Flood Coincidence Risk Analysis UsingMultivariate Copula Functions

Lu Chen, S.M.ASCE1; Vijay P. Singh, F.ASCE2; Guo Shenglian3; Zenchao Hao4; and Tianyuan Li5

Abstract: The coincidence of flood flows of the mainstream and its tributaries may determine flood peaks. This study analyzed the risk offlooding as a result of such flood coincidences by considering flood magnitudes and time (dates) of occurrence. The Pearson Type III (P3) andlog-Pearson Type III (LP3) distributions were selected as the marginal distribution of flood magnitude for annual maximum flood series; themixed von Mises distribution was selected as the marginal distribution of flood occurrence dates. Two four-dimensional (4D) copula func-tions were developed for the joint distribution of flood magnitudes and occurrence dates. The upper Yangtze River in China and the ColoradoRiver in the United States were selected to evaluate the method of computing risk. The coincidence probabilities of flood magnitudes anddates were calculated, and the conditional probabilities for the Three Gorges Reservoir (TGR) were analyzed. Results show that the vonMisesdistribution can fit the observed flood dates data well. The X-Gumbel copula was selected for risk analysis. On the basis of the proposedmodel, the coincidence and conditional probabilities for any return period were obtained. DOI: 10.1061/(ASCE)HE.1943-5584.0000504.© 2012 American Society of Civil Engineers.

CE Database subject headings: Risk management; Yangtze River; Colorado River; Floods; China; United States.

Author keywords: Coincidence risk; Copula function; Upper Yangtze River; Upper Colorado River.

Introduction

Disastrous floods can be caused by unusual combinations of hydrometeorological factors and river basin conditions. Topography, landcover, and temporal and spatial distribution of rainfall play dom-inant roles in the generation of floods, which can be reflected in thecontributions that major tributaries make to the mainstream flow.The coincidence of flood flows of the mainstream and its tributariesmay determine peak flow. Therefore, the risk of flooding as a resultof the combination of flood flows from different rivers is importantfor hydraulic design. The combination risk arises when large floodsoccur simultaneously in the mainstream and in its tributaries, andthis risk is characterized in terms of flood magnitude, and occur-rence date. Traditional methods focus only on the flood magni-tudes, and a more realistic approach is therefore needed.

The traditional approach to the risk assessment entails a deter-mination of the probability that a preselected value of the floodcharacteristic will be exceeded, which is equivalent to determiningthe return period (Prohaska et al. 2008). This approach is based on aunivariate frequency analysis. However, when the river receives asignificant tributary, the previously mentioned approach does notyield reliable estimates of flood properties (Prohaska et al.2008). The risk of combining floods involves at least two sitesin the mainstream and its tributary or in two tributaries. This sug-gests that a multivariate hydrological analysis, which considers thedependence between flood variables, is needed.

Prohaska et al. (2008) used a multivariate probability distribu-tion function to evaluate the coincidence of floods on three adjacentstreams, on the assumption that floods followed a log-normal dis-tribution. The use of the log-normal distribution for representingthe frequency distribution of the peak flow is not supported byhydrologic practices in many countries. For example, the PearsonType III (P3) distribution is assumed for frequency analysis of floodpeaks in China [Ministry of Water Resources (MWR) 1993], log-Pearson Type III (LP3) in the United States [Interagency AdvisoryCommittee on Water Data (IACWD) 1982], and the generalizedlogistic (GL) distribution in the United Kingdom (Robson andReed 1999). Further more, although Prohaska’s study involvedthree variables, the three-dimensional (3D) probability distributionwas not established directly. The graph-analytical scheme method,which was based on a two-dimensional (2D) probability distribu-tion, was implemented. This method is difficult to extend to ahigher dimension and leads to inaccurate results.

For these reasons, a new multivariate model based on thecopula function was applied in this study. In recent years, copulashave been used for multivariate hydrologic analyses. For example,they have been used for rainfall frequency analysis (De Micheleand Salvadori 2003; Grimaldi and Serinaldi 2006a; Kao andGovindaraju 2007; Zhang and Singh 2007a; and Kuhn et al.2007), flood frequency analysis (Favre et al. 2004; Shiau et al.2006; Zhang and Singh 2006; Renard and Lang 2007; Karmakar

1State Key Laboratory of Water Resources and Hydropower Engineer-ing Science, Wuhan Univ., Wuhan, 430072, China, Dept. of Biological &Agricultural Eng., Texas A & M Univ., College Station, TX 77843-2117(corresponding author). E-mail: chl8505@126.com

2Caroline & William N. Lehrer Distinguished Chair in Water Engineer-ing and Professor, Dept. of Biological. & Agricultural Eng. and Professor,Dept. of Civil & Environmental Eng., Texas A &M Univ., College Station,TX 77843-2117. E-mail: vsingh@tamu.edu

3Professor, State Key Laboratory of Water Resources and HydropowerEngineering Science, Wuhan Univ., Wuhan 430072, China. E-mail:slguo@whu.edu.cn

4Dept. of Biological & Agricultural Eng., Texas A & M Univ., CollegeStation, TX 77843-2117.

5State Key Laboratory of Water Resources and Hydropower Engineer-ing Science, Wuhan Univ., Wuhan, 430072, China. E-mail: tyli1986@sina.com

Note. This manuscript was submitted on May 13, 2011; approvedon September 13, 2011; published online on September 15, 2011. Discus-sion period open until November 1, 2012; separate discussions mustbe submitted for individual papers. This paper is part of the Journal ofHydrologic Engineering, Vol. 17, No. 6, June 1, 2012. ©ASCE, ISSN1084-0699/2012/6-742–755/$25.00.

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and Simonovic 2009; and Chen et al. 2010; Wang et al. 2009),drought frequency analysis (Shiau 2006; Kao and Govindaraju2010; Song and Singh 2010a, b), rainfall and flood analysis (Singhand Zhang 2007; Xiao et al. 2009; Keef et al. 2009), sea stormanalysis (De Michele et al. 2007), return-period analysis (Salvadoriand De Michele 2004; Salvadori 2004), multivariate simulation(AghaKouchak et al. 2010a, b), and some other theoretical analysesof multivariate extreme problems (Salvadori et al. 2007; Salvadoriand De Michele 2010, 2011). Detailed theoretical background anddescriptions for the use of copulas can be found in Nelsen (2006)and Salvadori et al. (2007).

Most of these studies involve bivariate copulas (Kao andGovindaraju 2010). De Michele and Salvadori (2003) used 2-Dcopulas to model the joint probability distribution of storm durationand average storm intensity. Favre et al. (2004) proposed an ap-proach based on copulas for bivariate frequency analysis. Shiauet al. (2006) used copulas for analyzing bivariate frequencies offlood peak and volume. Dupuis (2007) discussed bivariate modelingof extreme tails of correlated hydrological random variables. ZhangandSingh (2006, 2007b)presentedbivariate frequencyanalyse usingcopulas for flood and rainfall events, respectively. Xiao et al. (2008)proposed anewdesign floodhydrographmethodbased on abivariatejoint distribution. Nazemi and Elshorbagy (2011) applied a bivariatejoint copula to assess the long-term behavior of reconstructed water-sheds. Trivariate copula functions also have been used. Grimaldi andSerinaldi (2006b) applied the Archimedean copula to model the tri-variate joint distribution of floods. Serinaldi andGrimaldi (2007) de-scribed an inference procedure to carry out a trivariate frequencyanalysis by means of asymmetric Archimedean copulas. Zhangand Singh (2007a, c) applied the Archimedean copulas to trivariatefrequencyanalyses of rainfall and flood events.KaoandGovindaraju(2008) applied Plackett copulas to trivariate statistical analyses ofextreme rainfall events (e.g., Song and Singh 2010b). Song andSingh (2010b) modeled the joint probability distribution of droughtduration, severity, and interarrival time using a trivariate Plackettcopula. Applications of 4D copula functions in hydrological fieldshave also been recently reported.DeMichele et al. (2007) introduceda method for constructing multivariate distributions, given two cop-ulas for each bivariate marginal law and applied the method to pro-videa4Dcharacterizationof sea state statistics.Serinaldi et al. (2009)used a 4D student copula to analyze drought probabilistic character-istics. Because more variables are involved, 4D copulas will be usedin this study.

The objective of this study was to apply a multivariate copula toanalyze the coincidence of flood risk of rivers. The upper YangtzeRiver and Colorado River were selected as case studies. Daily flowdata from four sites at the upper Yangtze River and Colorado Riverwere chosen; and 4D copula functions were applied to construct thejoint distribution of flood occurrence dates and magnitudes. Thevon Mises distribution was used to describe the flood occurrencedates, while the Pearson Type III and log-Pearson Type III distribu-tions were selected as the marginal distribution of annual maximum(AM) flood peaks. The coincidence probabilities of flood magni-tudes and occurrence dateswere analyzed. The conditional probabil-ities for the Three Gorges Reservoir (TGR) were calculated.

Methodology

Copula Functions

The problem of specifying a probability model for dependentmultivariate observations can be simplified by expressing the cor-responding n dimensional joint cumulative distribution (Salvadori

and Michele 2010). Following Sklar (1959) and Nelsen (2006), ifF1;2 ;…;n ðx1; x2; :::; xnÞ is a multivariate distribution function ofn correlated random variables of X1;X2;…;Xn with respectivemarginal distributions F1ðx1Þ;F2ðx2Þ;…;FnðxnÞ, then it is possibleto write an n-dimensional cumulative distribution function withunivariate margins F1ðx1Þ;F2ðx2Þ; :::;FnðxnÞ as followsHðx1; x2; � � � ; xnÞ ¼ C½F1ðx1Þ;F2ðx2Þ; � � �FnðxnÞ� ¼ Cðu1; � � � ; unÞ

ð1Þ

where FkðxkÞ ¼ uk for k ¼ 1;…; n, with Uk ∼ Uð0; 1Þ. Grimaldiand Serinaldi (2006b) introduced the fully nested Archimedeancopula and indicated that the fully nested n-dimensional copulascan be expressed as

Cðu1; u2;…; unÞ ¼ C1fun;C2½un�1;…;Cn�1ðu2; u1Þ…�g¼ φ½�1�

1 ðφ1ðunÞþ φ1fφ½�1�

2 ðun�1Þ þ � � �þ φ½�1�

n�1½φn�1ðu2Þ þ φn�1ðu1Þ� � � �g ð2Þ

where function φ is referred to as the generator of the copula. Itis noted that when φ1;…;φn�1 ¼ φ, Eq. (2) turns out to be anArchimedean symmetric n-copula, which can be written as

CðuÞ ¼�φ�1

Xnk¼1

φðukÞ�

ð3Þ

Because several families of Archimedean copulas, includingFrank, Clayton, and Gumbel, have been popular choices fordependence models because of their simplicity and generationproperties (Nelson 2006), the Archimedean copula was considered.In the context of extrapolation in frequency analysis, it is of greatimportance to be able to carefully model the extreme dependencecarefully (Poulin et al. 2007; de Waal et al. 2007). The Gumbelcopula, representing a sort of standard multivariate extreme valueand belonging to the Archimedean family, was ultimately used forthis study.

The four-variate symmetric Gumbel copula is given as

Cðu1; u2; u3; u4Þ ¼ expf�½ð� ln u1Þθ þ ð� ln u2Þθþ ð� ln u3Þθ þ ð� ln u4Þθ�1θg ð4Þ

where θ = parameter of four-variate symmetric Gumbel copulafunction with θ ≥ 1.

The four-variate asymmetric Gumbel copula is given as

Caðu1; u2; u3; u4Þ ¼ exp½�½ð� ln u4Þθ1þ fðð� ln u1Þθ3 þ ð� ln u2Þθ3 �θ2∕θ3þ ð� ln u3Þθ2Þθ1∕θ2g1∕θ1 � ð5Þ

where θ1, θ2, and θ3 are parameters of the four-variate asymmetricGumbel copula function with 1 ≤ θ1 ≤ θ2 ≤ θ3:

Salvadori and de Michele (2010) outlined a procedure for intro-ducing a suitable number of extra parameters in a given copulamodel. In this method, a family of d-copulas can be defined as

CaðuÞ ¼ AðuaÞ · Bðu1�aÞ ¼ Aðua11 ; � � � ; uann Þ · Bðua11 ; � � � ; uann Þð6Þ

where A and B = d-copulas; a ¼ ða1;…; anÞ and ∈ In = set of nparameters.

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The X-Gumbel model, which is four-variate version of thesymmetric Gumbel model given in a previous section, can be extraparameterized by means of Eq. (6) as

CXðu1; u2; u3; u4Þ ¼ Cξðua11 ; ua22 ; ua33 ; ua44 Þ· Cχðu1�a1

1 ; u1�a22 ; u1�a3

3 ; u1�a44 Þ ð7Þ

where ξ and χ = parameters of copulas with Gumbel parameters ξ,χ ≥ 1, and extra parameters a1, a2, a3, a4 ∈ I.

Marginal Distribution of Flood Occurrence Dates

If the single response is not scalar but is angular or directional, thedata are called directional data. Directional data include vectors,points on a unit sphere, angular direction, and circular or periodicdata. The dates of flood occurrence can be described by thedirectional statistics method. Observations arising from the meas-urement of times can be converted into angles according tothe periodicity of time, such as days or years (Otieno andAnderson-Cook 2006). This method is based on defining individualdates of flood occurrence as a directional variable by using thefollowing equation:

αi ¼ Di2πL; 0 ≤ αi ≤ 2π ð8Þ

where L = length of flood season; and Di = flood occurrencedate.

The x and y coordinates of the flood dates described by theangles can be determined as

ðai; biÞ ¼ ðcosαi; sinαiÞ ð9Þ

�a ¼Xmi¼1

cos xi∕m; �b ¼Xmi¼1

sin xi∕m ð10Þ

where m = sample size.The mean direction of the circular data (denoted by �θ) can be

estimated as

�θ ¼

8>>>>>>>>><>>>>>>>>>:

arctan �b∕�a �a > 0; �b > 0

2π þ arctan �b∕�a �a > 0; �b < 0

π þ arctan �b∕�a �a < 0

π∕2 �a ¼ 0; �b > 0

3π∕2 �a ¼ 0; �b < 0

unkown �a ¼ 0; �b ¼ 0

ð11Þ

A measure of the variability of flood occurrences about themean date can be determined by defining the mean resultant vectoras

�r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 þ b2

p; 0 ≤ �r ≤ 1 ð12Þ

where �r describes the dispersion measure (Black and Werritty1997).

Because the distribution of dates is on a circle rather than alonga line the use of a normal distribution is no longer appropriate.Therefore, the von Mises distribution was employed instead of anormal distribution to describe seasonal data with a single peak.Fisher (1993) referred to the von Mises distribution as the naturalanalog of normal distribution for seasonal data with a single peak.It is commonly used and has some similar characteristics as the

normal distribution (Mardia 1972). The probability density func-tion (PDF) of the von Mises distribution is given as

f ðxÞ ¼ expк cosðx�μÞ

2πI0ðкÞ0 ≤ x ≤ 2π; 0 ≤ μ ≤ 2π; к ≥ 0 ð13Þ

It is symmetric and unimodal, with a mean direction at μ andthe dispersion given by a concentration parameter к, к ¼ A�1ðrÞ,where A�1ðrÞ represents the inverse function of A and. I0ðкÞ isthe modified Bessel function of zero order. For large values of к,the distribution is concentrated around the mean direction. Whenк ¼ 0, the density gives the uniform distribution on [0, 2].

The von Mises distribution is only a unimodal distribution.Because the annual maximum floods may be generated by differentmechanisms, the flood occurrence date series often obeys a multi-modal distribution. Thus, a mixed von Mises distribution, whichcan describe the multimodal character, is comprised of a finitemixture of von Mises distributions. The probability density func-tion for a mixture of N von Mises distributions (vM-PDF) takes onthe following form:

f XðxÞ ¼XNi¼1

pi2πI0ðκiÞ

exp½κi cosðx�μiÞ�;

0 ≤ x ≤ 2π; 0 ≤ μi ≤ 2π; κi ≥ 0

ð14Þ

where pi = mixing proportion; ui = mean direction, and κi =concentration parameter.

Various methods can be used to estimate the 3N parameters onwhich the mixture of N vM-PDFs depends (Carta and Ramírez2007), and the maximum likelihood estimate (MLE) methodwas used for parameter estimation.

Marginal Distribution of Flood Magnitude

For the AM flood series, the Pearson Type III has been recom-mended by the Chinese Ministry of Water Resources (MWR1993) as a uniform procedure for flood frequency analysis inChina. The probability density function of the P3 distributioncan be written as

f PðyÞ ¼βα

ΓðαÞ ðy� δÞα�1 exp½�βðy� δÞ�α > 0;

β > 0; δ ≤ y < ∞ ð15Þwhere α, β, and δ = shape, scale, and location parameters, respec-tively; and ΓðαÞ = gamma function. The L-moment method wasused to estimate parameters of the P3 distributions.

The log-Pearson Type III has been recommended by the work-ing group of the Water Resource Council on flow frequency meth-ods for flood frequency analysis in the United States. (Singh et al.1986). The probability density function of the LP3 distribution canbe written as

f LPðyÞ ¼1

ayΓðbÞ�ln y� c

a

�b�1

exp

���ln y� c

a

��ð16Þ

where a, b, and c are the scale, shape, and location parameters,respectively.

Estimation of Flood Risk

The Flood Estimation Handbook (Reed 1999) states that the flood-risk assessment is to estimate the risk of a flood occurrence. TheEnvironment Agency’s Strategy for Flood-Risk Management[Environmental Agency (EA) 2003] states that one task of flood-risk estimation is to estimate the chance of a probability of aparticular flood event. A methodology is presented in this paper

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for the estimation of a kind of special flood event, namely thecoincidence of flood flows in the main river and its tributaries.The term coincidence is used to denote the simultaneous occur-rence of floods at two (or more) rivers (Prohaska et al. 2008).The degree of coincidence is measured by the probability of floodevents. The theoretical background draws from the practical appli-cation of a multivariate probability distribution function or its con-ditional probabilities (Prohaska et al. 2008). As flood events arecharacterized by flood occurrence dates and magnitudes, both ofthe two factors should be considered. This study considered thequantitative characteristics of simultaneous floods on the main riverand its tributaries and the flood dates of simultaneous floods.

First, flood magnitude was selected as a reference variable foranalysis (Favre et al. 2004). The exceedance probability of coincid-ing flood volumes considered in flow profiles is defined as

PTQn

¼ PðQ1 > qT1 ;Qi > qTi ; � � � ;Qn > qTn Þ ð17Þ

where PTQn

= exceedance probability of coinciding flood magni-tudes; i = ith gauge station; n = number of variables and can beequal to 2, 3, and 4 in this study; Q1;Qi…;Qn = flow magnitudes;and qT1 ; q

Ti ;…; qTn = design flood volume for the return period T .

Second, a flood date was selected as a reference variable foranalysis. In this study, if annual maximum floods occur withindt days, the floods were defined as contemporary temporal occur-rences. The coincidence probability of flood dates at two or moreconsidered inflow profiles is defined as

Ptn ¼ Ptðtk < Ti ≤ tkþ1; tk � dtij < Tj

≤ tkþ1 þ dtij; � � � ; tk � dtin < Tn ≤ tkþ1 þ dtinÞ ð18Þwhere i and j = any river in the data set and gauge station j is locateddownstream of the catchment; Ti = random variable of floodoccurrence dates; and dt = time interval and is equal to 1 day inthis study. The flood travel time between the two sites also should

be considered. Eq. (18) can compute the probabilities of simulta-neous floods for two or three rivers in the basin.

Third, both the flood magnitudes and flood dates were selectedas reference variables for risk analysis. Assuming that the floodoccurrence dates are independent of flood magnitudes and thatflood peaks occur simultaneously at two or more rivers in the samebasin, the flood coincidence probabilities of rivers for given floodmagnitudes were estimated as

PTn ¼

XNt¼1

Ptn · PðQ1 > qT1 ;Qi > qTi ; � � � ;Qn > qTn Þ ð19Þ

Data

The proposed method was used on two rivers. One is the upperYangtze River, and the other is the upper Colorado River. Thereason for choosing the Yangtze River is that the Three GorgesReservoir is located in that river. Studying this river has a signifi-cant practical value. The reason for selecting the Colorado River isthat the dependence in that river is relatively large. It is a good casein which to demonstrate proposed model.

The upper Yangtze River, which is the longest river in China andthird longest in the world, was selected as a case study. The ThreeGorges Project is located on the Yangtze River. Flooding by theYangtze River occures periodically in central and eastern China,has often caused considerable destruction of property and lossof life. Among the most recent major flood events are those of1870, 1931, 1954, 1998, and 2010. For example, in 1998, the entireYangtze River basin suffered from tremendous flooding; it wasthe largest flood since 1954, which led to the economic loss of166 billion Chinese yuan (or US$20 billion) (Yin and Li 2001).The flood was caused by unusually high precipitation between June

Fig. 1. Locations of regional tributary rivers and gauging stations

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and August. Summer is the main flooding season because of theheavy monsoon rainfalls. Floods in the middle and lower reachesof the Yangtze River mainly stem from the upper region of theYichang site, which is also the control site for the TGP. Usually,the flood volume of upper Yichang site is approximately 50%of the total flow volume of the Yangtze River, approximately90% of the Jingjiang River reach, which is regarded as the mostkey area for flood prevention.

The upper Yangtze River comprises a complex of tributaries,principally the Yalong River, Min River, and Jialing River onthe left bank, and the Wu River on the right bank. A schematicof the regional main tributary rivers and gauging stations is shownin Fig. 1. Some basic features of the available data were given inTable 1. The Yalong River joins the Jinsha River, which is alsorecognized as part of the Yangtze River. Therefore, the JinshaRiver, instead of the Yalong River, was used in this study. Relativefrequencies of annual maximum floods in these rivers were calcu-lated, as graphed in Fig. 1. It is shown that large floods alwaysoccur in July and August, except in the Wu River, in which thehighest relative frequency (RF) occurs in the middle of June. There-fore, it is more likely that floods in the Jinsha River, Min River, andJialing River occur simultaneously. Therefore, the Jinsha, Min andJialing Rivers were selected in this study. The Yichang site, as thelocation of TGP, is an important site on the Yangtze River and wasalso selected.

The Colorado River, in the southwestern United States andnorthwestern Mexico, has a length of approximately 2,330 km(1,450 mi) (Munro 1992) and was also selected as a case study.The Colorado River above Lee’s Ferry is defined as the upper Colo-rado River basin, with an area of approximately 46;000 km2

(17;800 mi2). The Colorado River originates in the mountains ofcentral Colorado and flows about 370 km (230 mi) southwest intoUtah. There are some tributaries in the upper Colorado River basin,principally the Green River, Gunnison River, and San Juan River.Some basic features of the available data were given in Table 2. TheGreen River, located in the western United States, is the chief tribu-tary of the Colorado River. The watershed of the river, the GreenRiver basin, covers parts of Wyoming, Utah, and Colorado. It isonly slightly smaller than the Colorado when the two rivers merge.The average yearly mean flow of the river at Green River, Utah, is173:3 m3∕s (6;121 ft3∕s). The Gunnison River is a significanttributary of the Colorado River, 264 km (164 mi) long, in theSouthwest state of Colorado (U.S. Geological Survey 2011). Itis the fifth largest tributary of the Colorado River, with a mean flowof 122 m3∕s (4;320 ft3∕s). The San Juan River is a tributary of theColorado River in the southwestern United States, approximately616 km (383 mi) long, the mean flow of which is about 62:4 m3∕s(2;205 ft3∕s) at its mouth (U.S. Geological Survey 2011). Com-pared with the other two major tributaries, the mean flow of theSan Juan River is relatively smaller. Therefore, only the GreenRiver and Gunnison River were considered in this study. As Lee’sFerry is the division site between the upper and lower ColoradoRiver, this site was considered. The site near Grand Junction

was selected for analyzing the flow above Cameo (named aboveCameo subsequently) of the Colorado River. Therefore, four sitesin the Colorado River basin were considered in this study.

Pairwise dependence structures of the four stations in the tworiver basins were estimated. Empirical estimates of the bivariateKendall’s τ of flood magnitudes and occurrence dates for all thepairs of interest are given in Tables 3 and 4. The coefficient betweenthe Beibei and Yichang stations in the upper Yangtze River wasnegative, but it was very small and close to zero. This means thatthe association between the two variables can be negligible, and theGumbel copula was therefore used.

Table 3. Values of Kendall’s τ of Flood Magnitudes and Occurrence Datesfor All Pairs of the Four Stations in the Upper Yangtze River

Stations Pingsha Gaochang Beibei Yichang

Pingsha 1.00 0.11 −0.08 0.28

Gaochang 0.07 1.00 0.03 0.21

Beibei 0.08 0.08 1.00 0.32

Yichang 0.19 0.18 0.34 1.00

Note: The upper triangular matrix is the Kendall’s τ of flood magnitude; thelower triangular matrix is the Kendall’s τ of flood dates; the meaning is thesame hereafter.

Table 4. Values of the Kendall’s τ of Flood Magnitudes and OccurrenceDates for All Pairs of the Four Stations in the Upper Colorado River

StationsAboveCameo

GreenRiver

GunnisonRiver

Lee'sFerry

Above Cameo 1.00 0.68 0.66 0.49

Green River 0.37 1.00 0.58 0.42

Gunnison River 0.19 0.32 1.00 0.50

Lee's Ferry 0.19 0.19 0.13 1.00

Table 5. RMSE Values of Different Probability Distributions of FloodOccurrence Dates in the Upper Yangtze River (%)

Distribution Pingshan Gaochang Beibei Yichang

Mixed von Mises 1.898 1.413 1.725 2.067Generalized logistic (GLO) 4.028 3.291 3.646 7.148

Generalized pareto (GP) 3.688 3.911 2.574 3.465

Pearson Type III(P3) 3.184 2.773 2.725 5.591

Generalized extreme value (GEV) 2.927 2.688 2.654 5.988

Gamma distribution 4.539 2.806 4.450 6.280

Normal distribution 3.560 2.887 3.021 7.393

Gumbel distribution 6.641 4.525 4.255 5.985

Wakeby distribution 2.796 2.566 1.848 3.465

Kappa distribution 2.734 2.664 2.102 2.195

Exponential distribution 10.432 8.735 8.426 6.340

Note: The bold characters mean the minimum value of each column.

Table 2. Major Tributaries to the Upper Colorado River

Rivers Catchment area (km2) Record of length

Colorado River (above Cameo) 20,800 1933–2011Green River 44,850 1933–2011Gunnison River 20,533 1933–2011San Juan River 12,000 1933–2011Colorado River (Lee's Ferry) 111,800 1933–2011

Table 1. Rivers Connected to the Upstream Yangtze River

Rivers Catchment area (km2) Record of length

Yalong River 144,200 1951–2007Jinsha River 485,099 1951–2007Min River 135,400 1951–2007Jialing River 157,900 1951–2007Wu River 87,920 1951–2007Yangtze River (Yichang) 1,005,501 1951–2007

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Application

Estimation of Marginal Distributions

To show the validity of the mixed von Mises distribution, otherdistributions, such as Gumbel, normal, and P3 distributions, wereselected as possible marginal distributions for the upper YangtzeRiver. Parameters of the mixed von Mises distribution were

estimated by the maximum likelihood method. Parameters of otherdistributions were estimated by the L-moment method. Thesedistributions were then fitted to the data and were compared withthe mixed von Mises distribution. The best-fitted distributions wereselected using the root mean square error (RMSE) values shown inTable 5 (Zhang and Singh 2007b). It was found that mixed vonMises distribution had the smallest RMSE values for the flooddates at all four stations in the upper Yangtze River. The values

Fig. 2. Frequency histograms of flood occurrence dates fitted by the mixed von Mises distribution for the four stations in the upper Colorado River:(a) above Cameo; (b) Green River; (c) Gunnison River; and (d) Lee's Ferry

Table 6. Parameters and Hypothesis Test Results of Margin Distributions

`

Mixed von Mises distribution P3 distribution

ui Кi pi K-S α β δ χ2

Pingshan 4.33 67.99 0.17 0.042 (0.176) 0.36 (3.84)

5.19 8.89 0.44 10.41 0.0008 3,279.29

3.46 3.70 0.39

Gaochang 4.29 3.04 0.60 0.039 (0.176) 0.33 (3.84)

3.65 300.00 0.16 4.726 0.0005 6,660.33

2.88 7.90 0.24

Beibei 5.44 0.00 0.21 0.035 (0.176) 0.57 (5.99)

2.96 7.20 0.46 330.6 0.002 �110;580

5.29 4.25 0.33

2.99 13.48 0.50 0.045 (0.176) 0.92 (3.84)

Yichang 4.20 6.63 0.28 156.25 0.0016 �49;825

5.35 11.21 0.23

Above Cameo 3.32 3.69 1.0 0.064 (0.155) 4.919 0.0896 3.783 0.13 (3.84)

Green River 3.02 2.38 1.00 0.057 (0.155) 3.493 0.1050 3.966 0.003 (3.84)

Gunnison River 2.57 2.25 0.96 0.031 (0.155) 30.451 0.0503 2.418 0.54 (3.84)

4.29 4.97 0.04

Lee's Ferry 2.29 198.29 0.09 0.144 (0.155)

3.52 1.00 0.74 10.058 0.0952 3.584 0.79 (3.84)

5.84 260.56 0.17

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Fig. 3. Frequency curves of flood occurrence dates based on AM samples for four stations in the Yangtze River: (a) Pengshan; (b) Gaochang;(c) Beibei; and (d) Yichang

Fig. 4. Frequency curves of flood magnitudes based on AM samples: (a) Pengshan; (b) Gaochang; (c) Beibei; and (d) Yichang

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of the estimated parameters of the von Mises distribution of bothriver basins are listed in Table 6. The Kolmogorov-Smirnov (KS)test was selected as the goodness-of-fit test to evaluate the validityof the assumption that the flood occurrence dates followed themixed vonMises distribution. The results shown in Table 6 indicatethat this assumption could not be rejected at the 5% significancelevel. The frequency histograms of the flood occurrence datesfitted by the mixed von Mises distribution for AM sample seriesin the upper Colorado River are shown in [Fig. 2(a)–2(d)]. Themarginal distribution curves of flood occurrence dates in the upperYangtze River are shown in Fig. 3, in which the line represents thetheoretical distribution and the crosses the empirical frequencies ofobservations. Figs. 2 and 3 indicate that all the theoretical distri-butions fitted the observed data reasonably well.

The values of the estimated parameters of the P3 and LP3 dis-tributions are given in Table 6. A chi-squared goodness-of-fit testwas performed to test the assumption, H0, that the flood magnitudefollowed the P3 or LP3 distribution. It is shown that the P3 or LP3distribution is valid for flood magnitudes at the four sites studied

with a critical value = 0.05. The marginal distribution frequencycurves of flood magnitudes in the upper Yangtze River were shownin Fig. 4. It is seen that graphically, the P3 distribution fitted theempirical distribution.

Estimation of Joint Distributions

Four-variate symmetric Gumbel, asymmetric Gumbel, and X-Gumbel copulas were used for modeling the dependence amongthe four stations. A pseudolikelihood technique involving the ranksof the data was used for estimating parameters of the four-variatesymmetric Gumbel and asymmetric Gumbel copulas. For theYangtze River, the value of the estimated parameter of the symmet-ric Gumbel was θ̂ ¼ 1:14 for flood magnitudes and θ̂ ¼ 1:20 forflood occurrence dates. Estimates of parameters of the asymmetricGumbel were θ̂1 ¼ 1:06, θ̂2 ¼ 1:16, and θ̂3 ¼ 1:46 for floodmagnitudes and θ̂1 ¼ 1:18, θ̂2 ¼ 1:20, and θ̂3 ¼ 1:38 for the floodoccurrence dates. For the Colorado River, the value of the estimatedparameter of the symmetric Gumbel was θ̂ ¼ 1:99 for floodmagnitudes and θ̂ ¼ 1:22 for flood occurrence dates. Estimates

Table 7. Parameters of X-Gumbel Joint Distributions

Rivers Parameters a1 a2 a3 a4 x s

Upper Yangtze River Magnitude 0.039 1.0 0.92 0.63 2.99 1.46

Dates 0.999 0.132 0.137 0.576 1.09 2.19

Upper Colorado River Magnitudes 0.707 0.773 0.725 0.268 3.267 2.763

Dates 0.221 0.396 1.000 0.193 1.181 3.372

Fig. 5. Plots of empirical and fitted Pickland’s dependence functions of flood magnitude for all pairs of stations and the three models

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Fig. 6. Plots of empirical and fitted Pickand’s dependence functions of flood occurrence dates for all pairs of stations and the three models

Fig. 7. Joint distribution and empirical probabilities of observed combinations: (a) and (b) flood magnitudes and flood occurrence dates in upperYangtze River; (c) and (d) flood magnitudes and flood occurrence dates in upper Colorado River

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of parameters of the asymmetric Gumbel were θ̂1 ¼ 1:82,θ̂2 ¼ 2:35, and θ̂3 ¼ 2:72 for flood magnitudes and θ̂1 ¼ 1:13,θ̂2 ¼ 1:30, θ̂3 ¼ 1:54 for the flood occurrence dates. Pickand’sdependence function, which was recommended by Salvadori andde Michele (2011), was used for estimating parameters of theX-Gumbel copula. The values of parameters of X-Gumbel for floodmagnitudes and flood dates were given in Table 7. The empiricalPickand's functions plotted in Figs. 5 and 6 as circles were calcu-lated using the method proposed by Genest and Segers (2009). Thefitted Pickand's function for all pairs of stations and three copulamodels of flood magnitudes in the two basins are plotted in Figs. 5and 6, respectively. The dashed boundary lines were also drawn inFigs. 5 and 6. The symmetric Gumbel dependence functions are thesame in all of the plots. The asymmetric Gumbel dependence func-tions are different corresponding to different pairs. The asymmetricGumbel copula fits better than the symmetric one. The X-Gumbelprovides a better fit than the other two models. The observed datawere arranged in ascending order, and empirical copulas were cal-culated as plotted in Fig. 7, in which the x-axis represents the cor-responding order number of a combination of observed data. Thetheoretical joint probabilities, F, of the real occurrence combina-tions of x were estimated. The empirical joint probabilities wereplotted against theoretical probabilities, as shown in Fig. 7, whichshows that no significant difference between empirical and theoreti-cal joint probabilities can be detected.

Analysis of Flood Combination Risk

Coincidence Probabilities AnalysisAccording to the analysis mentioned previously, the X-Gumbelcopula was used for the flood coincidence risk analysis. Theexceedance probabilities of coinciding T-year flood volumesat two and three considered inflow profiles were calculated asshown in Tables 8 and 9. The average exceedence probabilitiesof 100-year, 50-year, 10-year, 5-year, and 2-year floods for the foursites are 0.0075, 0.015, 0.0763, 0.1561, and 0.4196, respectively.

The coincidence probabilities of flood dates in two, three,and four rivers (Pt

2, Pt3 and Pt

4) were evaluated and shown inFigs. 8(a)–8(e), respectively. For the Jinsha and Min Rivers, thehigher coincidence probabilities occur in late July and the middleof August. According to the observed data, there are seven timesthat the flood occurred simultaneously in the two rivers, five ofwhich were within this period. For the Jinsha and Jialing Rivers,the curve demonstrates that the multimodal characteristic and

the higher coincidence probabilities occur in the middle of Julyand early September, which indicates that the flood control waterlevel (FCWL) of the TGR should not be raised too high and thatcertain flood control storage is needed for the TGR. For the Jialingand Min tributaries, the highest probability occurs in July. Six ofeight flood events that occurred simultaneously in the two rivers arewithin this period. For the three rivers in the upper Yangtze River,July has the highest coincidence probabilities. For the four stations,the higher probabilities also occur in July. It is indicated that in May

Table 8. Exceedance Probability of Coinciding T-Year Flood Volumes at Two Considered Inflow Profiles

Tributaries T 100 50 10 5 2

Upper Colorado and Green rivers 100 0.00746 0.00921 0.00997 0.00999 0.01000

50 0.00929 0.01495 0.01976 0.01995 0.019994

10 0.00998 0.01982 0.07607 0.09390 0.099553

5 0.00999 0.01997 0.09458 0.15562 0.196065

2 0.01000 0.02000 0.09969 0.19677 0.418765

Upper Colorado and Gunnison rivers 100 0.00751 0.00927 0.00997 0.00999 0.01000

50 0.00929 0.01505 0.01979 0.01996 0.02000

10 0.00998 0.01981 0.07654 0.09441 0.09962

5 0.00999 0.01996 0.09458 0.15647 0.19651

2 0.01000 0.02000 0.09966 0.19669 0.42025

Green and Gunnison rivers 100 0.00749 0.00930 0.00998 0.00999 0.01000

50 0.00925 0.01502 0.01982 0.01997 0.02000

10 0.00997 0.01978 0.07639 0.09467 0.09969

5 0.00999 0.01995 0.09420 0.15621 0.19682

2 0.01000 0.01999 0.09959 0.19632 0.41979

Table 9. Exceedance Probability of Coinciding T-Year Flood Volumes atThree Considered Inflow Profiles

UpperColorado Green

Gunnison

100 50 10 5 2

100 100 0.00671 0.00744 0.00751 0.00751 0.00751

50 0.00744 0.00899 0.00929 0.00929 0.00929

10 0.00749 0.00925 0.00996 0.00997 0.00998

5 0.00749 0.00925 0.00997 0.00999 0.00999

2 0.00749 0.00925 0.00997 0.00999 0.01000

50 100 0.00741 0.00896 0.00927 0.00927 0.00927

50 0.00904 0.01346 0.01505 0.01505 0.01505

10 0.00930 0.01502 0.01971 0.01980 0.01981

5 0.00930 0.01502 0.01978 0.01994 0.01996

2 0.00930 0.01502 0.01978 0.01995 0.01999

10 100 0.00746 0.00921 0.00996 0.00997 0.00997

50 0.00929 0.01495 0.01969 0.01979 0.01979

10 0.00997 0.01975 0.06874 0.07602 0.07654

5 0.00998 0.01982 0.07605 0.09207 0.09456

2 0.00998 0.01982 0.07639 0.09419 0.09944

5 100 0.00746 0.00921 0.00997 0.00999 0.00999

50 0.00929 0.01495 0.01976 0.01993 0.01996

10 0.00998 0.01982 0.07573 0.09180 0.09439

5 0.00999 0.01995 0.09247 0.14128 0.15632

2 0.00999 0.01997 0.09466 0.15613 0.19480

2 100 0.00746 0.00921 0.00997 0.00999 0.01000

50 0.00929 0.01495 0.01976 0.01995 0.01999

10 0.00998 0.01982 0.07607 0.09389 0.09939

5 0.00999 0.01997 0.09457 0.15553 0.19452

2 0.01000 0.02000 0.09954 0.19528 0.38732

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Fig. 8. Coincidence probabilities of flood dates on each day in the upper Yangtze River and its tributaries: (a) Jinsha and Min rivers; (b) Jinshaand Jialing rivers; (c) Min and Jialing tributaries; (d) Jinsha, Min, and Jialing tributaries; and (e) Jinsha, Min, Jialing, and Yangtze (Yichang station)rivers

Table 10. Coincidence Probabilities Considering Flood Magnitudes and Occurrence Dates in Two of the Tributaries

Rivers T 100 50 10 5 2

Upper Colorado and Green rivers 100 0.007456 0.00921 0.009967 0.009991 0.009998

50 0.009291 0.014947 0.019759 0.019947 0.019994

10 0.009978 0.019822 0.076068 0.093895 0.099553

5 0.009994 0.019966 0.094576 0.155616 0.196065

2 0.009998 0.019996 0.099686 0.196775 0.418765

Upper Colorado and Gunnison rivers 100 0.00006 0.00007 0.00007 0.00007 0.00007

50 0.00007 0.00011 0.00015 0.00015 0.00015

10 0.00007 0.00015 0.00057 0.00070 0.00074

5 0.00007 0.00015 0.00070 0.00116 0.00144

2 0.00007 0.00015 0.00074 0.00144 0.00287

Green and Gunnison rivers 100 0.00014 0.00018 0.00019 0.00019 0.00019

50 0.00018 0.00029 0.00038 0.00038 0.00038

10 0.00019 0.00038 0.00146 0.00181 0.00191

5 0.00019 0.00038 0.00180 0.00299 0.00377

2 0.00019 0.00038 0.00191 0.00376 0.00804

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and June, the coincidence probabilities are very small, which meansa low combination risk. Therefore, it is possible to raise the floodcontrol water level of the TGR in two months. All of the analysismentioned previously demonstrates that the calculated results are inaccordance with the historical data.

The coincidence probabilities of T-year design flood for twoand three tributaries were calculated on the basis of on Eq. (19),and the results are listed in Tables 10 and 11. The results are rea-sonable from the point of view that the coincidence probabilitiesincrease when the return period T is decreasing. The averagecoincidence probabilities of a 100-year and 10-year design floodin the two tributaries are 0.000143 and 0.001467, respectively.The coincidence probabilities of a 1,000-year and 500-year designflood in three tributaries are 3:63 × 10�6 and 3:71 × 10�5. FromTables 10 and 11, the coincidence probabilities of any other returnperiod can be obtained directly or by interpolation.

Conditional Probabilities AnalysisThe flood control standard of the TGR is 1,000 years. To analyzethe effect of the upper tributary streams on the TGR, the conditionalprobabilities were calculated. The conditional probabilities of theoccurrence of a T-year flood at the TGR, given the occurrence offloods in the upper tributaries, can be defined as

PðQn > qTn jQ1 > qT1 ; � � � ;Qn�1 > qTn�1Þ

¼ PðQ1 > qT1 ; � � � ;Qn > qTn ÞPðQ1 > qT1 ; � � � ;Qn�1 > qTn�1Þ

ð20Þ

where n = number of random variables and is from two to four;Q1;…;Qn = flow magnitudes in any of the two rivers; andqT1 ;…; qTn = T-year design flood. For the case of n = 2, the

conditional probabilities of T-year design flood for the YangtzeRiver at the TGR, given the flood volume in one of the upper tribu-taries by specifying, was obtained by Eq. (20) and listed in Table 12.In a similar manner, the conditional probabilities of Yichangstation, given the flood volume of two or three upper rivers, wereobtained. The calculated conditional probabilities are listed in theTables 13 and 14.

Table 12 shows that for a fixed return period in the upper rivers,the conditional probabilities show an increasing trend when the

Table 11. Coincidence Probabilities Considering Flood Magnitudes and Occurrence Dates in Three Tributaries of the Upper Yangtze River

Jinsha River Min River

Jialing River

100 50 10 5 2

100 100 3.63E-06 4.02E-06 4.06E-06 4.06E-06 4.06E-06

50 4.02E-06 4.86E-06 5.02E-06 5.02E-06 5.02E-06

10 4.05E-06 5E-06 5.38E-06 5.39E-06 5.39E-06

5 4.05E-06 5E-06 5.39E-06 5.4E-06 5.4E-06

2 4.05E-06 5E-06 5.39E-06 5.4E-06 5.4E-06

50 100 4.00E-06 4.84E-06 5.01E-06 5.01E-06 5.01E-06

50 4.88E-06 7.27E-06 8.13E-06 8.13E-06 8.13E-06

10 5.03E-06 8.11E-06 1.06E-05 1.07E-05 1.07E-05

5 5.03E-06 8.11E-06 1.07E-05 1.08E-05 1.08E-05

2 5.03E-06 8.11E-06 1.07E-05 1.08E-05 1.08E-05

10 100 4.03E-06 4.98E-06 5.38E-06 5.39E-06 5.39E-06

50 5.02E-06 8.08E-06 1.06E-05 1.07E-05 1.07E-05

10 5.39E-06 1.07E-05 3.71E-05 4.11E-05 4.13E-05

5 5.39E-06 1.07E-05 4.11E-05 4.97E-05 5.11E-05

2 5.39E-06 1.07E-05 4.13E-05 5.09E-05 5.37E-05

5 100 4.03E-06 4.98E-06 5.38E-06 5.40E-06 5.40E-06

50 5.02E-06 8.08E-06 1.07E-05 1.08E-05 1.08E-05

10 5.39E-06 1.07E-05 4.09E-05 4.96E-05 5.10E-05

5 5.40E-06 1.08E-05 5.00E-05 7.63E-05 8.45E-05

2 5.40E-06 1.08E-05 5.11E-05 8.44E-05 1.05E-04

2 100 4.03E-06 4.98E-06 5.38E-06 5.40E-06 5.40E-06

50 5.02E-06 8.08E-06 1.07E-05 1.08E-05 1.08E-05

10 5.39E-06 1.07E-05 4.11E-05 5.07E-05 5.37E-05

5 5.40E-06 1.08E-05 5.11E-05 8.40E-05 1.05E-04

2 5.40E-06 1.08E-05 5.38E-05 1.06E-04 2.09E-04

Table 12. Conditional Probabilities PðQTY > qTY jQ1 > qT1 Þ of the TGR

under the Condition of the Flood Occurring in One of Upper YangtzeRivers

Yichang Return period 1,000 500 100 50 10

Jinsha River 1,000 0.35 0.47 0.71 0.78 0.89

500 0.24 0.35 0.62 0.71 0.86

100 0.07 0.12 0.36 0.48 0.74

50 0.04 0.07 0.24 0.36 0.66

10 0.01 0.02 0.07 0.13 0.41

Min River 1,000 0.27 0.37 0.58 0.65 0.79

500 0.18 0.27 0.50 0.58 0.75

100 0.06 0.10 0.28 0.38 0.62

50 0.03 0.06 0.19 0.28 0.55

10 0.01 0.02 0.06 0.11 0.34

Jialing River 1,000 0.39 0.53 0.77 0.83 0.93

500 0.26 0.39 0.68 0.77 0.90

100 0.08 0.14 0.40 0.54 0.79

50 0.04 0.08 0.27 0.40 0.72

10 0.01 0.02 0.08 0.14 0.44

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return period of the TGR decreases. For example, given theoccurrence of a 1,000-year design flood in the Jinsha River, theconditional probabilities of a 1,000-year and 10-year design floodin the TGR are 0.35 and 0.71, respectively. The conditional prob-abilities of the TGR given T-year design floods in three rivers isgreater than those given T-year design floods in two rivers; theconditional probabilities of the TGR given T-year design floodsin two rivers is greater than those only given T-year floods inone river. From these points of view, the results of the calculationare reasonable. It is shown in Table 12, that the Jialing River has themost significant impact on the flow of the TGR. The coefficient ofcorrelation between the Jialing River (at Beibei station) and theYangtze River (at Yichang station) is 0.318, the largest value inTable 4, which shows the close relationship between the two rivers.It is demonstrated that from Table 13, the higher conditional prob-abilities of the TGR were generally obtained when the flows of theJinsha and Jialing Rivers are known. Table 14 gives the conditionalprobabilities of the TGR when the T-year design floods in theupper three rivers were known. It can be seen that when the threeupper rivers have 1,000-year floods, the conditional probability ofthe TGR is 1.0.

Conclusions

The flood combination risk, which reflects the probability of thecoincidence of multidimensional flood peaks, is important for res-ervoir operation and flood management. The copula function wasused to establish the joint distribution of flood magnitudes andflood occurrence dates. The coincidence probabilities of flood

magnitudes and dates were calculated. The conditional probabil-ities of the TGR for different return periods were analyzed. Themain conclusions of this study were summarized as follows:1. Symmetric Gumbel, asymmetric Gumbel, and X-Gumbel

copula functions were used. For multivariables (greater thantwo), the symmetric Archimedean copula has only one para-meter, which forces all of the pairs of variables to share thesame dependence structure. The nested classes of the Archime-dean copula can only can model n� 1 dependence. On thecontrary, the selected X-Gumbel copula provides the best fitfor dependence structures, and therefore, it was used for thecombination risk analysis.

2. By analyzing the coincidence probabilities of flood magni-tudes and flood dates, this paper contributed to a betterpractical knowledge in the area of engineering hydrology, par-ticularly with regard to the assessment of flood events and theperformance of comprehensive flood-risk analyses. Accordingto the analysis results, it is possible to raise the flood controlwater level of the TGR in May and June. On the contrary, inSeptember, the FCWL of the TGR should not be raised toohigh, and certain flood control storage is needed for theTGR. Therefore, it is better to apply a seasonal flood controlwater level strategy for the TGR. Different FCWL valuesshould be employed for each subseason. The entire floodseason can be divided into three subseasons, namely the pre-flood season, main flood season, and postflood season. Themain flood season should not end by the middle of September.Using this FCWL, the flood control level of the Jinjiang reachwill be largely upgraded from a 10-year flood to a 100-yearflood. The flow in the Jialing River has the most significantimpact on the inflow in the TGR. If the three upper rivershave a 1,000-year design flood, the TGR also experiences a1,000-year flood. The coincidence probabilities or conditionalprobabilities of any other return period can be obtained directlyfrom Tables 8–14 or by interpolation.

Acknowledgments

The study is financially supported by the Ministry of Science andTechnology (2009BAC56B01; 2009BAC56B02; 2010CB428405)of China.

References

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Table 13. Conditional Probabilities PðQTY > qTY jQ1 > qT1 ;Q2 > qT2 Þ of the

TGR under the Condition of the Flood Occurring in Two of Upper YangtzeRivers

Yichang Return period 1,000 500 100 50 10

Jinsha and

Jialing rivers

1,000 0.99 1.00 1.00 1.00 1.00

500 0.97 0.98 1.00 1.00 1.00

100 0.72 0.81 0.93 0.98 1.00

50 0.45 0.62 0.80 0.88 1.00

10 0.05 0.10 0.31 0.40 0.76

Jinsha and

Min rivers

1,000 0.73 0.96 1.00 1.00 1.00

500 0.42 0.73 1.00 1.00 1.00

100 0.09 0.18 0.73 0.96 1.00

50 0.04 0.09 0.42 0.73 1.00

10 0.01 0.02 0.09 0.18 0.75

Jialing and

Min rivers

1,000 0.90 0.95 1.00 1.00 1.00

500 0.81 0.89 0.99 1.00 1.00

100 0.50 0.63 0.88 0.95 1.00

50 0.32 0.47 0.75 0.86 0.99

10 0.05 0.10 0.32 0.43 0.80

Table 14. Conditional Probabilities PðQTY > qTY jQ1 > qT1 ;Q2 > qT2 ;Q2 >

qT2 Þ of the TGR under the Condition of the Flood Occurring in Three ofUpper Yangtze Rivers

Return Period 1,000 500 100 50 10

1,000 1.00 1.00 1.00 1.00 1.00

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100 0.76 0.86 0.97 1.00 1.00

50 0.50 0.68 0.86 0.94 1.00

10 0.06 0.12 0.38 0.48 0.87

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