Flood hydrograph coincidence analysis of the upper Yangtze River and Dongting Lake, China

In hydrological research, flood events can be analyzed by flood hydrograph coincidence. The duration of the flood hydrograph is a key variable to calculate the flood hydrograph coincidence risk probability and determining whether flood hydrograph coincidence occurs, while the actual duration of the flood hydrograph is neglected in most of existing related research. This paper creatively proposes a novel method to analyze the flood hydrograph coincidence risk probability by establishing a five-dimensional joint distribution of flood volumes, durations and interval time for two hydrologic stations. More specifically, taking the annual maximum flood of the upper Yangtze River and input from Dongting Lake as an example, the Pearson Type III and the mixed von Mises distributions were used to establish the marginal distribution of flood volumes, flood duration and interval time. Subsequently, the five-dimensional joint distribution based on vine copula was established to analyze the flood hydrograph coincidence risk probability. The results were verified by comparison with a historical flood sequence, which show that during 1951–2002, the hydrograph coincidence probabilities corresponding to its flood event coincidence volumes of 2.00 × 1011 m3, 4.00 × 1011 m3, and 6.00 × 1011 m3 are 0.213, 0.123, and 0.049, respectively. It has provided theoretical support for flood control safety and risk management in the middle and lower Yangtze River. This study also demonstrates the significant beneficial role of regulation by the Three Gorges Water Conservancy Project in mitigating flood risk of the Yangtze River. The hydrograph coincidence probability corresponding to its flood event coincidence volume of 2.00 × 1011 m3 has decreased by 0.141.


Introduction
Floods are caused by the rapid increase in river water volume and water level in response to heavy or continuous rainfall, or snow-and ice-melt in a river basin. They can result in significant financial losses, the destruction of infrastructure, and the loss of human life, and as such are serious natural hazards (Brocca et al. 2011;Li et al. 2012;Stein et al. 2020). In the context of global climate change, extreme meteorological and hydrological events are becoming more frequent, and the risk of basin-wide flooding is increasing (Wang et al. 2019;Su 2020;Try et al. 2020;Yang et al. 2020). Therefore, more attention to basin-wide flooding mitigation is needed.
Flood characteristics at the basin exit are usually influenced by the coincidence of runoff from two or more subbasins in the catchment. For example, the peak or maximum flood volume of the main stream and one or more tributaries may reach the same river section at a given time (Chen et al. 2012;Feng et al. 2020). When runoff from two locations is coincident, the magnitude of the flood peak and flood volume increase primarily due to accumulation. Therefore, flood event coincidence analysis in a basin is of significance to the formulation of flood control strategies and the rational development and utilization of water resources, particularly in large river basins.
The traditional method to study flood event coincidence within a river basin is to conduct probabilistic statistical analysis on the historical flood event coincidence using synchronously collected flood data in the study area (Zhu et al. 2015;Gao et al. 2017;Zhang et al. 2018). However, the traditional approach only focuses on historical floods and thus cannot provide the coincidence risk probability of extremely large floods (often missing from the historic record), such as floods with 100-year or even 1000-year return periods. However, extreme flood events are particularly important in the design of dams and other hydrologic systems as well as the mapping of flood-prone areas. Considering that flood event coincidence is a typical multivariate frequency combination problem, multivariate analysis can be applied (Prohaska et al. 2008). Furthermore, flood sequences are usually characterized by a skewed distribution, rather than a perfectly normal distribution. For example, flood sequences in most parts of China generally follow a Pearson Type III (P-III) distribution (Ministry of Water Resources (MWR) 2006).
Copula functions are an effective method for constructing multivariable joint distributions. Due to their flexible structure (Favre et al. 2004), copulas have been widely used in hydrological multivariate analysis in recent years (Salvadori and Michele 2004;Zhang and Singh 2007;Karmakar and Simonovic 2009;Kao and Govindaraju 2010;Liu et al. 2011;Reddy and Ganguli 2012;Chen et al. 2013;Sraj et al. 2015). In analyzing flood event coincidence, Klein et al. (2010), Schulte and Schumann (2016) and Bing et al. (2018) used copulas to establish the joint distribution among different river flood peaks, and to calculate the probability of the simultaneous occurrence of different river flood peaks within the basin. However, they only considered flood magnitude and neglected flood occurrence time. Further, Chen et al. (2012), Peng et al. (2018), Feng et al. (2020), and Zhang et al. (2020) used copulas to analyze the flood peak coincidence considering the flood magnitude and occurrence time simultaneously.
While even when the flood peaks are not directly coincident, but the flood hydrographs overlap, the impacts may be catastrophic. For example, in 2020 Shexian County in China was impacted by a flood with a 50-year return period; it caused US $300 million in damages to the city and US $100 million in damages to the countryside (Cheng and Zhang 2020). The peaks of the two "floods" that caused this damage occurred on July 3 and July 7 1 3 and thus were not perfectly coincident. However, the two flood hydrographs overlapped to a large degree. When the latter flood reached the area, the flood stage in the river was still receding from the previous flood. As a result, the second flood event led to higher rising water levels than from a single event and therefore caused serious flood damage. As such, overlap in flood hydrographs should also be considered as a flood coincidence event.
Coincidence of different flood events can be analyzed by the coincidence of their flood hydrograph, and flood duration is an important factor. Huang et at. (2018) and Yan et al. (2013) established a three-dimensional joint distribution of the annual maximum 15-day flood volumes and their interval time to analyze flood hydrograph coincidence, while the duration of annual maximum flood (AMF) in the real condition is not always 15 days, which will depend on the actual condition of the basin. To address this problem, this paper creatively proposes a method to analyze flood hydrograph coincidence by establishing a five-dimensional joint distribution of flood volumes, durations and interval time at two stations in the basin. The approach will depend on the nature of the river being studied and can be applied to other basins with similar complex river systems and frequent flood hazards. Specifically, this study focuses on the analysis of flood hydrograph coincidence between the upper Yangtze River and input from Dongting Lake located downstream to establish a five-dimensional joint distribution (Fig. 1). The coincidence risk probability of the flood hydrograph of the upper Yangtze River and Dongting Lake was then calculated and analyzed through stochastic simulation, so that the curve of the flood hydrograph coincidence volume and flood event coincidence risk probability could be obtained. Moreover, the role of the Three Gorges Project on flood event coincidence risk prevention of the upper Yangtze River was assessed.

Study area
The middle and lower Yangtze River Basin is prone to frequent flood hazards. During the flood season, if the flood of the upper Yangtze River encounters a flood from Dongting Lake, which is in the south of the Jingjiang section of the river (Fig. 1), it will not only adversely affect the flood control of the middle Yangtze River, but also make flooding in the Jingjiang section more severe. The flood event coincidence risk probability of the upper Yangtze River and tributaries of the middle Yangtze River Basin is an important theoretical component of the Three Gorges Project to timely formulate reasonable flood control plans. Therefore, studying flood event coincidence risk probability of the upper Yangtze River and Dongting Lake is of great significance for flood control and disaster reduction in the middle and lower Yangtze River ).

Utilized data and approach
In this study, the annual maximum floods (AMFs) from 1951 to 2016 at the Yichang Station (Station 1) and Chenglingji Station (Station 2) were collected to study the flood event coincidence risk probability of the upper Yangtze River and Dongting Lake (Fig. 1). The Yichang Station is located at the boundary between the upper and middle Yangtze River and is a control station for water and sediment from the upper Yangtze River. It controls a drainage area of 1 × 10 6 km 2 , with an average annual runoff of 4.29 × 10 11 m 3 . The maximum annual runoff was 5.75 × 10 11 m 3 (in 1954), whereas the minimum annual runoff at the station was 2.85 × 10 11 m 3 (in 2006). Runoff during the flood season (May to October) accounts for 78% of the average annual runoff. The Chenglingji Station is an important control station for the discharge of water and sediment from Dongting Lake into the main stream of the Yangtze River (Fig. 1). It controls a drainage area of 2.59 × 10 5 km 2 , with an average annual runoff of 2.85 × 10 11 billion m 3 . The maximum annual runoff was 5.27 × 10 11 m 3 (in 1954), whereas the minimum annual runoff was 1.48 × 10 11 m 3 (in 2011). Runoff during the flood season (May to October) accounts for 73% of the annual average runoff. Annual daily discharge data at the Yichang and Chenglingji Stations from 1951 to 2016 were collected by the Changjiang Water Resources Commission of The Ministry of Water Resources of China to study the flood risk probability of the upper Yangtze River and Dongting Lake. The water storage operation of the Three Gorges Project has changed the local hydrological cycle of the Three Gorges Reservoir, which has changed the amount, duration and occurrence time of the upstream flood. These alterations have also impacted the flood hydrograph coincidence risk probability of downstream reaches. Therefore, this study conducts risk probability analysis of flood hydrograph coincidence between the upper Yangtze River and Dongting Lake during two periods for comparative purposes: (1) from 1951 to 2002 (Period 1: before the construction of the Three Gorges Project), and (2) from 2003 to 2016 (Period 2: after the construction of the Three Gorges Project).

Definitions of AMF event characteristics
To analyze the AMF hydrograph coincidence risk probability, an AMF event must be defined by specific characteristics that allow the occurrence of an AMF event to be objectively recognized (Tosunoglu et al. 2020). The following criteria were used to define and characterize an AMF event.
(1) The highest point in the annual daily discharge diagram was defined as the peak of the AMF event (Q); (2) The first point at which the daily increase in discharge exceeds a certain threshold (Q t ') on the rising limb of the hydrograph corresponding to the AMF peak is the starting point of the AMF event. Its corresponding time is the starting time of the AMF event (t'); (3) The last point at which the daily decline in discharge exceeds a certain threshold (Q t '') on the recessional limb of the hydrograph corresponding to the AMF peak is the ending point of the AMF event. Its corresponding time is the ending time of the AMF event (t''); (4) If the interval time between the AMF event peak and its adjacent flood event peak is no more than 7 days, the flood event corresponding to the adjacent flood peak is also considered as a part of the AMF event; (5) The volume of the AMF event (V) is the volume of runoff between t' and t''; (6) The duration of the AMF event (d) is the interval time between t' and t''; (7) The occurrence time of the AMF event (T) can be denoted by the starting time (t').

Determination of the two thresholds in details:
(1) Firstly, we selected 30 typical AMF events according to historical records. Their common characteristics are: (a) including the peak of annual maximum flow; (b) the starting and ending points of flood events are two inflection points, and the durations from AMF peak to its adjacent flood peaks are greater than seven days. (2) By analyzing 30 typical AMF events, it can be found that the daily increase at the starting point and the decrease at the ending point of the AMF event are both about 1000 (m 3 s −1 d −1 ) mostly. Therefore, this study sets the two thresholds as 1000 (m 3 s −1 d −1 ) for flood event extraction. By comparing the flood event extraction results to the actual flood history records, it is proved that the threshold values setting as 1000 (m 3 s −1 d −1 ) are reasonable.

3
The positions of the AMF event characteristics in an annual daily discharge diagram are shown in Fig. 2.

Flood hydrograph coincidence model
Recent research on flood hydrograph coincidence assumes that flood hydrograph coincidence occurs when the flood hydrographs from two stations overlap and the duration of the overlapping parts account for more than 1/2 of the flood duration of any station. The magnitudes of the two floods are represented by flood volumes (Yan et al. 2013;Huang et al. 2018). Assuming that there are two stations A and B, their flood volumes, flood durations and flood occurrence times are denoted by V 1 , V 2 , d 1 , d 2 , T 1 , and T 2 , whereas the interval time between the two stations is denoted by T d where: The definition of flood hydrograph coincidence is schematically shown in Fig. 3 for two conditions: (1) (1) When the flood duration of station A is longer than or equal to station B. In this case, , the duration of the overlapping part of the two floods accounts for more than 1/2 of the duration of the flood in station B, and a flood hydrograph coincidence occurs (Fig. 3a); (2) When the flood duration of station A is shorter than station B. In this case, if , the duration of the overlapping part of the two floods accounts for more than 1/2 of the duration of the flood in station A, and a flood hydrograph coincidence occurs (Fig. 3b).
Put in the form of Eq. (1), a flood hydrograph coincidence happens when: Therefore, when studying the hydrograph coincidence risk probability of two floods greater than specified magnitudes, it is necessary to calculate the probability (P fc ) that the flood volumes of the two stations are greater than the design values, and their interval time meets the above definition of flood hydrograph coincidence. This probability is expressed as: 2 are the design flood volumes of station A and B, respectively, and T d and T d are the upper and lower limits of the interval time (T d ), respectively. In light of Eq. (3), it is necessary to establish the five-dimensional joint distribution of the variables, V 1 , V 2 , d 1 , d 2 , and T d , for the flood hydrograph coincidence analysis. The upper and lower limits of T d are related to d 1 and d 2 at the same time. The form is complex, and it is difficult to simplify directly. In this paper, the idea of stochastic simulation is adopted to calculate the probability (Roo et al. 1992). According to this approach, you get M sets of simulation values associated with the joint distribution, and the value of P fc ' is: where m is the number of the data satisfying Eq. (3). We can assume that P � fc ≈ P fc , when M is large enough (generally greater than 5000).

Marginal distribution model
The Chinese MWR 2006 advocates the P-III distribution as the unified model of flood frequency analysis. The duration and interval time of the two stations' AMFs can be regarded as vectors with periodic changes. Thus, they can be described by the mixed von Mises distribution (Chen et al. 2012;Yan et al. 2013;Huang et al. 2018;Peng et al. 2019).
The P-III curve is an asymmetric and unimodally skewed curve. Its probability density function is expressed as: where Γ( ) is the gamma function of , and , , a 0 are parameters of the P-III curve that can be calculated as: where x is the average value; C v is the coefficient of variation; and C s is the coefficient of skewness. The curve-fitting method is often used to estimate the parameters of the P-III curve (Geyer 1940).
The von Mises distribution is known as the normal distribution on a circle; it is an important model to describe directional data (Fisher 1993). The probability density function of the unimodal von Mises distribution is: where is the mean direction; is the concentration parameter; and I 0 ( ) is the 0-order modified Bessel function of the first kind calculated by: The probability density function of the mixed von Mises distribution is: where G = 1 , 2 , … , p ; 1 , 2 , … , p ; 1 , 2 , … , p ; p is the mean direction; and is the mixing proportion. The maximum likelihood method is often used to estimate the mixed von Mises distribution parameters (Spurr and Koutbeiy 1991).

Copula functions
Copula a tool that connects the joint distribution with its marginal distributions (Sklar 1959). The traditional copula functions used in the construction of a multidimensional joint distribution are mainly multidimensional elliptic copula and Archimedean copula, but their structures are relatively fixed, and they require the same correlation structure among variables. Therefore, they cannot accurately describe the dependencies of higher-dimensional variables (Aas et al. 2006;Schepsmeier and Czado 2016;Yu et al. 2019;Jane et al. 2020;Tosunoglu et al. 2020;Wu et al. 2020;).
The vine copula was first proposed by Joe (1996). In comparison with two previous forms, the structure of vine copula has been greatly improved. Vine copula allow a combination of d(d − 1)∕2 specified copulas, and its principle is to decompose a multivariate probability density function into d(d − 1)∕2 two-dimensional copula density functions. There are three main types of vine copula: C-vine, D-vine and R-vine. Among them, the structure of the R-vine copula is relatively flexible and there are many forms. C-vine and D-vine are special forms of R-vine with fixed structures (Bedford and Cooke 2001;Cooke 2002;Cooke and Kurowicka 2006;Aas et al. 2006).

Goodness-of-fit of distributions
In order to quantitatively evaluate the fitting error and select the appropriate copula function, the root mean square error (RMSE) and the Akaike information criterion (AIC) were used: where m is the number of model parameters, n is the number of samples, P i represents the copula value of consecutive sample observations, and Pe i represents the corresponding multivariate empirical probability. AIC is a measure of the quality of the statistical model's fit to the data. For a particular copula function, the smaller the AIC value of the objective function, the better the copula function simulation. RMSE was defined as: where y i ( ) is the theoretical joint probability value, ỹ i represents the empirical observation, i is the serial number of the sample, and n is the total number of the samples. The range of RMSE is [0, ∞] ; RMSE is equal to 0 for a perfect model.

Marginal distributions
Advantages of using a copula function to establish the multidimensional joint distribution is that the marginal distribution can work in many forms, and the marginal distribution and joint distribution can be considered separately. The flood hydrograph coincidence model proposed in this paper includes five variables: V 1 , V 2 , d 1 , d 2 , and T d . The P-III distribution was used to establish the marginal distributions of AMF volumes at the Yichang and Chenglingji stations, whereas the mixed von Mises distribution was used to establish the marginal distributions of the AMF durations and their interval time at the Yichang and Chenglingji stations. The curve-fitting method and maximum likelihood method were (10) n used to estimate the parameters of the P-III distribution and mixed von Mises distribution, respectively. The distributions were subsequently analyzed using distribution fitting tests.
Results of the estimated parameters, along with the results of the Kolmogorov-Smirnov (K-S) and Chi-square tests of the P-III distributions of the AMF volumes, are shown in Table 1. When the significance level is 5%, the Chi-square test results and K-S test results are both less than the critical values, and the distributions pass the tests. The AMF volumes could be effectively described by the P-III distribution. The frequency curves provided in Fig. 4 show that the empirical data points fit the theoretical curves well. Table 2 provides the results of the estimated parameters, the K-S test results and the Chisquare test results of the mixed von Mises distributions of the AMF durations and interval times. When the significance level is 5%, the Chi-square test results and K-S test results are both less than the critical values, and the distributions pass the tests. We therefore assume that the AMF durations and interval times could be effectively described by the mixed von Mises distribution. The probability density fitting diagrams are shown in Fig. 5 and Fig. 6. It can be seen that the theoretical curves in Fig. 5 a, b, c and Fig. 6 fit well with the empirical histogram, which is reflected in the number of peaks, the location of peaks and the magnitude of peaks of them all fit well. There is a little discrepancy between the peak height of the theoretical curve and the empirical histogram in Fig. 5 d. This may be due to the fact that the measured sequence is too short and the distribution characteristics of the variable are not obvious. In general, the overall trend is consistent and has passed the hypothesis test of distribution fitting, so it can be accepted to be reasonable. After the completion and operation of the Three Gorges Project, the maximum peak value of the probability density of the AMF interval time between Yichang Station and Chenglingji Station changed from interval (0, 25] to interval (25,50], which explains the flood regulation and storage function of the Three Gorges Project in the basin and its beneficial impact on reducing the probability of flood coincidence risk to a certain extent.

Joint distributions
Vine copula parameter estimation and preferential selection were carried out using a routine in the R software package. The vine copula parameter estimation and preferential selection results were used for establishing the five-dimensional joint distributions in this study. The parameters of the R software package were set as follows: (1) The optional two-dimensional copula functions were all types of copula in the package, which included 37 two-dimensional copula functions in total; (2) The alternative joint distribution structures were C-vine, D-vine and R-vine copula; (3) AIC was the evaluation standard to select suitable two-dimensional copula functions; (4) The confidence level of independent hypothesis testing was set at 0.05; (5) The correlation between variables was expressed by the Kendall correlation coefficient. Table 3 shows the vine copula structures of the five-dimensional joint distributions of AMF hydrograph coincidence for the two study periods. After the establishment of the joint distributions, the accuracy and reliability of the joint distributions were tested. In this paper, 5000 sets of five-dimensional flood variables were simulated, and the scatter diagrams between the two variables of the simulated sequence and the historical sequence were compared (Figs. 7, 8). The comparison shows that the simulated sequence scatter diagrams not only cover almost all the historical sequence scatter diagrams, but also fully retain the shape and trend characteristics of the historical sequence scatter diagrams. Therefore, the simulated sequences retained the natural characteristics of the historical sequences, and the vine copula structure established above fully reflect the structural characteristics among the variables of the historical sequences. The established five-dimensional joint distributions in this study are therefore considered accurate and reliable and can be used for the risk probability analysis of AMF hydrograph coincidence.

Risk probability analysis of AMF hydrograph coincidence
As mentioned above, flood hydrograph coincidence refers to the overlap between flood hydrographs, and the duration of the overlapping parts account for more than 1/2 of the flood duration of any station. Equation (3) specifies the method to calculate the flood hydrograph coincidence risk probability. The stochastic simulation method mentioned above was used to calculate AMF hydrograph coincidence probability at the Yichang and Chenglingji stations. P ′ fc was calculated using the five joint distributions based on vine copula and the stochastic simulation of 100,000 sets of values of the five-dimensional flood variables. We assumed that P � fc ≈ P fc due to the large number of simulated data. Figure 9 shows the hydrograph coincidence risk probability of the AMFs greater than specified magnitudes of the simulated and historical sequences (in terms of the return period, R 1 ≥ R 0 ; R 2 ≥ R 0 ). During Period 1, the AMF hydrograph coincidence risk probabilities for each return period combination of the simulated and historical sequence were very close; the absolute value of errors was all less than 0.025 (Fig. 9c). During Period 2, the AMF hydrograph coincidence risk probabilities with a return period of 1-year were very similar between the simulated sequence and the historical sequence. Due to the short historical timeframe, there was no AMF hydrograph coincidence with a return period combination of more than 2-years, and the risk probability values corresponding to the simulated sequence were very small (less than 1/14). Therefore, the AMF hydrograph coincidence risk probability calculated by the simulated sequence was basically the same as that of the historical sequence; thus, this method can be used to calculate the AMF hydrograph coincidence risk probability with extremely large magnitudes.
In the AMF hydrograph coincidence risk probability analysis, six (6) return periods of the AMF were chosen at each station, and the hydrograph coincidence risk probability with 36 combinations (R 1 ≥ R 01 , R 2 ≥ R 02 ) was calculated (Table 4), along with the AMF hydrograph coincidence volume (Table 5). Subsequently, the calculation results of the AMF hydrograph coincidence risk probability and volume were used to develop the hydrograph coincidence volume-risk probability curves (Fig. 10).
The flood hydrograph coincidence volume-risk probability curves in Fig. 10 show that the AMF hydrograph coincidence risk probability of the upper Yangtze River and Dongting Lake decreased with an increase in the hydrograph coincidence volume. For example, during the 1951-2002 period, the hydrograph coincidence probabilities corresponding to hydrograph coincidence volumes of 2.00 × 10 11 m 3 , 4.00 × 10 11 m 3 and 6.00 × 10 11 m 3 were 0.213, 0.123, and 0.049, respectively. During the 2003-2016 period, the hydrograph coincidence probabilities corresponding to hydrograph coincidence volumes of 2.00 × 10 11 m 3 , 4.00 × 10 11 m 3 and 6.00 × 10 11 m 3 were 0.072, 0.028 and 0.005, respectively. At the same time, a comparison of the flood hydrograph coincidence volume-risk probability curves in Fig. 10 show that the AMF hydrograph coincidence risk probabilities of the upper Yangtze River and Dongting Lake were greatly reduced by operation of the Three Gorges Project. For example, from 1951 to 2002, the hydrograph coincidence probabilities corresponding to hydrograph coincidence volumes of 2.00 × 10 11 m 3 , 4.00 × 10 11 m 3 and 6.00 × 10 11 m 3 were 0.213, 0.123, and 0.049, respectively, whereas during the 2003-2016 period, these values decreased to 0.072, 0.028 and 0.005, respectively.

Fig. 10
The AMF hydrograph coincidence volume-risk probability curves. (Excel XL Toolbox was used to create this artwork) coincidence, the AMF event was extracted to determine the actual AMF hydrograph coincidence in this study.
(2) In view of the fact that flood duration is usually neglected in the analysis of the flood hydrograph coincidence in a basin, this study proposed a method for flood hydrograph coincidence risk probability analysis by establishing a five-dimensional joint distribution of the flood volumes, durations and interval time at two stations. This model takes the flood factors into account and was able to present the actual flood coincidence more comprehensively. Therefore, the risk probability analysis in this study can describe the hydrological process of flood event coincidence more reasonable and therefore it can provide useful support tool for flood control planning. (3) In recent studies of flood event coincidence, traditional copulas were used to establish the joint distribution of flood peaks among stations, and to calculate the risk probability of flood peak coincidence. However, when there are certain differences in correlation structures among variables, the use of traditional copulas to establish high-dimensional joint distributions has limitations. In this paper, a vine copula was used to establish the five-dimensional joint distributions of the flood variables. It was then shown using historical sequences that the joint distributions established in this study were reliable. Therefore, when the correlation structures among hydrological variables under study are not identical and the joint distribution dimension is high, the vine copula function is recommended. (4) The results of this study show that the regulation of the Three Gorges Project during the flood season reduced the flood hydrograph coincidence risk probability and alleviated the flood control pressure in the middle and lower Yangtze River. At the same time, the flood hydrograph coincidence model developed in this study can more comprehensively analyze the flood event coincidence risk probability of the main stem of the Yangtze River and its tributaries, which provides a useful reference for the Three Gorges Project operation in terms of flood mitigation.

Conclusions
This study proposed a method to analyze the flood hydrograph coincidence risk probability using a five-dimensional joint distribution of flood volumes, durations and interval time for two hydrologic stations. The upper Yangtze River and Dongting Lake flood hydrograph coincidence was selected as a case study. The risk probabilities and volumes of flood hydrograph coincidence with different return period combinations were obtained. Flood duration was used in this method based on the flood hydrograph, and result verification showed that the risk analysis was reasonable and consistent with actual situations. Finally, by analyzing the flood hydrograph coincidence risk probability of the upper Yangtze River and Dongting Lake before and after operation of the Three Gorges Project, the positive effect of the Three Gorges Project on flood risk mitigation was demonstrated. For example, during the 1951-2002 and 2003-2016 period (before and after the Three Gorges dam operation), the hydrograph coincidence probabilities corresponding to its flood event coincidence volume of 2.00 × 10 11 m 3 are 0.213 and 0.072, respectively. In future work, a flood disaster model will be set up and combined with this study to further quantify flood risk, thereby providing a more intuitive theoretical basis for regional flood control and disaster reduction.