基于Archimedean Copulas函数的多变量干旱频率及空间分析
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摘要
干旱在人类历史上从古至今一直影响和制约着经济与社会的发展。我国由于其特殊的地理、气候、水文特征情况，更是深受其害。水资源的短缺以及因此导致的用水量供需不平衡更是加剧了人与自然和谐相处的矛盾，对于工农业生产、经济建设和社会发展产生了极为不利的影响，造成局部地区生态环境极度脆弱。据此，对我国干旱半干旱地区的干旱特征进行理论分析就显得很有必要。本文以新疆地区和黑河流域为代表，通过对其月平均降水提取气象干旱特征变量以及月平均径流资料提取水文干旱特征变量，构建多维干旱变量的联合分布，并进行深度的干旱分析。相关研究对于探讨区域和流域的干旱演变规律具有十分重要的意义。
选取具有代表性的新疆41个气象站的月降水数据、黑河流域21个气象站的月降水数据提取气象干旱变量；选取黑河流域上游莺落峡、梨园站和正义峡三个水文站的月径流资料提取水文干旱变量，基于Archimedean Copulas函数建立了二维、三维气象和水文干旱变量的联合分布，经拟合优度评价，选取最佳的Copula函数用以刻画干旱演变规律，以期为区域、流域内旱灾防治和控制、用水量合理调配、水资源可持续发展以及水利建筑工程规划建设提供一定的理论支撑。主要取得以下结论：
（1）基于游程理论从新疆地区和黑河流域长系列的月平均降水以及黑河流域长系列的月平均径流资料中提取干旱历时、干旱烈度和干旱烈度峰值三个干旱特征变量。由相对离差平方和最小准则（WLS）判断三个干旱特征变量的分布类型。结果表明：新疆地区41个站点以及黑河流域21个站点的干旱历时服从指数分布；干旱烈度服从Weibull分布；烈度峰值服从广义Pareto分布。黑河上游莺落峡、梨园站和正义峡的干旱历时服从广义Pareto分布，干旱烈度服从Weibull分布，烈度峰值服从Gamma分布。经单变量边缘分布的Kolmogorov-Smirnov（K-S）拟合度检验，剔除新疆地区干旱特征变量拟合情况不佳的6个站点（哈密市，吐鲁番市，且末县，和田县和民丰县）；以及黑河流域拟合情况不佳的7个站点（马鬃山，玉门镇，永昌，阿拉善右旗，拖勒，刚察和门源）。
（2）干旱特征变量间必须具有一定的相关性方可采用Copulas函数构建联合分布。通过变量间相关性度量得出：新疆地区的干旱历时和干旱烈度之间的相关性最好，干旱烈度和烈度峰值之间的相关性次之，干旱历时和烈度峰值之间的相关性最弱；黑河流域21个气象站和三个水文站的干旱烈度和烈度峰值之间的相关性最好，干旱历时和干旱烈度之间的相关性次之，而干旱历时和烈度峰值之间的相关性最弱。
（3）对于新疆地区和黑河流域气象和水文干旱特征变量联合分布的Copulas函数参数估计：选用适线法和极大似然法对20种二维Archimedean Copulas函数和Plackett、Farlie-Gumbei-Morgenstern(FGM) Copulas函数以及4种对称的、5种非对称的三维Archimedean Copulas函数的参数进行估计，适线准则采取离差平方和最小准则。采用AIC准则、均方根误差（RMSE）准则以及Bias准则进行拟合度评价。拟合优度表明：对于新疆地区以及黑河流域整体来讲，适线法的Copulas函数的参数估计效果优于极大似然法，故采用前者进行Copulas函数的参数估计。
（4）经拟合优度评价优选最佳的Copulas函数。对于整个新疆地区的气象干旱而言：Frank Copula函数对干旱历时和干旱烈度、干旱历时和烈度峰值的二维联合分布的拟合度最好；Clayton Copula函数对于干旱烈度和烈度峰值的二维联合分布以及干旱历时、干旱烈度和烈度峰值的三维联合分布拟合效果最佳。对于黑河流域的气象干旱特征：FGM Copula函数对干旱历时和干旱烈度的拟合度最好；Nelsen No20Copula函数对于干旱历时和烈度峰值的拟合度最佳；而Nelsen No14Copula函数对于干旱烈度和烈度峰值的拟合效果最好。Clayton Copula函数对于干旱历时、干旱烈度和烈度峰值的三维联合分布拟合效果最佳。对于黑河莺落峡、梨园站和正义峡的水文干旱而言：FGM Copula函数对干旱历时和干旱烈度、干旱历时和烈度峰值的二维联合分布的拟合度最好；NelsenNo20Copula函数对干旱烈度和烈度峰值的拟合度最好。同样Clayton Copula函数对于干旱历时、干旱烈度和烈度峰值的三维联合分布拟合效果最好。
（5）基于优选的Copulas函数构建新疆地区和黑河流域的二、三维干旱特征变量的联合分布模型，计算干旱变量的联合概率及重现期。新疆地区和黑河流域的二、三维重现期分布具有大致类似的规律：单变量的重现期介于二维、三维变量联合重现期与同现重现期之间；二维、三维变量的同现重现期大于相应的联合重现期；三维变量的同现重现期大于相应的二维变量的同现重现期，而三维变量的联合重现期小于相应的二维变量的联合重现期。
（6）由二维、三维干旱特征变量的特定条件概率：P(S≤50|D≥4)、P(M≤10|D≥4)、P(S≤50|M≥10)以及P(D≤4∪S≤50∪M≤10)绘制新疆地区和黑河流域水文和气象干旱特征的空间分布图。多维干旱特征变量空间分布规律大体为，对于新疆地区：特定干旱事件发生概率从北到南递增，即新疆的南疆比北疆更易发生带有特定条件概率的干旱事件；对于黑河流域气象干旱分布特征：黑河下游比上游更容易发生特定条件概率的干旱事件。
Drought affactes and restricts the economic and social development a lot in humanhistory. Because of its special geography, climate and hydrology characteristics in China,drought has hurt coutries economic development seviouly. Especially in recent years, theshortage of water resources led to unbalance between supply and demand of waterconsumption, this increasingly aggravates the conflict between human and nature in harmonywith each other. Drought has very disadvantageous effects on industry, agricultureproduction,economic construction and social development. Drought also causes the extremelyfragile ecology environment of local area. Accordingly, the theoretical analysis of drought isvery necessary in the arid and semi-arid area in China. Taking Xinjiang region and HeiheRiver basin as representations, meteorological drought variables were extracted by monthlyrainfall datas and hydrology drought variables were extracted by monthly runoff datas toconstruct multivariate joint distributions of drought variables to further do drought analysis.The related research is very important to the study of the evolution rules of drought events.
Two and three-dimension joint distributions of drought variables were analyzed based onArchimedean Copulas functions for monthly precipitation at41representative weatherstations in Xinjiang province and at21weather stations in Heihe River basin. The monthlyrunoff data of3hydrologic stations, i.e., Yingluoxia, Liyuan and Zhengyixia in the upstreamregion of Heihe River basin, were also selected for hydrological drought analysis. The bestCopula function was selected by evaluations of goodness of fit to describe rules of droughtevents and to provide theoretical supports for drought prevention and control, reasonableallocation of water consumption, water resources sustainable development, waterconservancy construction project planning and construction in the locality. The followingmain conclusions were drawn:
（1）Three drought variables, i.e., drought duration, drought severity and drought peak,were selected by run-length theory from long sequence of average monthly rainfall data inXinjiang region and Heihe River basin as well as average monthly runoff data in Heihe Riverbasin. Relative squared residuals minimization rule（WLS）showed that, for41weatherstations in Xinjiang and21weather stations in Heihe River basin: drought durations areexponential distributed; drought severities are Weibull distributed, while drought peaks are generally Pareto distributed. And for3hydrologic stations in Heihe River basin: droughtdurations of are Pareto distributed; drought severities are subjected to Weibull distribution,while drought peaks are subjected to Gamma distribution. The stations that have a poordegree of fitting were rejected by Kolmogorov-Smirnov （K-S） test of marginaldistributions.For Xinjiang region:Hami city,Tulufan city,Qiemo county,Hetian county andMinfeng county stations were rejected.While for Heihe River basin:Mazongshan,Yumentown,Yongchang,Alashanyouqi,Tuole,Gangcha and Menyuan stations were rejected.
（2） The characteristic drought variables must be correlated between each other to useCopulas function to construct theire joint distributions. Through correlation measurement, forXinjiang region, the correlation was strongest between drought duration and drought severity,the next was drought severity and drought peak, while the correlation was weakest betweendrought duration and drought peak. For Heihe River basin based on the basic data at or the21weather stations and3hydrologic stations: the correlation was strong between droughtseverity and drought peak, the next was drought duration and drought severity, while thecorrelation was weakest between drought duration and drought peak.
（3） The parameter estimation of Copulas functions for the meteorological and thehydrology drought variables for Xinjiang region and Heihe River basin was conducted.20kinds of two-dimension Archimedean Copulas, Plackett Copulas, Farlie-Gumbei-Morgenstern(FGM) Copulas functions as well as4kinds of symmetric,5kinds of asymmetricthree-dimension Archimedean Copulas functions were analyzed based on fitting curvemethod and maximum likelihood method. The standard of fitting curve method was minimumstandards of squared residuals. AIC, RMSE and Bias as goodness-of-fit assessment indiceswere taken to compare the fitting effects. The results of parameter estimation showed thatfitting curve method was better than maximum likelihood method for Xinjiang region andHeihe River basin. So it was selected to estimate the parameters of Copulas functions.
（4）The best Copula function was selected by evaluation of goodness degree for fitting.For Xinjiang region: Frank Copula was the best when it was applied to two-dimension jointdistribution for drought duration-drought severity, and drought duration-drought peak.Clayton Copula was the best for drought severity-drought peak and three-dimension jointdistributions of the three drought variables. For Heihe River basin: FGM Copula was the bestwhen it was applied to two-dimension joint distributions for drought duration-droughtseverity. Nelsen No20Copula was the best for drought duration-drought peak. Nelsen No14Copula was best for drought severity-drought peak and Clayton Copula was the best forthree-dimension joint distribution of the three drought variables.While for3hydrologic stations-Yingluoxia, Liyuan and Zhengyixia in the upstream of Heihe River basin: FGMCopula was the best when it was applied to two-dimension joint distributions for droughtduration-drought severity, and drought duration-drought peak. Nelsen No20Copula wasthe best for drought severity-drought peak and Clayton Copula was the best forthree-dimension joint distribution of the three drought variables.
（5）Two and three-dimension joint distribution models were constructed based on thebest Copulas functions, and the associated probability and return period of drought variableswere calculated for Xinjiang region and Heihe River basin. Two-and three-dimensional returnperiods were similar for Xinjiang region and Heihe River basin: the return period of singlevariable ranged between two-dimension joint return period and co-occurrence return period.The co-occurrence return period of the two-dimension, three-dimension was longer than jointreturn period. The co-occurrence return period of the three-dimension was longer than that oftwo-dimension, while the joint return period of the three-dimension was shorter than that oftwo-dimension.
（6）Spatial distributions of drought feature variables in Xinjiang and Heihe River basinwere drawn through the conditional probabilities of two-dimensionl and three-dimensionaldrought variables: P(S≤50|D≥4), P(M≤10|D≥4), P(S≤50|M≥10) and P(S≤50|M≥10,D≥4).The spatial distribution characteristics of multivariate drought variables showed that, forXinjiang region,the conditional probabilities of drought events increased from the north to thesouth, meaning the conditional probability of drought events in southern Xinjiang were morelikely to happen than the northern Xinjiang. While for Heihe River basin, the conditionalprobabilities of drought events in the downstream region were more likely to happen than thatof upstream region.
选取具有代表性的新疆41个气象站的月降水数据、黑河流域21个气象站的月降水数据提取气象干旱变量；选取黑河流域上游莺落峡、梨园站和正义峡三个水文站的月径流资料提取水文干旱变量，基于Archimedean Copulas函数建立了二维、三维气象和水文干旱变量的联合分布，经拟合优度评价，选取最佳的Copula函数用以刻画干旱演变规律，以期为区域、流域内旱灾防治和控制、用水量合理调配、水资源可持续发展以及水利建筑工程规划建设提供一定的理论支撑。主要取得以下结论：
（1）基于游程理论从新疆地区和黑河流域长系列的月平均降水以及黑河流域长系列的月平均径流资料中提取干旱历时、干旱烈度和干旱烈度峰值三个干旱特征变量。由相对离差平方和最小准则（WLS）判断三个干旱特征变量的分布类型。结果表明：新疆地区41个站点以及黑河流域21个站点的干旱历时服从指数分布；干旱烈度服从Weibull分布；烈度峰值服从广义Pareto分布。黑河上游莺落峡、梨园站和正义峡的干旱历时服从广义Pareto分布，干旱烈度服从Weibull分布，烈度峰值服从Gamma分布。经单变量边缘分布的Kolmogorov-Smirnov（K-S）拟合度检验，剔除新疆地区干旱特征变量拟合情况不佳的6个站点（哈密市，吐鲁番市，且末县，和田县和民丰县）；以及黑河流域拟合情况不佳的7个站点（马鬃山，玉门镇，永昌，阿拉善右旗，拖勒，刚察和门源）。
（2）干旱特征变量间必须具有一定的相关性方可采用Copulas函数构建联合分布。通过变量间相关性度量得出：新疆地区的干旱历时和干旱烈度之间的相关性最好，干旱烈度和烈度峰值之间的相关性次之，干旱历时和烈度峰值之间的相关性最弱；黑河流域21个气象站和三个水文站的干旱烈度和烈度峰值之间的相关性最好，干旱历时和干旱烈度之间的相关性次之，而干旱历时和烈度峰值之间的相关性最弱。
（3）对于新疆地区和黑河流域气象和水文干旱特征变量联合分布的Copulas函数参数估计：选用适线法和极大似然法对20种二维Archimedean Copulas函数和Plackett、Farlie-Gumbei-Morgenstern(FGM) Copulas函数以及4种对称的、5种非对称的三维Archimedean Copulas函数的参数进行估计，适线准则采取离差平方和最小准则。采用AIC准则、均方根误差（RMSE）准则以及Bias准则进行拟合度评价。拟合优度表明：对于新疆地区以及黑河流域整体来讲，适线法的Copulas函数的参数估计效果优于极大似然法，故采用前者进行Copulas函数的参数估计。
（4）经拟合优度评价优选最佳的Copulas函数。对于整个新疆地区的气象干旱而言：Frank Copula函数对干旱历时和干旱烈度、干旱历时和烈度峰值的二维联合分布的拟合度最好；Clayton Copula函数对于干旱烈度和烈度峰值的二维联合分布以及干旱历时、干旱烈度和烈度峰值的三维联合分布拟合效果最佳。对于黑河流域的气象干旱特征：FGM Copula函数对干旱历时和干旱烈度的拟合度最好；Nelsen No20Copula函数对于干旱历时和烈度峰值的拟合度最佳；而Nelsen No14Copula函数对于干旱烈度和烈度峰值的拟合效果最好。Clayton Copula函数对于干旱历时、干旱烈度和烈度峰值的三维联合分布拟合效果最佳。对于黑河莺落峡、梨园站和正义峡的水文干旱而言：FGM Copula函数对干旱历时和干旱烈度、干旱历时和烈度峰值的二维联合分布的拟合度最好；NelsenNo20Copula函数对干旱烈度和烈度峰值的拟合度最好。同样Clayton Copula函数对于干旱历时、干旱烈度和烈度峰值的三维联合分布拟合效果最好。
（5）基于优选的Copulas函数构建新疆地区和黑河流域的二、三维干旱特征变量的联合分布模型，计算干旱变量的联合概率及重现期。新疆地区和黑河流域的二、三维重现期分布具有大致类似的规律：单变量的重现期介于二维、三维变量联合重现期与同现重现期之间；二维、三维变量的同现重现期大于相应的联合重现期；三维变量的同现重现期大于相应的二维变量的同现重现期，而三维变量的联合重现期小于相应的二维变量的联合重现期。
（6）由二维、三维干旱特征变量的特定条件概率：P(S≤50|D≥4)、P(M≤10|D≥4)、P(S≤50|M≥10)以及P(D≤4∪S≤50∪M≤10)绘制新疆地区和黑河流域水文和气象干旱特征的空间分布图。多维干旱特征变量空间分布规律大体为，对于新疆地区：特定干旱事件发生概率从北到南递增，即新疆的南疆比北疆更易发生带有特定条件概率的干旱事件；对于黑河流域气象干旱分布特征：黑河下游比上游更容易发生特定条件概率的干旱事件。
Drought affactes and restricts the economic and social development a lot in humanhistory. Because of its special geography, climate and hydrology characteristics in China,drought has hurt coutries economic development seviouly. Especially in recent years, theshortage of water resources led to unbalance between supply and demand of waterconsumption, this increasingly aggravates the conflict between human and nature in harmonywith each other. Drought has very disadvantageous effects on industry, agricultureproduction,economic construction and social development. Drought also causes the extremelyfragile ecology environment of local area. Accordingly, the theoretical analysis of drought isvery necessary in the arid and semi-arid area in China. Taking Xinjiang region and HeiheRiver basin as representations, meteorological drought variables were extracted by monthlyrainfall datas and hydrology drought variables were extracted by monthly runoff datas toconstruct multivariate joint distributions of drought variables to further do drought analysis.The related research is very important to the study of the evolution rules of drought events.
Two and three-dimension joint distributions of drought variables were analyzed based onArchimedean Copulas functions for monthly precipitation at41representative weatherstations in Xinjiang province and at21weather stations in Heihe River basin. The monthlyrunoff data of3hydrologic stations, i.e., Yingluoxia, Liyuan and Zhengyixia in the upstreamregion of Heihe River basin, were also selected for hydrological drought analysis. The bestCopula function was selected by evaluations of goodness of fit to describe rules of droughtevents and to provide theoretical supports for drought prevention and control, reasonableallocation of water consumption, water resources sustainable development, waterconservancy construction project planning and construction in the locality. The followingmain conclusions were drawn:
（1）Three drought variables, i.e., drought duration, drought severity and drought peak,were selected by run-length theory from long sequence of average monthly rainfall data inXinjiang region and Heihe River basin as well as average monthly runoff data in Heihe Riverbasin. Relative squared residuals minimization rule（WLS）showed that, for41weatherstations in Xinjiang and21weather stations in Heihe River basin: drought durations areexponential distributed; drought severities are Weibull distributed, while drought peaks are generally Pareto distributed. And for3hydrologic stations in Heihe River basin: droughtdurations of are Pareto distributed; drought severities are subjected to Weibull distribution,while drought peaks are subjected to Gamma distribution. The stations that have a poordegree of fitting were rejected by Kolmogorov-Smirnov （K-S） test of marginaldistributions.For Xinjiang region:Hami city,Tulufan city,Qiemo county,Hetian county andMinfeng county stations were rejected.While for Heihe River basin:Mazongshan,Yumentown,Yongchang,Alashanyouqi,Tuole,Gangcha and Menyuan stations were rejected.
（2） The characteristic drought variables must be correlated between each other to useCopulas function to construct theire joint distributions. Through correlation measurement, forXinjiang region, the correlation was strongest between drought duration and drought severity,the next was drought severity and drought peak, while the correlation was weakest betweendrought duration and drought peak. For Heihe River basin based on the basic data at or the21weather stations and3hydrologic stations: the correlation was strong between droughtseverity and drought peak, the next was drought duration and drought severity, while thecorrelation was weakest between drought duration and drought peak.
（3） The parameter estimation of Copulas functions for the meteorological and thehydrology drought variables for Xinjiang region and Heihe River basin was conducted.20kinds of two-dimension Archimedean Copulas, Plackett Copulas, Farlie-Gumbei-Morgenstern(FGM) Copulas functions as well as4kinds of symmetric,5kinds of asymmetricthree-dimension Archimedean Copulas functions were analyzed based on fitting curvemethod and maximum likelihood method. The standard of fitting curve method was minimumstandards of squared residuals. AIC, RMSE and Bias as goodness-of-fit assessment indiceswere taken to compare the fitting effects. The results of parameter estimation showed thatfitting curve method was better than maximum likelihood method for Xinjiang region andHeihe River basin. So it was selected to estimate the parameters of Copulas functions.
（4）The best Copula function was selected by evaluation of goodness degree for fitting.For Xinjiang region: Frank Copula was the best when it was applied to two-dimension jointdistribution for drought duration-drought severity, and drought duration-drought peak.Clayton Copula was the best for drought severity-drought peak and three-dimension jointdistributions of the three drought variables. For Heihe River basin: FGM Copula was the bestwhen it was applied to two-dimension joint distributions for drought duration-droughtseverity. Nelsen No20Copula was the best for drought duration-drought peak. Nelsen No14Copula was best for drought severity-drought peak and Clayton Copula was the best forthree-dimension joint distribution of the three drought variables.While for3hydrologic stations-Yingluoxia, Liyuan and Zhengyixia in the upstream of Heihe River basin: FGMCopula was the best when it was applied to two-dimension joint distributions for droughtduration-drought severity, and drought duration-drought peak. Nelsen No20Copula wasthe best for drought severity-drought peak and Clayton Copula was the best forthree-dimension joint distribution of the three drought variables.
（5）Two and three-dimension joint distribution models were constructed based on thebest Copulas functions, and the associated probability and return period of drought variableswere calculated for Xinjiang region and Heihe River basin. Two-and three-dimensional returnperiods were similar for Xinjiang region and Heihe River basin: the return period of singlevariable ranged between two-dimension joint return period and co-occurrence return period.The co-occurrence return period of the two-dimension, three-dimension was longer than jointreturn period. The co-occurrence return period of the three-dimension was longer than that oftwo-dimension, while the joint return period of the three-dimension was shorter than that oftwo-dimension.
（6）Spatial distributions of drought feature variables in Xinjiang and Heihe River basinwere drawn through the conditional probabilities of two-dimensionl and three-dimensionaldrought variables: P(S≤50|D≥4), P(M≤10|D≥4), P(S≤50|M≥10) and P(S≤50|M≥10,D≥4).The spatial distribution characteristics of multivariate drought variables showed that, forXinjiang region,the conditional probabilities of drought events increased from the north to thesouth, meaning the conditional probability of drought events in southern Xinjiang were morelikely to happen than the northern Xinjiang. While for Heihe River basin, the conditionalprobabilities of drought events in the downstream region were more likely to happen than thatof upstream region.
引文
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肖义，郭生练，刘攀.2007.基于Copula函数的设计洪水过程线方法.武汉大学学报(工学版).40（4）：13-14.
熊立华，郭生练，肖义.2005. Copula联结函数在多变量水文频率分析中的应用.武汉大学学报(工学版).38（6）：16-17.
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姚玉璧，张存杰，邓振镛.2007.气象、农业干旱指标综述．干旱地区农业研究.25（1）：185-186.
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袁文平，周广胜.2004.干旱指标的理论分析与研究展望.地球科学进展.19（6）：982-991.
曾少聪.2010.生态人类学视野中的西南干旱—以云南旱灾为例.贵州社会科学.251（11）：24-29.
张景书.1993.干旱的定义及其逻辑分析.干旱地区农业研究.11(3)：3-4.
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张娜，郭生练，闫宝伟.2008. Copula函数在分期设计洪水中的应用研究.水文.28（5）：28-29.
张娜，郭生练，刘攀.2008.基于Copula函数法推求分期设计洪水和汛限水位.武汉大学学报(工学
版).41（6）：33-34.
张雨，宋松柏.2010. Copulas函数在多变量干旱联合分布中的应用.灌溉排水学报.29（3）：63-64.
张钰，徐德辉.2001.关于干旱与旱灾概念的探讨.发展.9：14-16.
张涛，赵春伟，雒文生.2008.基于Copula函数的洪水过程随机模拟.武汉大学学报(工学版).41（4）：1-2.
张世法，苏逸深，宋德敦.2008.中国历史干旱（1949-2000）.南京：河海大学出版社：204.
Artem Prokhorov.2008.A Goodness-of-fit Test for Copulas. Computational Statistics&Data Analysis.9.
Annie Poulin，David Huard，Anne-Catherine Favre Ph.D.2007. Importance of Tail Dependence in BivariateFrequency Analysis DOI:10.1061/ASCE，1084-0699.12（4）:394.
Box G，Cox D．1964.An analysis of transformatiens[J].Journal of the Royal Statistical Society.B26：211-252.
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