFRST

Safe event

Dangerous event

p2 1

evaluation and Monte-Carlo-

1

2

Return periods (T)

based uncertainty assessment

T

2

1

T

 1 Risk  1  1    T

1 F  AMF  IFRST 

n

Case 1 (AMF > DF) OR (Pr > pr)

T

Pr

Pr pPr

pPr

qPr=1-pPr

qPr=1-pPr

qPr=1-pPr

AMF qAMF= 1-pAMF

Return periods (T)

uncertainty assessment

(AMF

Case 2 IFRST) OR (Pr

AMF

AMF

qAMF= 1-pAMF

pAMF

qAMF= 1-pAMF

pAMF

Return periods (T)

1 1  C  FAMF  DF  ,FPr  pr  

and Monte-Carlo-based

pr)

(AMF

Case 2 IFRST) AND (Pr

1  FAMF  DF   FPr  pr   C  FAMF  DF  ,FPr  pr  

pr)

(AMF

pPr

AMF pAMF

Return periods (T)

qAMF= 1-pAMF

AMF pAMF

Return periods (T)

1 FAMF  IFRST   FPr  pr   C  FAMF  IFRST  ,FPr  pr  

Return periods (T)

C  FAMF  IFRST  ,FPr  pr  

C  FAMF  DF  ,FPr  pr  

Conditional unexceedance probability of AMF given Pr=pr

 1  FAMF DF Pr  pr

FAMF IFRST Pr  pr 

C  FAMF  IFRST  ,FPr  pr  

under given Pr scenarios and Monte-Carlo-based uncertainty assessment

AMF

Case 2

Conditional exceedance probability of AMF > DF given Pr=pr FPr  pr 

qAMF= 1-pAMF

FPr  pr 

1 C  FAMF  IFRST  ,FPr  pr  

Case 1

FAMF DF Pr  pr  1 

pr)

qPr=1-pPr

pPr

pAMF

Case 2 IFRST) | (Pr

Pr

qPr=1-pPr

pPr

Conditional statistics of AMF

1  FPr  pr 

Pr

qPr=1-pPr

pAMF

Return periods (T)

1 1  FAMF  DF   FPr  pr   C  FAMF  DF  ,FPr  pr  

Pr

qAMF= 1-pAMF

n

Case 1 (AMF > DF) |(Pr > pr)

Case 1 (AMF > DF) AND (Pr > pr)

Pr

Bivariate flood risk evaluation

1 F  AMF  IFRST or AMF  DF 

 1 Reliability  1  Risk  1    T

pPr

2

Return periods (T)

Return periods (T)

1 F  AMF  DF 

event

Dangerous event time t (years)

p2

time t (years)

time t (years)

Univariate flood risk

Safe IFRST

Case 3 Conditional probability that IFRST
FIFRST AMF DF Pr  pr  FAMF DF Pr  pr  FAMF IFRST Pr  pr

FPr  pr 

IFRST

 FAMF IFRST Pr  pr

(a)

AMF ...

Dangerous event

p1 DF q=1-p1

(b)

q=1-p2

Safe

event

Safe

event

...

AMF

...

IFRST Dangerous event

p2 (c)

AMF ...

Dangerous event

p1 DF

q=1-p1-p2

Safe

...

event

IFRST p2

Dangerous event

1

2

3

4

time t (years)

Dangerous event

Safe event Case 1 (AMF > DF) AND (Pr > pr)

Case 1 (AMF > DF) OR (Pr > pr)

Case 1 (AMF > DF) | (Pr > pr)

Pr

Pr

Pr

pPr

pPr

pPr

qPr =1-pPr

qPr =1-pPr

qPr =1-pPr

qAMF=1pAMF

(AMF

Case 2 IFRST) OR (Pr

AMF pAMF

qAMF=1pAMF

pr)

(AMF

Case 2 IFRST) AND (Pr

AMF

qAMF=1pAMF

pAMF

pr)

(AMF

Pr

Pr

qPr=1-pPr

qPr=1-pPr

pPr

pAMF

qAMF=1q=1-p pAMF

pr)

Pr

qPr=1-pPr

pPr

Case 2 IFRST) | (Pr

AMF pAMF

pPr

AMF pAMF

qAMF=1q=1-p pAMF

X pAMF

qAMF=1q=1-p pAMF

X

Table 1. Correlation coefficients between AMF and Pr. Bold numbers denote the extreme precipitation events most closely correlated with AMF. Region UCX UCH

Correlation coefficient Spearman Kendall Spearman Kendall

(AMF, Pr1) (AMF, Pr2) (AMF, Pr3) (AMF, Pr4) (AMF, Pr5) 0.3195* 0.2059* 0.2250 0.1586

0.7214** 0.5394** 0.3350* 0.2369*

0.6325** 0.4610** 0.5214** 0.3756**

0.5855** 0.4134** 0.4729** 0.3175**

0.5742** 0.4334** 0.4793** 0.3320**

Note: * and ** indicate that correlation coefficients are significant at the 95% and 99% confidence level, respectively.

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Table 2. Parameters of the Student t copula and goodness-of-fit test values. Region UCX UCH

Parameter

Cramér–von Mises test

ρ

v

Sn

p-value

0.7393 0.5658

4.2369 15.0086

0.0171 0.0228

0.8986 0.6778

Note: ρ represents the correlation parameter; v denotes the degrees-of-freedom parameter.

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Table 3. Return period, risk and reliability of different flood-related events in the UCX and UCH. Case 1, Case 2 and Case 3 denote (AMF>5760 m3/s), (AMF ≤ 2500 m3/s), and (AMF>5760 m3/s OR AMF ≤ 2500 m3/s), respectively. Region Event

Probability

Return period (years)

Risk

UCX

0.03 [0.01, 0.07] 0.66 [0.56, 0.77] 0.69 [0.62, 0.78] 0.05 [0.01, 0.09] 0.47 [0.36, 0.58] 0.52 [0.42, 0.60]

29.15 [13.40, 117.90] 1.51 [1.30, 1.77] 1.44 [1.28 1.61] 22.14 [11.22, 76.49] 2.12 [1.73 2.77] 1.93 [1.66 2.39]

0.29 [0.08, 0.54] 0.71 [0.46, 0.92] 0.96 [0.92, 0.99] 0.04 [0.01, 0.08]

UCH

Case 1 Case 2 Case 3 Case 1 Case 2 Case 3

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Reliability

0.37 [0.12, 0.61] 0.63 [0.39, 0.88] 0.85 [0.74, 0.92] 0.15 [0.08, 0.26]

Table 4. Bivariate return periods of different (AMF, Pr) pairs and their 95% CIs for flood control and sediment transport in the UCX and UCH. Flood control Region

UCX

UCH

Event

Sediment transport Return period (years)

18.80 [9.53 59.99] 56.11 (AMF>5760) AND (Pr2>58.12) [23.05 251.73] 2.06 (AMF>5760) |(Pr2>58.12) [1.37 4.10] 12.55 (AMF>5760) OR (Pr3>52.96) [7.15 32.65] 69.21 (AMF>5760) AND (Pr3>52.96) [25.61 279.95] 3.39 (AMF>5760) |(Pr3>52.96) [1.80 8.05] (AMF>5760) OR (Pr2>58.12)

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Event

Return period (years)

1.31 [1.17 1.48] 1.78 (AMF<2500) AND (Pr2<28.89) [1.48 2.21] 1.18 (AMF<2500) |(Pr2<28.89) [1.10 1.32] 1.60 (AMF<2500) OR (Pr3<26.95) [1.37 1.94] 3.18 (AMF<2500) AND (Pr3<26.95) [2.38 4.61] 1.49 (AMF<2500) |(Pr3<26.95) [1.28 1.89] (AMF<2500) OR (Pr2<28.89)

Flood risk analysis framework is developed for flood control and sediment transport Monte Carlo based method is proposed to explore uncertainty of flood risk evaluation This framework is exemplarily applied for catchments of the Loess Plateau, China

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