Archimedean Copulas函数在干旱分析中的应用
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摘要
基于Copulas函数理论,选取渭河流域90个气象站的月降水资料,应用Archimedean Copulas函数构建了干旱特征变量的联合分布,分析了该流域的干旱发生规律,对于渭河水资源合理配置,水利工程规划、设计和管理以及抗旱减灾具有重要的研究意义和指导作用。本文主要取得如下结论:
(1)通过边缘分布模型的优选和Kolmogorov-Smirnov(K-S)检验,渭河流域干旱特征变量的边缘分布类型分别为:干旱历时以服从指数分布,干旱烈度服从Gamma分布,烈度峰值服从广义Pareto分布为主。
(2)通过相关性度量分析,渭河流域各站的干旱烈度和烈度峰值以及干旱历时和干旱烈度之间存在着较高的相关性;干旱历时和烈度峰值之间的相关性较低。
(3)分别采用极大似然法和适线法对20种二维Archimedean Copulas函数和4种三维Archimedean Copulas函数进行了参数估计,并以RMSE、AIC和Bias为拟合优度评价指标,结果表明:适线法的参数估计效果优于极大似然法。
(4)优选了出对渭河流域适用的Archimedean Copula函数。对于干旱历时和干旱烈度的联合分布,Gumbel Copula函数在铜川、渭南、西安、咸阳等地区的的拟合效果最优,Frank Copula函数在宝鸡地区的拟合效果最优,对于整个渭河流域90个气象站而言,Gumbel Copula函数的适用性最好,拟合效果最优。对于干旱历时和烈度峰值的联合分布,Nelsen No 20 Copula函数在整个渭河流域的拟合效果最好,拟合效果次之的依次是Nelsen 13 Copula和Frank Copula函数。对于干旱烈度和烈度峰值的联合分布,Nelsen No 20 Copula函数对宝鸡和咸阳地区的拟合程度最优,但是,对于整个渭河流域而言,Clayton Copula函数的拟合效果最优。对于干旱历时、干旱烈度和烈度峰值之间的三维联合分布,Clayton Copula函数的拟合效果最优。
(5)通过优选出的Archimedean Copula函数分别建立了相应的干旱变量的联合分布模型,计算了二维情况和三维情况下的联合概率、联合重现期、同现重现期以及条件重现期,并绘制了相应的图形,从图上可查出不同组合条件下的概率和重现期,可以为系统分析渭河流域的干旱特征提供依据。
(6)对于二维分布情况,相同条件下的同现重现期大于联合重现期;对于三维分布情况,各干旱变量相同取值下的同现重现期大于联合重现期,给定某一干旱变量大于等于某确定值时的其它两个干旱变量的条件联合重现期大于给定这一干旱变量小于等于相同的确定值时的其它两个干旱变量的条件联合重现期。
(7)以多年月平均降水量为截取水平,绘制了渭河流域干旱变量条件概率,显示了典型组合条件下概率的空间分布规律。条件概率P(S≥100/D≥3),P(M≥50/D≥3)的总体空间分布特征为由东南向西北递增;条件概率P(S≥100/M≥50)的空间分布特征为以渭河支流泾河上游的合水、庆阳、西峰、镇原,以及甘肃的华亭、庄浪、张家川和宁县的泾源等地区为中心,条件发生概率向四周呈递增趋势,其中西北地区的增长趋势比东南地区的增长趋势明显。
Based on Copula theory, monthly precipitation data of 90 Gauging stations in Wei River Basin were selected. Archimedean Copulas were applied to construct the joint distribution of drought variables and analyzed the characteristics of drought occurrences in the basin, which has great significance and guidance to water resources rational allocation; the planning, design and management of water conservancy project, as well as to drought and disaster reductions. Main results in this study are as followed:
(1) Through the optimal selection of marginal distributions and Kolmogorov-Smirnov(K-S) test, the type of marginal distribution of drought variables in Wei River basin are followed: most of the drought durations are subjected to exponential distribution; most of the drought severities are subjected to Gamma distribution, while most of the severity peaks are subjected to generalized Pareto Distribution.
(2) Though correlation measurement, the correlation between drought duration and drought severity and that between drought severity and severity peak in Wei River Basin are high, while the correlation between drought duration and severity peak is low.
(3) Maximum likelihood method and fitting curve method were applied to the parametesr estimation to 20 kinds of two dimensional Archimedean Copulas and 4 kinds of three dimensional Archimedean Copulas respectively. Take RMSE, AIC and Bias as goodness-of-fit assessment index to make comparison, the results show that fitting curve method is better than maximum likelihood method.
(4) The most appropriate Archimedean Copulas for Wei River Basin were optimaly selected: for the joint distribution of drought duration and drought severity, Gumbel Copula shows the best result to TongChuan, WeiNan, Xi’an, XianYang and etc.,while Frank Copula shows the best result to BaoJi area, however, for the whole Wei River basin, Gumbel Copula gives the best applicability and shows the best result; for the joint distribution of drought duration and drought severity, Nelsen No 20 Copula shows the best result of goodness-of-fit to the whole basin, next to Nelsen No 20 Copula which show better results are Nelsen No 13 Copula and Frank Copula; for the joint distribution of drought severity and severity peak, Nelsen No 20 Copula shows best result to BaoJi and XianYang regions, but to the whole basin, Clayton copula shows the best result; for the three dimensional joint distribution of drought duration, drought severity and severity peak, Clayton Copula also shows the best result.
(5) The joint distribution model for drought variables were established by the optimal selected Archimedean Copulas, joint probability, joint return period, co-occurrence return period and conditional return period were calculated, corresponding figures were drawn out as well. From those figures, different probabilities and return periods can be given, which can provide basis for systemic analyzing of drought characteristics in Wei River Basin.
(6) For two dimensional distribution, co-occurrence return period is much greater than the joint return period under the same condition. For three dimensional distribution, the co-occurrence return period is much greater than the joint return period when the drought variables take the same value; the joint return period of two drought variables under the condition that other drought variable is greater than a certain value is longer than that under the condition of the other drought variable being smaller than the same certain value.
(7) Taking the monthly average rainfall in years as truncation level, draw the conditional probabilities of drought variables in Wei River Basin, which shows the spatial distribution characteristics of the probability under typical combinational conditions. The general spatial distribution characteristic of conditional probability P(S≥100/D≥3),P(M≥50/D≥3) is that it is increasing from southeast to northwest; for the conditional probability P(S≥100/M≥50), it takes HeShui, QingYang, XiFeng and ZhenYuan which lies in the upper stream of Jing River (which is a branch of Wei River) and HuaTing, ZhuangLang, ZhangJiaChuan in GanSu and JingYuan in NingXia as center, and it increases to the around, there into, the increasing tend in northwest area is much more significant than that in southeast area.
(1)通过边缘分布模型的优选和Kolmogorov-Smirnov(K-S)检验,渭河流域干旱特征变量的边缘分布类型分别为:干旱历时以服从指数分布,干旱烈度服从Gamma分布,烈度峰值服从广义Pareto分布为主。
(2)通过相关性度量分析,渭河流域各站的干旱烈度和烈度峰值以及干旱历时和干旱烈度之间存在着较高的相关性;干旱历时和烈度峰值之间的相关性较低。
(3)分别采用极大似然法和适线法对20种二维Archimedean Copulas函数和4种三维Archimedean Copulas函数进行了参数估计,并以RMSE、AIC和Bias为拟合优度评价指标,结果表明:适线法的参数估计效果优于极大似然法。
(4)优选了出对渭河流域适用的Archimedean Copula函数。对于干旱历时和干旱烈度的联合分布,Gumbel Copula函数在铜川、渭南、西安、咸阳等地区的的拟合效果最优,Frank Copula函数在宝鸡地区的拟合效果最优,对于整个渭河流域90个气象站而言,Gumbel Copula函数的适用性最好,拟合效果最优。对于干旱历时和烈度峰值的联合分布,Nelsen No 20 Copula函数在整个渭河流域的拟合效果最好,拟合效果次之的依次是Nelsen 13 Copula和Frank Copula函数。对于干旱烈度和烈度峰值的联合分布,Nelsen No 20 Copula函数对宝鸡和咸阳地区的拟合程度最优,但是,对于整个渭河流域而言,Clayton Copula函数的拟合效果最优。对于干旱历时、干旱烈度和烈度峰值之间的三维联合分布,Clayton Copula函数的拟合效果最优。
(5)通过优选出的Archimedean Copula函数分别建立了相应的干旱变量的联合分布模型,计算了二维情况和三维情况下的联合概率、联合重现期、同现重现期以及条件重现期,并绘制了相应的图形,从图上可查出不同组合条件下的概率和重现期,可以为系统分析渭河流域的干旱特征提供依据。
(6)对于二维分布情况,相同条件下的同现重现期大于联合重现期;对于三维分布情况,各干旱变量相同取值下的同现重现期大于联合重现期,给定某一干旱变量大于等于某确定值时的其它两个干旱变量的条件联合重现期大于给定这一干旱变量小于等于相同的确定值时的其它两个干旱变量的条件联合重现期。
(7)以多年月平均降水量为截取水平,绘制了渭河流域干旱变量条件概率,显示了典型组合条件下概率的空间分布规律。条件概率P(S≥100/D≥3),P(M≥50/D≥3)的总体空间分布特征为由东南向西北递增;条件概率P(S≥100/M≥50)的空间分布特征为以渭河支流泾河上游的合水、庆阳、西峰、镇原,以及甘肃的华亭、庄浪、张家川和宁县的泾源等地区为中心,条件发生概率向四周呈递增趋势,其中西北地区的增长趋势比东南地区的增长趋势明显。
Based on Copula theory, monthly precipitation data of 90 Gauging stations in Wei River Basin were selected. Archimedean Copulas were applied to construct the joint distribution of drought variables and analyzed the characteristics of drought occurrences in the basin, which has great significance and guidance to water resources rational allocation; the planning, design and management of water conservancy project, as well as to drought and disaster reductions. Main results in this study are as followed:
(1) Through the optimal selection of marginal distributions and Kolmogorov-Smirnov(K-S) test, the type of marginal distribution of drought variables in Wei River basin are followed: most of the drought durations are subjected to exponential distribution; most of the drought severities are subjected to Gamma distribution, while most of the severity peaks are subjected to generalized Pareto Distribution.
(2) Though correlation measurement, the correlation between drought duration and drought severity and that between drought severity and severity peak in Wei River Basin are high, while the correlation between drought duration and severity peak is low.
(3) Maximum likelihood method and fitting curve method were applied to the parametesr estimation to 20 kinds of two dimensional Archimedean Copulas and 4 kinds of three dimensional Archimedean Copulas respectively. Take RMSE, AIC and Bias as goodness-of-fit assessment index to make comparison, the results show that fitting curve method is better than maximum likelihood method.
(4) The most appropriate Archimedean Copulas for Wei River Basin were optimaly selected: for the joint distribution of drought duration and drought severity, Gumbel Copula shows the best result to TongChuan, WeiNan, Xi’an, XianYang and etc.,while Frank Copula shows the best result to BaoJi area, however, for the whole Wei River basin, Gumbel Copula gives the best applicability and shows the best result; for the joint distribution of drought duration and drought severity, Nelsen No 20 Copula shows the best result of goodness-of-fit to the whole basin, next to Nelsen No 20 Copula which show better results are Nelsen No 13 Copula and Frank Copula; for the joint distribution of drought severity and severity peak, Nelsen No 20 Copula shows best result to BaoJi and XianYang regions, but to the whole basin, Clayton copula shows the best result; for the three dimensional joint distribution of drought duration, drought severity and severity peak, Clayton Copula also shows the best result.
(5) The joint distribution model for drought variables were established by the optimal selected Archimedean Copulas, joint probability, joint return period, co-occurrence return period and conditional return period were calculated, corresponding figures were drawn out as well. From those figures, different probabilities and return periods can be given, which can provide basis for systemic analyzing of drought characteristics in Wei River Basin.
(6) For two dimensional distribution, co-occurrence return period is much greater than the joint return period under the same condition. For three dimensional distribution, the co-occurrence return period is much greater than the joint return period when the drought variables take the same value; the joint return period of two drought variables under the condition that other drought variable is greater than a certain value is longer than that under the condition of the other drought variable being smaller than the same certain value.
(7) Taking the monthly average rainfall in years as truncation level, draw the conditional probabilities of drought variables in Wei River Basin, which shows the spatial distribution characteristics of the probability under typical combinational conditions. The general spatial distribution characteristic of conditional probability P(S≥100/D≥3),P(M≥50/D≥3) is that it is increasing from southeast to northwest; for the conditional probability P(S≥100/M≥50), it takes HeShui, QingYang, XiFeng and ZhenYuan which lies in the upper stream of Jing River (which is a branch of Wei River) and HuaTing, ZhuangLang, ZhangJiaChuan in GanSu and JingYuan in NingXia as center, and it increases to the around, there into, the increasing tend in northwest area is much more significant than that in southeast area.
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