湖泊科学   2022, Vol. 34 Issue (4): 1319-1334.  DOI: 10.18307/2022.0423 0

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Li Zhen, Li Xianghu, Zhang Dan, Lin Yaling. Copula based hydrological drought probability analysis in the Lake Dongting-catchment-Yangtze River system. Journal of Lake Sciences, 2022, 34(4): 1319-1334. DOI: 10.18307/2022.0423
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2021-05-31 收稿
2021-11-17 收修改稿

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(1: 中国科学院南京地理与湖泊研究所, 中国科学院流域地理学重点实验室, 南京 210008)
(2: 中国科学院大学, 北京 100049)

Copula based hydrological drought probability analysis in the Lake Dongting-catchment-Yangtze River system
Li Zhen1,2 , Li Xianghu1,2 , Zhang Dan1 , Lin Yaling1,2
(1: Key Laboratory of Watershed Geographic Sciences, Nanjing Institute of Geography and Limnology, Chinese Academy of Sciences, Nanjing 210008, P.R.China)
(2: University of Chinese Academy of Sciences, Beijing 100049, P.R.China)
Abstract: Lake Dongting is an important river-connected lake with a complicated water system pattern in the middle reaches of the Yangtze River. Under the dual influence of climate change and human activities, the river-lake relationship has changed and extreme drought events have occurred frequently in recent years. Based on the measured data of the hydrological stations of Lake Dongting, the catchment of Lake Dongting and the tributary of the Yangtze River, this study identifies hydrological drought events through the standard water level index (SWI) and the standard runoff index (SRI). Copula function is employed to analyze the characteristics of joint probability distribution of concurrent hydrological drought in the lake-catchment-river system. The results revealed that there were 9 hydrological drought events in Lake Dongting and the probability of hydrological drought was 14.01% from 1964 to 2016. Meanwhile, the joint probability of concurrent hydrological drought in the Lake Dongting-catchment system and Lake Dongting-Yangtze River system were 9.65% and 8.58%, which indicates that the inflow from the basin has a greater impact on the hydrological drought of Lake Dongting on the annual scale. On the seasonal scale, the joint probability of hydrological drought in the spring of the Lake Dongting basin system is the highest, and the number of hydrological drought events occurring concurrently is the most, indicating that the hydrological drought in the spring of Lake Dongting is closely related to the decrease in the recharge to the basin. The Lake Dongting-Yangtze River system has the highest joint probability of hydrological drought in autumn, especially since 2003. On the one hand, it is affected by the reduction of autumn precipitation and human activities in the basin. On the other hand, it is also one of the important reasons that the autumn impoundment of the Three Gorges Reservoir reduces the water level of the middle and lower mainstream of the Yangtze River, which weakened the blocking effect of the Yangtze River.
Keywords: Hydrological drought    Copula    Lake Dongting-catchment-Yangtze River    standard water level index (SWI)    standard runoff index (SRI)

1 研究区域与方法 1.1 研究区域与数据资料

 图 1 洞庭湖-流域-长江系统水系和水文站、气象站分布 Fig.1 Locations of river network, hydrological stations and meteorological stations in the Lake Dongting-catchment-Yangtze River system
1.2 研究方法 1.2.1 标准化水位/径流指数(SWI/SRI)

1.2.2 边缘分布函数

1.2.3 Copula函数

Copula理论最初由Sklar[47]提出，目前已广泛应用于二元和多元干旱频率分析. Copula函数根据随机变量之间的依赖结构，连接一维边缘分布以形成概率区间在[0, 1]上的多元联合分布[48]. 在研究中常用的Copula函数分为4种类型：阿基米德型(Frank、Clayton、Gumbel)，椭圆型(t、Gaussian)，极值型(Husler-Reiss、t-EV)和混合型(Plackett)[32]. 本文采用Gumbel、Clayton、Frank、Gaussion、t和Plackett 6种Copula函数进行拟合，同时根据相关性指标法，建立起Kendall秩相关系数τ与Copula函数的参数θ之间的关系来进行参数估计[49].

 $F(x, y)=P(X \leqslant x, Y \leqslant y)=C_{\theta}\left(F_{X}(x), F_{Y}(y)\right)$ (1)

Kendall秩相关系数[50]计算公式如下：

 $\tau=\frac{2}{n(n-1)} \sum\limits_{i=1}^{n-1} \sum\limits_{j=i+1}^{n} \operatorname{sgn}\left[\left(x_{i}-x_{j}\right)\left(y_{i}-y_{j}\right)\right]$ (2)

 $T(X \leqslant x, Y \leqslant y)=\frac{s}{C\left(F_{X}(x), F_{Y}(y)\right)}$ (3)

1.2.4 边缘分布函数和Copula函数的优选

 $D=\max _{i j}\left|F\left(X_{(i)}\right)-G\left(Y_{(j)}\right)\right|$ (4)
 $MSE = \frac{1}{n}\sum\limits_{ij = 1}^n {{{\left[ {F\left( {{X_{(i)}}} \right) - G\left( {{Y_{(j)}}} \right)} \right]}^2}}$ (5)
 $A I C=2 k+n \ln (M S E)$ (6)

2 结果分析 2.1 边缘分布函数和Copula函数的选择 2.1.1 边缘分布函数的选择

2.1.2 Copula联合分布函数的选择

2.2 各区域水文干旱概率分布特征

 图 2 年尺度洞庭湖SWI概率分布(a)；流域SRI概率分布(b)；长江SRI概率分布(c) Fig.2 Annual scale probability distribution of the SWI of Lake Dongting(a); the probabilities of the SRI of the catchment(b) and Yangtze River(c)

 图 3 季节尺度(a~d)洞庭湖SWI概率分布(a~d)；流域SRI概率分布(e~h)；长江SRI概率分布(i~l) Fig.3 Seasonal scale probability distribution of the SWI of Lake Dongting (a-d); the probabilities of the SRI of the catchment(e-h) and Yangtze River(i-l)

2.3 洞庭湖-流域-长江系统水文干旱联合概率分布特征

 图 4 年尺度1964-2016年洞庭湖-流域系统水文干旱联合概率(a)；洞庭湖-长江系统水文干旱联合概率(b) Fig.4 Annual scale isoline of the joint probability distribution of the concurrent droughts between Lake Dongting and its catchment(a); the joint probability between Lake Dongting and the Yangtze River(b)

 图 5 季节尺度1964-2016年洞庭湖-流域系统水文干旱联合概率 (a~d分别表示春、夏、秋、冬季) Fig.5 Seasonal joint probability of concurrent drought between Lake Dongting and its catchment from 1964 to 2016 (a-d illustrate probabilities in spring, summer, autumn and winter, respectively)

 图 6 季节尺度上洞庭湖-长江系统水文干旱联合概率 (a~d分别表示春、夏、秋、冬季) Fig.6 Seasonal joint probability of concurrent drought between Lake Dongting and the Yangtze River from 1964 to 2016 (a-d illustrate probabilities in spring, summer, autumn and winter, respectively)
3 讨论

 图 7 洞庭湖流域SRI与降水量季节分布 (a~d分别表示春、夏、秋、冬季) Fig.7 Seasonal distribution of SRI and precipitation in Lake Dongting catchment (a-d illustrate probabilities in spring, summer, autumn and winter, respectively)

4 结论

1) 年尺度上，洞庭湖流域发生水文干旱的概率最高，但洞庭湖及长江在2003年以后发生水文干旱的频次明显较多，所占比重较大. 季节尺度上，洞庭湖和长江2003年之后秋季水文干旱加剧，洞庭湖流域春季水文干旱概率远高于其它季节.

2) 年尺度上, 洞庭湖-流域系统水文干旱联合概率大于洞庭湖-长江系统，但2003年后洞庭湖-长江系统联合水文干旱频次明显多于洞庭湖-流域系统. 季节尺度上，洞庭湖-流域系统春季水文干旱联合概率最高，且两者同时发生水文干旱事件的次数最多；而洞庭湖-长江系统，其秋季水文干旱联合概率最大，并且一半以上的水文干旱事件发生在2003年以后.

3) 洞庭湖春季水文干旱与流域的水文干旱具有较好的同步性，流域入湖补给减少对洞庭湖春季水文干旱的影响更大. 而洞庭湖-长江系统秋季水文干旱自2003年以后更加极端和频发，这一方面受秋季降水减少和流域内人类活动的影响，造成流域径流对湖泊的补给减弱，另一方面三峡水库秋季蓄水使长江中下游干流水位降低，长江对湖泊顶托作用减弱也是重要原因之一.

5 参考文献