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Analysis of runoff and sediment characteristics of Datong Station based on Copula function
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摘要: 长江口受人类活动影响较为强烈，其水沙序列一致性遭到破坏。以大通站1965—2019年逐月径流量和来沙量资料为基础，采用滑动窗口算法结合Copula理论的方法，研究长江口来水来沙变化特征及丰枯一致性问题，并分析水沙联合分布模型边缘函数选取不确定性问题。结果表明：（1）受人类活动的影响，大通站水沙组合在1965—2019年间均呈减少趋势，并且在1979和2000年发生突变，水沙序列1965—1979年和1980—2000年阶段最优函数模型均为Clayton模型，2001—2019年则为Frank模型；（2）大通水文站3个阶段水沙序列同步频率分别是84.52%、84.40%和83.20%，远大于丰枯异步频率，揭示了长江上游来水来沙条件具有较强的一致性；（3）PE3-PE3边缘分布组合的95%置信区间小于PE3-GPD组合，说明采用PE3-PE3组合可以减少函数选取的不确定性。通过探讨大通站来水来沙联合变化特征，可为长江口水域水资源管理、河道整治等工作提供参考。Abstract: The Yangtze River estuary has been strongly influenced by human activities, and the consistency of its runoff-sediment sequence has been damaged. Based on the monthly runoff and sediment data from 1965 to 2019 at Datong Station, a sliding window algorithm based on Copula is applied to investigate the variation characteristics of runoff and sediment in the Yangtze estuary and the consistency of its abundance-depletion, and the uncertainty of the marginal distribution selection for the joint runoff-sediment distribution model is also analyzed. The results show that: (1) The runoff-sediment combination at Datong Station shows a decreasing trend from 1965 to 2019 due to human activities, and changes abruptly in 1979 and 2000. The optimal function models for the runoff-sediment series in 1965—1979 and 1980—2000 are Clayton, while 2001—2019 is best modeled by Frank; (2) the synchronous frequencies of the runoff-sediment series in the three periods are 84.52%, 84.40% and 83.20% respectively, which are much greater than the asynchronous frequencies of abundance-depletion, revealing the strong consistency of the incoming runoff-sediment in the upper reaches of the Yangtze River; (3) the 95% confidence interval of the PE3-PE3 marginal distribution combination is smaller than that of the PE3-GPD combination, indicating that the PE3-PE3 combination would reduce the uncertainty of function selection. By exploring the joint variation characteristics of incoming runoff and sediment at Datong station, this study provides a reference for water resources management and river regulation in the Yangtze estuary.
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图 2 变化滑动窗口下大通站径流量与来沙量相关系数
Figure 2. Correlation coefficient of runoff and sediment discharge under the changing sliding window of Datong Station
图 3 不同阶段大通站径流量与来沙量关系
Figure 3. Relationship between runoff and sediment discharge in different periods of Datong Station
图 4 3个阶段水沙联合分布及联合重现期等值线
Figure 4. Joint runoff-sediment distribution and joint recurrence contours in three periods
图 5 3个阶段不同边缘分布组合对应的50%、75%和95%置信区间面积
Figure 5. Areas of 50%, 75%, and 95% confidence intervals for different combinations of marginal distributions in three periods
表 1 4种不同的Copula 函数
Table 1. Four different Copula functions
类型 函数表达式 Clayton $C\left(u,v;\theta \right)={({u}^{-\theta }+{v}^{-\theta }-1)}^{(-1/\theta )},\theta \in (0,\infty )$ Frank $C\left(u,v;\theta \right)=-\dfrac{1}{\theta }\mathrm{ln}\left\{1+\dfrac{\left[\mathrm{exp}\left(-\theta u\right)-1\right]\left[\mathrm{exp}\left(-\theta v\right)-1\right]}{\mathrm{exp}\left(-\theta \right)-1}\right\},\theta \in R$ G-H $C\left(u,v;\theta \right)=\mathrm{exp}\left\{ {-[{(-\mathrm{l}\mathrm{n}u)}^{\theta }+{(-\mathrm{l}\mathrm{n}v)}^{\theta }]}^{\frac{1}{\theta } }\right\},\theta \in [1,\infty ]$ Gaussian $C\left(u,v;\theta \right)={\displaystyle\int }_{-\infty }^{ {{\textit{ø}} }^{-1}\left(u\right)}{\displaystyle\int }_{-\infty }^{ {{\textit{ø}} }^{-1}\left(v\right)}\dfrac{1}{2{\text{π} } \sqrt{1-{\theta }^{2} } }{\rm{exp} }\left[\dfrac{ {x}_{1}^{2}-2\theta {x}_{1}{x}_{2}+{x}_{2}^{2} }{2\left(1-{\theta }^{2}\right)}\right]{\rm {d} }{x}_{1}{ {\rm {d} }x}_{2},\theta \in [-\mathrm{1,1}]$ 注：表中Copula函数参数θ的取值均采用Kendall秩相关系数方法求解。 
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表 2 水沙关系突变前后特征值分析
Table 2. Analysis of eigenvalues before and after the mutation of runoff-sediment relationship
设计重现期/a 月径流量/亿m3 月输沙量/万t 1965—1979年 1980—2000年 2001—2019年 1965—1979年 1980—2000年 2001—2019年 2 677.72 720.23 657.74 2652.03 2389.41 837.82 5 927.69 1048.81 986.92 6068.86 5241.05 1962.10 10 1082.74 1255.17 1201.09 8668.20 7215.87 2825.71 20 1224.00 1444.40 1401.03 11273.70 9046.30 3700.80 50 1397.71 1678.36 1651.86 14723.83 11262.24 4875.47 100 1522.38 1846.96 1834.63 17336.35 12796.82 5777.79
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表 3 水沙边缘分布选取
Table 3. Selection of marginal distribution of runoff and sediment
阶 段 变量 评价准则 PE3 GPD GLO 1965—1979年 径流 AIC −299.285 −249.523 −297.387 RMSE 0.428 0.492 0.431 来沙量 AIC −292.955 −280.809 −290.845 RMSE 0.436 0.451 0.438 1980—2000年 径流 AIC −444.970 −289.279 −417.867 RMSE 0.409 0.557 0.431 来沙量 AIC −405.640 −463.132 −445.233 RMSE 0.442 0.394 0.409 2001—2019年 径流 AIC −409.383 −252.618 −371.375 RMSE 0.402 0.567 0.437 来沙量 AIC −368.871 −463.444 −444.361 RMSE 0.440 0.357 0.373
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表 4 水沙联合模型选取
Table 4. Selection of runoff-sediment joint model
阶 段 Copula函数 参数θ 欧氏距离 1965—1979年 Clayton 8.151 0.0457 Frank 18.497 0.0692 G-H 5.076 0.0666 Gaussian 0.953 0.3157 1980—2000年 Clayton 8.066 0.0456 Frank 18.324 0.0573 G-H 5.033 0.0644 Gaussian 0.952 0.2516 2001—2019年 Clayton 7.066 0.0745 Frank 16.302 0.0526 G-H 4.533 0.0580 Gaussian 0.941 0.4963
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表 5 大通水文站水沙丰枯遭遇计算
Table 5. Calculation of flood and drought of runoff and sediment at Datong Hydrological Station
类型 组合 1965—1979年 1980—2000年 2001—2019年 水沙异步
频率/%X≥${x_{ {p_{\rm{f} } } } },\;{y_{ {p_{\rm{k} } } } }$<Y<${y_{ {p_{\rm{f} } } } } $ 5.70 5.74 4.20 X≥${x_{ {p_{\rm{f} } } } } $, Y≤${y_{ {p_{\rm{k} } } } } $ 0 0 0 ${x_{ {p_{\rm{k} } } } } $<X<${x_{ {p_{\rm{f} } } } } $, Y≥${y_{ {p_{\rm{f} } } } } $ 5.70 5.74 4.20 ${x_{ {p_{\rm{k} } } } } $<X<${x_{ {p_{\rm{f} } } } } $, Y≤${y_{ {p_{\rm{k} } } } } $ 2.04 2.06 4.20 X≤${x_{ {p_{\rm{k} } } } } $, Y≥$ {y_{ {p_{\rm{f} } } } } $ 0 0 0 X≤${x_{ {p_{\rm{k} } } } } $, ${y_{ {p_{\rm{k} } } } } $<Y<${y_{ {p_{\rm{f} } } } } $ 2.04 2.06 4.20 总计 15.48 15.60 16.80 水沙同步
频率/%X≥${x_{ {p_{\rm{f} } } } } $, Y≥$ {y_{ {p_{\rm{f} } } } } $ 19.30 19.26 20.80 ${x_{ {p_{\rm{k} } } } } $<X<${x_{ {p_{\rm{f} } } } } $, ${y_{ {p_{\rm{k} } } } } $<Y<$ {y_{ {p_{\rm{f} } } } } $ 42.26 42.20 41.60 X≤${x_{ {p_{\rm{k} } } } } $, Y≤${y_{ {p_{\rm{k} } } } } $ 22.96 22.94 20.80 总计 84.52 84.40 83.20
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表 6 边缘分布选取不确定性评价指标的95%置信区间取值
Table 6. Values of 95% confidence interval of uncertainty evaluation index selected by marginal distribution
阶 段 组合 参数θ 参数
变幅/%置信区间面积
S95% /(亿 m3·万 t)1965—1979年 PE3-PE3 [8.248, 8.379] 1.588 2.643×105 PE3-GPD [8.220, 8.346] 1.533 1.432×106 1980—2000年 PE3-PE3 [8.078, 8.183] 1.300 1.404×105 PE3-GPD [8.082, 8.187] 1.299 1.923×106 2001—2019年 PE3-PE3 [16.288, 16.434] 0.896 4.756×105 PE3-GPD [16.296, 16.442] 0.896 6.983×105
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