# Copula-Based Multivariate Simulation Approach for Flood Risk Transfer of Multi-Reservoirs in the Weihe River, China

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Copula Method

#### 2.2. Common Copula Functions

#### 2.3. Pair-Copula Method

- 1.
- Step 1: The first layer pair-copula sequence is constructed using bivariate copula functions of one random variable with other random variables as follows:

- 2.
- Step 2: The second layer pair-copula sequence is constructed from distribution functions of Step 1 as new random variable sequences as follows:

- 3.
- Step 3: Step 2 is repeated until the last bivariate copula is obtained:

- 4.
- Finally, the joint density function of x
_{1}, x_{2}, …, x_{n}can be described as:

_{t}is the probabilistic marginal distribution function of x

_{j}.

#### 2.4. Correlation Analysis Method

#### 2.4.1. Kendall’s (K-) Plots

#### 2.4.2. Chi-Plot

#### 2.5. Verification Method

## 3. Case Study

#### 3.1. Overview of the Weihe River

^{2}. In this study, three reservoir sites comprising a multi-reservoirs system in the city of Weinan were considered: Xianyang, Huaxian County, and Zhangjiashan. The flood control standard of the urban and agricultural sections of Weinan city is once-in-a-century and once-in-half-a-century, respectively.

^{3}/s during 1960–2010. The related data of these cases were obtained through analyses of multiple representative reports on the regional watershed. The flood volume capacity of the Huaxian County site has sharply decreased, and its flood level has increased in recent years. This is primarily because the lower reaches of the Weihe River experienced severe sediment accumulation, and the riverbed has risen continuously since the construction of the Sanmenxia reservoir. For instance, the water level of the Huaxian County site in 2000 (1890 m

^{3}/s) was 0.27 m higher than in 1981 (5380 m

^{3}/s). At the Huaxian County site, the record level (occurring in 2003) was 0.15% higher than its second-highest level (occurring in 1996). Further, the historic peak water levels of the Xianyang and Huaxian County reservoirs were 5340 and 3570 m

^{3}/s, respectively. Therefore, the flood control situation is dire.

#### 3.2. Multi-Reservoir Joint Distribution and Data Collection

## 4. Results and Discussion

^{3}/s in the 50-year, 40-year, 30-year, 20-year, 16-year, 11-year, 5-year, and 3-year return periods, respectively, in Huaxian County. Therefore, Zhangjiashan would face high flood risks from Huaxian County to maintain the low flood risk of Xianyang. Moreover, a small probability event at Zhangjiashan could occur when Xianyang has a low flood risk. However, Huaxian County was found to be at a high flood risk. Zhangjiashan would only have a low flood risk when Xianyang and Huaxian Country also show low flood risks.

^{3}/s in the 50-year, 40-year, 30-year, 20-year, 16-year, 11-year, 5-year, and 3-year return periods, respectively, at Huaxian County. This result indicates that Zhangjiashan would not share risks when the flood risk of Xianyang is lower than the low flood risk of Huaxian County. Therefore, the flood risk of Xianyang has little influence on the flood risk of Zhangjiashan. For example, the flood peak volume of Zhangjiashan was 2909, 3066, and 3540 m

^{3}/s for the 20-year return period at Huaxian County in the 10-year, 50-year, and 10-year return periods, respectively, at Xianyang. These results correspond to the relative geographic positions of Xianyang and Zhangjiashan.

^{9}, 1.24 × 10

^{9}, 1.12 × 10

^{9}, 1.12 × 10

^{9}, 1.12 × 10

^{9}, 0.93 × 10

^{9}, 0.71 × 10

^{9}, and 0.71 × 10

^{9}m

^{3}in the 50-year, 40-year, 30-year, 20-year, 16-year, 11-year, 5-year, and 3-year return periods, respectively, at Huaxian County. Moreover, the 1-day, 3-day, and 5-day flood volumes would be similar to the flood peak volume of Zhangjiashan. In contrast, the 9-day and 12-day flood volumes showed fewer variations with different risk levels at the Huaxian County site. Therefore, the long-term cumulative flood volume was strongly related to other regions that shared similar risks.

## 5. Conclusions

^{3}during the 10-year return period at Xianyang and the 5-year return period at Huaxian County. The complex relationship among the flood volumes of the three reservoirs was also analysed. The study results provide constructive suggestions for the Zhangjiashan reservoir construction project. Furthermore, the flood risk distribution of multi-reservoirs using a copula-based approach can provide new insights into reservoir joint disposal and risk control. However, there are many factors and indicators that pertain to reservoir construction (e.g., sediment amount, regional rainfall). Integrating these factors and indicators into a copula-based approach presents an interesting potential avenue for future research.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 12.**Flood probability of Zhangjiashan under different return periods of Xianyang and Huaxian County.

**Figure 13.**Flood peak volume of Zhangjiashan under different return periods of Xianyang and Huaxian County.

**Figure 14.**Flood volume of Zhangjiashan under different return periods of Xianyang and Huaxian County.

Copula | Generating Function $(\mathit{\phi}(\mathit{\theta})=)$ | ${\mathit{C}}_{}^{\mathit{A}\mathit{c}\mathit{r}}(\mathit{u},\mathit{v})$ | Parameter $(\mathit{\theta}\in )$ |
---|---|---|---|

Gumbel copula | ${(-\mathrm{ln}t)}^{1/\theta}$ | $C(u,v)={e}^{-{[{(-\mathrm{ln}u)}^{1/\theta}+{(-\mathrm{ln}v)}^{1/\theta}]}^{\theta}}$ | [1, ∞) |

Clayton copula | ${t}^{-\theta}-1$ | $C(u,v)={({u}^{-\theta}+{v}^{-\theta}-1)}^{-1/\theta}$ | (0, ∞) |

Frank copula | $-\mathrm{ln}\frac{{e}^{-\theta t}-1}{{e}^{-\theta t}-1}$ | $C(u,v)=-\frac{1}{\theta}\mathrm{ln}\left[1+\frac{({e}^{-\theta u}-1)({e}^{-\theta v}-1)}{{e}^{-\theta}-1}\right]$ | R |

**Table 2.**Selected copulas for multi-reservoir joint flood risk distribution with minimum AIC and BIC.

Multi-Reservoirs | Chosen Copula | AIC | BIC | |
---|---|---|---|---|

1-day volume | 1,2 | Gaussian | −4372.627 | −4356.78 |

2,3 | Gaussian | −13,322.1 | −13,314.2 | |

1,3:2 | Clayton | −40,068.585 | −40,060.7 | |

Full | −10,036.55 | −10,023.8 | ||

3-day volume | 1,2 | Gaussian | −5542.449 | −5526.6 |

2,3 | Clayton | −13,780.09 | −13,772.2 | |

1,3:2 | Gaussian | −43,701.22 | −43,693.3 | |

Full | −11,746.95 | −11,734.2 | ||

5-day volume | 1,2 | Gaussian | −6262.168 | −6246.32 |

2,3 | Gaussian | −14,147.22 | −14,139.3 | |

1,3:2 | Rotated Gumbel | −45,910.489 | −45,902.6 | |

Full | −12,766.7 | −12,754 | ||

9-day volume | 1,2 | Gaussian | −7023.314 | −7007.46 |

2,3 | Gaussian | −14,858.63 | −14,850.7 | |

1,3:2 | Frank | −48,641.955 | −48,634 | |

Full | −12,603.55 | −12,590.8 | ||

12-day volume | 1,2 | Gaussian | −7376.352 | −7360.5 |

2,3 | Rotated Joe | −15,311.32 | −15,303.4 | |

1,3:2 | Frank | −49,965.467 | −49,957.5 | |

Full | −303,895.2 | −303,883 | ||

Flood peak volume | 1,2 | Clayton | −4372.627 | −4356.78 |

2,3 | Gaussian | −13,322.1 | −13,314.2 | |

1,3:2 | Clayton | −40,068.585 | −40,060.7 | |

Full | −3546.92 | −3534.21 |

Return Period of Xianyang Site | ||||||||
---|---|---|---|---|---|---|---|---|

50 | 40 | 30 | 20 | 16 | 11 | 5 | 3 | |

Flood peak volume | 0.242 | 0.265 | 0.315 | 0.351 | 0.235 | 0.138 | 0.175 | 0.108 |

1-day flood volumes | 0.012 | 0.090 | 0.359 | 0.321 | 0.283 | 0.168 | 0.051 | 0.012 |

3-day flood volumes | 0.059 | 0.164 | 0.253 | 0.278 | 0.130 | 0.059 | 0.095 | 0.021 |

5-day flood volumes | 0.012 | 0.090 | 0.359 | 0.321 | 0.283 | 0.168 | 0.051 | 0.012 |

9-day flood volumes | 0.169 | 0.106 | 0.042 | 0.029 | 0.016 | 0.010 | 0.029 | 0.042 |

12-day flood volumes | 0.160 | 0.097 | 0.033 | 0.021 | 0.008 | 0.002 | 0.021 | 0.033 |

**Table 4.**Flood peak volume of Zhangjiashan under different return periods of Xianyang and Huaxian County.

Return Period | Flood Peak Volume (m^{3}/s) | |
---|---|---|

Xianyang | Huaxian County | Zhangjiashan |

10 | 50 | 4915 |

40 | 4325 | |

30 | 3641 | |

20 | 2909 | |

16 | 2536 | |

11 | 2144 | |

5 | 1856 | |

3 | 1856 | |

50 | 50 | 3211 |

40 | 3110 | |

30 | 3066 | |

20 | 3066 | |

16 | 3066 | |

11 | 3066 | |

5 | 3066 | |

3 | 3066 | |

100 | 50 | 3583 |

40 | 3558 | |

30 | 3547 | |

20 | 3540 | |

16 | 3540 | |

11 | 3540 | |

5 | 3540 | |

3 | 3540 |

**Table 5.**Flood volume of Zhangjiashan under different return periods of Xianyang and Huaxian County.

Return Period | Flood Volume (m^{3}) | |||||
---|---|---|---|---|---|---|

Xianyang | Huaxian County | Zhangjiashan Site | ||||

10 | 50 | 15,080 | 17,710 | 25,200 | 29,910 | 35,970 |

40 | 14,330 | 17,200 | 25,200 | 29,910 | 35,970 | |

30 | 12,930 | 16,514 | 24,484 | 29,910 | 35,970 | |

20 | 12,930 | 15,410 | 23,027 | 29,910 | 35,970 | |

16 | 12,930 | 14,756 | 21,530 | 29,910 | 35,970 | |

11 | 10,807 | 13,410 | 18,390 | 29,910 | 35,970 | |

5 | 8,200 | 11,230 | 17,746 | 29,910 | 24,176 | |

3 | 8,200 | 10,364 | 15,727 | 29,910 | 24,176 | |

50 | 50 | 29,190 | 30,000 | 35,300 | 38,124 | 49,598 |

40 | 25,710 | 29,250 | 32,388 | 37,124 | 49,307 | |

30 | 23,835 | 26,353 | 30,260 | 36,208 | 49,307 | |

20 | 23,835 | 25,684 | 28,480 | 35,290 | 49,307 | |

16 | 23,835 | 25,684 | 28,480 | 35,290 | 49,307 | |

11 | 23,835 | 25,684 | 28,480 | 35,290 | 49,307 | |

5 | 23,835 | 25,684 | 28,480 | 35,290 | 49,307 | |

3 | 23,835 | 25,684 | 28,480 | 35,290 | 49,307 |

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**MDPI and ACS Style**

Wang, S.; Wu, J.; Wang, S.; Xie, X.; Fan, Y.; Lv, L.; Huang, G. Copula-Based Multivariate Simulation Approach for Flood Risk Transfer of Multi-Reservoirs in the Weihe River, China. *Water* **2022**, *14*, 2676.
https://doi.org/10.3390/w14172676

**AMA Style**

Wang S, Wu J, Wang S, Xie X, Fan Y, Lv L, Huang G. Copula-Based Multivariate Simulation Approach for Flood Risk Transfer of Multi-Reservoirs in the Weihe River, China. *Water*. 2022; 14(17):2676.
https://doi.org/10.3390/w14172676

**Chicago/Turabian Style**

Wang, Shen, Jing Wu, Siyi Wang, Xuesong Xie, Yurui Fan, Lianhong Lv, and Guohe Huang. 2022. "Copula-Based Multivariate Simulation Approach for Flood Risk Transfer of Multi-Reservoirs in the Weihe River, China" *Water* 14, no. 17: 2676.
https://doi.org/10.3390/w14172676