湖泊科学   2021, Vol. 33 Issue (3): 879-892.  DOI: 10.18307/2021.0320. 0

### 引用本文 [复制中英文]

[复制中文]
Luo Yun, Dong Zengchuan, Liu Yuhuan, Zhong Dunyu, Guan Xike, Zhou Qiang, Chen Xia, Yang Jie. Safety design for rivers-connected lake flood control based on Copula function: A case study of Lake Hongze. Journal of Lake Sciences, 2021, 33(3): 879-892. DOI: 10.18307/2021.0320.
[复制英文]

2020-05-10 收稿
2020-08-11 收修改稿

### 码上扫一扫

(1: 河海大学水文水资源学院, 南京 210098)
(2: 江苏省水利厅, 水土保持生态环境监测总站, 南京 210029)

Safety design for rivers-connected lake flood control based on Copula function: A case study of Lake Hongze
Luo Yun1 , Dong Zengchuan1 , Liu Yuhuan1 , Zhong Dunyu1 , Guan Xike1 , Zhou Qiang2 , Chen Xia2 , Yang Jie1
(1: College of Hydrology and Water Resource, Hohai University, Nanjing 210098, P. R. China)
(2: Soil and Water Conservation Ecological Environment Monitoring Station, Water Resources Department of Jiangsu Province, Nanjing 210029, P. R. China)
Abstract: As a kind of water storage unit, especially a large water-carrying lake, the lake is a typical plain reservoir. It has many similarities with the valley reservoir in function. However, due to its special geographical topography, the flood processes of the lake have a big difference compared with the valley reservoir. For valley reservoirs, the dam site flood which is the whole reservoir inflow flood is concerned in the flood control safety design. However, for lakes, more attention should be paid to the flood processes of each sub-region, which plays an important role in the lake area flood evolution. In this paper, according to the flood process characteristics of large water-carrying lakes, Copula function is used to construct multiple joint distribution functions, and the method that figures up the confidence interval of each sub-region flood process based on the whole flood process is proposed accordingly. Taking Lake Hongze as an example, the results show that: 1) If the joint return period is known, this method can determine the 95% confidence interval of the flood volume and flood peak of the whole flood process; 2) The rivers entering Lake Hongze were merged and clustered by runoff correlation analysis, which not only considers the natural hydrological and hydraulic connections between the rivers, but also avoids the problem of excessively high dimensions of the joint distribution function; 3) When the whole flood volume is confirmed, this method can determine the 95% confidence interval for the flood volume allocation. This method has a strong statistical and theoretical foundation and expands the application range of multivariable flood frequency analysis technology in the practice of water conservancy projects.
Keywords: Water-carrying lake    plain reservoir    Copula function    flood control safety    design flood    Lake Hongze

1 研究区与数据 1.1 流域概况

 图 1 洪泽湖地理位置及入湖河流和水文站点分布 Fig.1 Location of the Lake Hongze and its rivers and gauging stations
1.2 数据来源

2 方法 2.1 Copula函数的理论与基本方法

Copula函数是定义在[0, 1]均匀分布的多维联合分布函数，根据Sklar理论[38]，令H为一个n维分布函数，各变量的边缘分布为F1, F2, …, Fn，那么存在一个n-Copula函数C，使得对任意的x属于Rn，满足：

 $H\left(x_{1}, x_{2}, \cdots, x_{n}\right)=C_{\theta}\left(F_{1}\left(x_{1}\right), F_{2}\left(x_{2}\right), \cdots, F_{n}\left(x_{n}\right)\right)$ (1)
 $c_{\theta}\left(F_{1}\left(x_{1}\right), F_{2}\left(x_{2}\right), \cdots, F_{n}\left(x_{n}\right)\right)=\\\partial^{n} C_{\theta}\left(F_{1}\left(x_{1}\right), F_{2}\left(x_{2}\right), \cdots, F_{n}\left(x_{n}\right)\right) / \partial F_{1}\left(x_{1}\right) \partial F_{2}\left(x_{2}\right) \cdots \partial F_{n}\left(x_{n}\right)$ (2)

 $A I C=2 k+n \ln \left(\frac{1}{n-1} \sum\limits_{i=1}^{n}\left(P_{e i}-P_{i}\right)^{2}\right)$ (3)
 $B I C=\ln (n) k+n \ln \left(\frac{1}{n-1} \sum\limits_{i=1}^{n}\left(P_{e i}-P_{i}\right)^{2}\right)$ (4)
 $R M S E=\frac{1}{n} \sqrt{\sum\limits_{i=1}^{n}\left(P_{e i}-P_{i}\right)^{2}}$ (5)

2.2 入湖洪水的总峰和总量组合分析

2.2.1 基于Copula函数推求总峰和总量联合分布

 $F\left(q_{t}, w_{t}\right)=C_{\theta}\left(F_{Q_{t}}\left(q_{t}\right), F_{W_{t}}\left(w_{t}\right)\right)$ (6)

2.2.2 总峰和总量的联合重现期

 $P\left(q_{t}, w_{t}\right)=\{Q \geqslant q\} \cup\{W \geqslant w\}$ (7)
 $T\left(q_{t}, w_{t}\right)=1 /\left(1-P\left(q_{t}, w_{t}\right)\right)=1 /\left(1-F\left(q_{t}, w_{t}\right)\right)$ (8)

2.2.3 联合重现期下的总峰和总量置信区间估计

 $C_{\theta}\left(F_{Q_{t}}\left(q_{t}\right), F_{W_{t}}\left(w_{t}\right)\right)=1-1 / Z$ (9)
 $F_{W_{t}}\left(w_{t}\right)=F\left(F_{Q_{t}}\left(q_{t}\right), 1-1 / Z\right)$ (10)
 $w_{t}=F_{W_{t}}^{-1}\left(F\left(F_{Q_{t}}\left(q_{t}\right), 1-1 / Z\right)\right)$ (11)

 $f\left(q_{t}, w_{t}\right)=c\left(F_{Q_{t}}\left(q_{t}\right), F\left(F_{Q_{t}}\left(q_{t}\right), 1-1 / Z\right)\right) \cdot f_{Q_{t}}\left(q_{t}\right) \cdot f_{W_{t}}\left(F_{W_{t}}^{-1}\left(F\left(F_{Q_{t}}\left(q_{t}\right), 1-1 / Z\right)\right)\right)$ (12)
 $S_{Q_{t}}=\int \limits_{0}^{+\infty} c\left(F_{Q_{t}}\left(q_{t}\right), F\left(F_{Q_{t}}\left(q_{t}\right), 1-1 / Z\right)\right) \cdot f_{Q_{t}}\left(q_{t}\right) \cdot f_{W_{t}}\left(F_{W_{t}}^{-1}\left(F\left(F_{Q_{t}}\left(q_{t}\right), 1-1 / Z\right)\right)\right) \mathrm{d} q_{t}$ (13)
 $\varphi_{q}\left(q_{t}\right)=f\left(q_{t}, w_{t}\right) / S_{Q_{t}}$ (14)

2.3 基于径流相关性的地区聚类

2.4 入湖洪水总量的组成分析

2.4.1 基于Copula函数推求各分区洪量的联合分布

Wi表示各分区入湖洪水的洪量，对应的设计值分别为wi，其边缘分布函数分别为FWi(wi)，那么分区间洪量Wi的联合分布函数表示为：

 $F({w_1},{w_2}, \cdots ,{w_n}) = {C_\theta }({F_{{W_1}}}({w_1}),{F_{{W_2}}}({w_2}), \cdots ,{F_{{W_n}}}({w_n}))$ (15)

2.4.2 洪水总量确定下分区洪量分配的置信区间估计

 $f\left(w_{1}, w_{2}\right)=c\left(F_{W_{1}}\left(w_{1}\right), F\left(W_{\text {total }}-w_{1}, 1-1 / Z\right)\right) \cdot f_{W_{1}}\left(w_{1}\right) \cdot f_{W_{2}}\left(F_{W_{2}}^{-1}\left(F\left(W_{\text {total }}-w_{1}, 1-1 / Z\right)\right)\right)$ (16)
 $S_{w_{1}}=\int \limits_{0}^{W_{\text {total }}} c\left(F_{W_{1}}\left(w_{1}\right), F\left(W_{\text {total }}-w_{1}, 1-1 / Z\right)\right) \cdot f_{W_{1}}\left(w_{1}\right) \cdot f_{W_{2}}\left(F_{W_{2}}^{-1}\left(F\left(W_{\text {total }}-w_{1}, 1-1 / Z\right)\right)\right) \mathrm{d} w_{1}$ (17)
 $\varphi_{w}\left(w_{1}\right)=f\left(w_{1}, w_{2}\right) / S_{w_{1}}$ (18)

3 结果和分析 3.1 实例结果 3.1.1 基于径流相关性的地区聚类结果

 图 2 洪泽湖入湖河道的径流相关关系 Fig.2 The correlation results of runoff between rivers flowing into Lake Hongze
 图 3 洪泽湖与入湖洪水河流的拓扑关系 Fig.3 The topological relationship between Lake Hongze and flood rivers entering the lake
3.1.2 联合分布函数拟合结果

 图 4 联合分布函数P-P图 Fig.4 The probability-probability plots of joint distribution function

 $C\left(F_{Q_{t}}\left(q_{t}\right), F_{w_{t}}\left(w_{t}\right)\right)=\exp \left\{-\left[\left(-\ln F_{Q_{t}}\left(q_{t}\right)\right)^{5.32}+\left(-\ln F_{W_{t}}\left(w_{t}\right)\right)^{5.32}\right]^{1 / 5.32}\right\}$ (19)
 $C\left(F_{W_{1}}\left(w_{1}\right), F_{W_{2}}\left(w_{2}\right), F_{W_{3}}\left(w_{3}\right)\right)=\left[F_{W_{1}}\left(w_{1}\right)^{-2.39}+F_{W_{2}}\left(w_{2}\right)^{-2.39}+F_{W_{3}}\left(w_{3}\right)^{-2.39}-2\right]^{-1 / 2.39}$ (20)
3.1.3 入湖洪水的总峰和总量置信区间结果

 图 5 入湖洪水总峰与总量的95 % 置信区间 Fig.5 The 95 % confidence interval of the flood volume and flood peak of the whole flood process
3.1.4 入湖洪水总量分配方案结果

 图 6 分区洪量分配的95 % 置信区间：(a) 洪量263亿m3；(b) 洪量253亿m3 (空间上的每一个球形点代表一套洪量分配方案，球形点的颜色代表该方案发生的概率) Fig.6 The 95 % confidence interval of flood volume allocation: (a) 263 million m3; (b) 253 million m3

3.2 设计结果合理性分析

3.2.1 入湖洪水总峰与总量置信区间的合理性分析

 图 7 洪泽湖总入湖设计洪水过程线 Fig.7 The whole designed flood hydrograph of Lake Hongze

3.2.2 地区聚类结果的合理性分析

 图 8 各入湖河道年径流量箱线图 Fig.8 The box-plot of annual runoff for each river
3.2.3 分区洪量分配置信区间的合理性分析

 图 9 典型年法分区洪量分配方案：(a) 洪量263亿m3；(b) 洪量253亿m3 (红色区域为分区洪量分配方案的95 % 置信区间，黑色球形点代表采用典型年计算的洪量分配方案) Fig.9 The flood volume allocation schemes are computed by typical years method: (a) 263 million m3; (b) 253 million m3

4 结论与展望

1) 在根据洪峰和洪量等两变量分析入湖洪水频率时，对于给定的联合重现期水平，存在满足防洪标准的无穷多种峰、量组合，但并非所有的组合都符合水文事件的内在规律，只有在一定范围内的取值才是合理的；

2) 大型过水性湖泊入湖洪水入湖情况复杂，通过径流相关性分析对入湖河道合并聚类，形成分区入湖过程，既能够考虑河道间天然的水文、水力联系，也可以避免联合分布函数维度过高的问题；

3) 在入湖总洪量已知的情况下，基于分区洪水的洪量联合分布函数，推求分区洪量95 % 置信区间的方法具有一定的合理性，其可以有效避免传统洪水地区分配方法的随机性和不确定性.

5 参考文献