水文频率新型计算理论与应用研究
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摘要
水文频率分析是水文学研究的重要内容，旨在研究和分析水文随机现象，揭示其中蕴含的统计规律，并对未来可能的情势做出预估，满足水利工程规划、设计、管理以及水资源利用等工作的需要。
本文针对现行水文频率分析存在的若干问题，综述了近年来国内外学者围绕这些问题的研究进展，总结了目前研究中存在的一些不足之处。选用陕西省典型区域水文气象资料，从单变量水文频率分析和多变量水文频率分析两方面入手，研究和提出了一些新的理论方法，并用于实例分析，主要开展了以下方面的研究。
1.研究尾部指数估计理论在水文事件分析中的应用。对研究区月降水序列是否服从重尾分布进行了判别，并采用不同的尾部指数估计量对序列的极值指数和尾部指数进行了估计。
2.研究非一致分布年径流序列的计算理论方法与应用。将研究区的非一致性年径流序列以变异点为节点划分为若干子序列，分别推导了序列理论频率、经验频率、矩和分位数的计算公式，选用P-III分布对研究区年径流序列的子序列进行拟合合成，评价结果的优劣。
3.研究部分概率权重矩法（PPWM）和高阶概率权重矩法（HPWM）的计算理论与应用。采用GEV分布的PPWM法，通过给定不同的删失水平，分别估计年径流序列和年最大洪峰流量序列的分布参数，对序列较大值的拟合效果进行评价，并采用MonteCarlo模拟对PPWM法的统计性能进行了分析。推导了P-III分布的HPWM公式，分别采用GEV分布的HPWM法和P-III分布的HPWM法估计年径流序列和年最大洪峰流量序列的分布参数，对序列较大值的拟合效果进行评价，并采用Monte Carlo模拟对HPWM法的统计性能进行了分析。
4.研究贝叶斯理论在P-III分布参数估计中的应用。推导了基于贝叶斯方法的P-III分布参数估计公式，采用三种马尔科夫链蒙特卡洛（MCMC）抽样方法获得研究区年最大洪峰流量序列的参数样本，并评价了不同抽样方法的优劣。
5.研究分层阿基米德copulas函数在干旱事件多变量联合分布构建中的应用。针对对称阿基米德copulas函数变量可互换、不考虑变量间相关性差异的缺点，根据干旱历时、干旱烈度和烈度峰值三个干旱特征变量间相关性的大小，选择Clayton copulas函数和Frank copulas函数进行组合，分层构建出三变量阿基米德copulas函数，推导了其概率密度及条件概率表达式，给出了三维copulas函数的拟合优度检验方法，进而用于研究区干旱分析。
通过上述研究，本文取得以下主要结论。
1.月降水序列的样本分布属于重尾分布，可以应用尾部指数估计理论对其分布尾部特性进行度量；常用的Hill估计量、Pickands估计量和矩估计量均能方便地估计极值指数和尾部指数，用Sum-plot作图法选取阈值k可以获得较好的效果。
2.提出了非一致分布年径流序列计算方法。该法具有物理意义明确、数学基础严格的特点，实例拟合计算结果较为理想，且该方法无需进行资料的“还原”、“还现”计算，应用方便。
3. PPWM法和HPWM法均能有效改善研究区洪水序列较大流量的拟合效果，并显著降低设计值的计算误差。其中，GEV分布的PPWM法和P-III分布的HPWM法计算出的设计值误差相对较小，GEV分布的HPWM法计算出的设计值误差相对略大，但其计算十分简便，亦有可取之处。
4.对于研究区年径流序列，PPWM法和HPWM法均无法改善序列较大值的拟合效果，而采用P-III分布普通PWM法进行参数估计，对序列的拟合效果较好。
5.基于贝叶斯方法估计P-III分布参数时，不同的抽样算法对MCMC方法的性能有一定的影响，其中AM算法具有收敛速度快、计算效率高、参数估计效果好等优点，优于MH法和DR法。
6.三维Clayton/Frank组合分层copulas函数能够描述研究区干旱历时、干旱烈度和烈度峰值的三变量联合概率分布。与对称的三维阿基米德copulas函数相比，Clayton/Frank组合分层copulas函数更为合理地反映了干旱变量间不同的相依结构，且结构组合灵活，能够较为全面地描述干旱变量的联合分布特性。
Hydrologic frequency analysis is an important content in hydrology research. Aims atstudy and analysis the hydrological random phenomena, reveal the statistic laws that beingcontained in hydrological phenomena, in order to forecast the possible hydrological regime inthe future, and satisfy the requirements of hydraulic engineering planning, designing,management and water resources utilization.
Aiming at several problems existing in current hydrologic frequency analysis research,this dissertation reviewed the present researches around these issues both at home and abroad,summarized some disadvantages existing in the present study. The hydrometeorological dataof typical region in Shaanxi province were used for case studies. Univariate and multivariatehydrologic frequency analysis were studied respectively, several new theoretical method wereemployed and proposed, and were used for case studies.
The main contents of this dissertation are as follows.
1. Research on the application of tail-index estimation theory in hydrological events.Whether the monthly precipitation series obey the heavy-tailed distribution was judged. Threekinds of estimators were selected to estimate the extreme index and tail index of the data.
2. Research on the principle of nonidentically distributed hydrologic data and itsapplication in annual runoff series. By dividing the nonidentically distributed annual runoffseries into several sub-series according to the change points, the formulas of the theoreticalprobability, the empirical probability, the moments and the quantile were deduced. P-IIIdistribution was chosen for application and assessment.
3. Research on the method of partial probability weighted moments, the method ofhigher-order probability weighted moments and their application in annual runoff series andannual maximum flows. Giving different censoring levels, the parameters of GEV distributionwere estimated using partial probability weighted moments. The effects of fitting the largervalue of data were evaluated. Monte Carlo simulation was carried to analyze the statisticalproperties of partial probability weighted moments. The expressions of higher-orderprobability weighted moments for P-III distribution were derived. The parameters of GEVdistribution and P-III distribution were estimated separately. The effects of fitting the largervalue of data were evaluated. Monte Carlo simulation was carried to analyze the statistical properties of higher-order probability weighted moments.
4. Research on the application of Bayesian theory in parameter estimation for P-IIIdistribution. The estimators of parameters for P-III distribution based on Bayesian theorywere presented. Three kinds of MCMC sampling methods were used to obtain the parametersamples of annual maximum flows, and their properties were evaluated in application.
5. Research on the application of hierarchical Archimedean copulas in multivariatedraught analysis. Commonly-used Archimedean copulas are distribution functions of severalexchangeable uniform random variates, and take no account of dependence differencebetween variates. For this reason, these copulas suffer from a very limited dependencestructure. Contrapose this disadvantages of ordinary multivariate Archimedean copulas, weconstructed trivariate hierarchical Archimedean copulas (HAC) that combines Claytoncopulas with Frank copulas to model multivariate joint distributions according to thedependence of draught duration, draught severity and severity peak. The expressions ofdensity, cumulative distribution functions and conditional copulas of the trivariate HAC werederived. A goodness-of-fit test for trivariate copulas was conducted. This kind of trivariateHAC was used for draught analysis.
The main conclusions are shown as follows.
1. The distribution of monthly precipitation series belongs to heavy-tailed distribution,and the tail-index estimation theory is suitable to measure its tail behavior. All of thecommonly-used Hill estimator, Pickands estimator and moment estimator can estimate theextreme index and the tail index conveniently. Sum-plot was used for selecting the best k, andwas justified its good effect and high accuracy.
2. The formulas of nonidentically distributed hydrologic data have definite physicalsignificance and strict mathematical foundation. Case studies show that this method can befitted by arbitrary common distributions and is convenient to use.
3. Both method of partial probability weighted moments and method of higher-orderprobability weighted moments can improve the fitting effect of larger value of flood dataeffectively, and can significantly reduce the error of design values. The errors of design valuesthat calculated using PPWM for GEV distribution and higher-order PWMs for P-IIIdistribution are relatively smaller. The errors of design values that calculated usinghigher-order PWMs for GEV distribution are relatively larger. However, the calculation ofhigher-order PWMs for GEV distribution is extremely simple.
4. Regarding annual runoff series, neither method of partial probability weightedmoments nor method of higher-order probability weighted moments can improve the fittingeffect of larger value of runoff data. Method of PWMs for P-III distribution can fit the annual runoff data quite well.
5. In estimating the parameters of P-III distribution based on Bayesian method, differentsampling method has some influence on the performance of the MCMC method. Fast inconvergence rate, high in computational efficiency and good effects in parameter estimation,AM algorithm was much better than MH and DR algorithm.
6. Trivariate hierarchical Archimedean copulas that combine Clayton copulas with Frankcopulas can describe the joint distribution of drought duration, drought severity and severitypeak. Comparing with the symmetrical trivariate Archimedean copulas, trivariate HACreflected the different dependence structures among draught variates more reasonably, andhad more flexible structure. This kind of copulas describes the joint distributioncharacteristics of drought variates comprehensively.
本文针对现行水文频率分析存在的若干问题，综述了近年来国内外学者围绕这些问题的研究进展，总结了目前研究中存在的一些不足之处。选用陕西省典型区域水文气象资料，从单变量水文频率分析和多变量水文频率分析两方面入手，研究和提出了一些新的理论方法，并用于实例分析，主要开展了以下方面的研究。
1.研究尾部指数估计理论在水文事件分析中的应用。对研究区月降水序列是否服从重尾分布进行了判别，并采用不同的尾部指数估计量对序列的极值指数和尾部指数进行了估计。
2.研究非一致分布年径流序列的计算理论方法与应用。将研究区的非一致性年径流序列以变异点为节点划分为若干子序列，分别推导了序列理论频率、经验频率、矩和分位数的计算公式，选用P-III分布对研究区年径流序列的子序列进行拟合合成，评价结果的优劣。
3.研究部分概率权重矩法（PPWM）和高阶概率权重矩法（HPWM）的计算理论与应用。采用GEV分布的PPWM法，通过给定不同的删失水平，分别估计年径流序列和年最大洪峰流量序列的分布参数，对序列较大值的拟合效果进行评价，并采用MonteCarlo模拟对PPWM法的统计性能进行了分析。推导了P-III分布的HPWM公式，分别采用GEV分布的HPWM法和P-III分布的HPWM法估计年径流序列和年最大洪峰流量序列的分布参数，对序列较大值的拟合效果进行评价，并采用Monte Carlo模拟对HPWM法的统计性能进行了分析。
4.研究贝叶斯理论在P-III分布参数估计中的应用。推导了基于贝叶斯方法的P-III分布参数估计公式，采用三种马尔科夫链蒙特卡洛（MCMC）抽样方法获得研究区年最大洪峰流量序列的参数样本，并评价了不同抽样方法的优劣。
5.研究分层阿基米德copulas函数在干旱事件多变量联合分布构建中的应用。针对对称阿基米德copulas函数变量可互换、不考虑变量间相关性差异的缺点，根据干旱历时、干旱烈度和烈度峰值三个干旱特征变量间相关性的大小，选择Clayton copulas函数和Frank copulas函数进行组合，分层构建出三变量阿基米德copulas函数，推导了其概率密度及条件概率表达式，给出了三维copulas函数的拟合优度检验方法，进而用于研究区干旱分析。
通过上述研究，本文取得以下主要结论。
1.月降水序列的样本分布属于重尾分布，可以应用尾部指数估计理论对其分布尾部特性进行度量；常用的Hill估计量、Pickands估计量和矩估计量均能方便地估计极值指数和尾部指数，用Sum-plot作图法选取阈值k可以获得较好的效果。
2.提出了非一致分布年径流序列计算方法。该法具有物理意义明确、数学基础严格的特点，实例拟合计算结果较为理想，且该方法无需进行资料的“还原”、“还现”计算，应用方便。
3. PPWM法和HPWM法均能有效改善研究区洪水序列较大流量的拟合效果，并显著降低设计值的计算误差。其中，GEV分布的PPWM法和P-III分布的HPWM法计算出的设计值误差相对较小，GEV分布的HPWM法计算出的设计值误差相对略大，但其计算十分简便，亦有可取之处。
4.对于研究区年径流序列，PPWM法和HPWM法均无法改善序列较大值的拟合效果，而采用P-III分布普通PWM法进行参数估计，对序列的拟合效果较好。
5.基于贝叶斯方法估计P-III分布参数时，不同的抽样算法对MCMC方法的性能有一定的影响，其中AM算法具有收敛速度快、计算效率高、参数估计效果好等优点，优于MH法和DR法。
6.三维Clayton/Frank组合分层copulas函数能够描述研究区干旱历时、干旱烈度和烈度峰值的三变量联合概率分布。与对称的三维阿基米德copulas函数相比，Clayton/Frank组合分层copulas函数更为合理地反映了干旱变量间不同的相依结构，且结构组合灵活，能够较为全面地描述干旱变量的联合分布特性。
Hydrologic frequency analysis is an important content in hydrology research. Aims atstudy and analysis the hydrological random phenomena, reveal the statistic laws that beingcontained in hydrological phenomena, in order to forecast the possible hydrological regime inthe future, and satisfy the requirements of hydraulic engineering planning, designing,management and water resources utilization.
Aiming at several problems existing in current hydrologic frequency analysis research,this dissertation reviewed the present researches around these issues both at home and abroad,summarized some disadvantages existing in the present study. The hydrometeorological dataof typical region in Shaanxi province were used for case studies. Univariate and multivariatehydrologic frequency analysis were studied respectively, several new theoretical method wereemployed and proposed, and were used for case studies.
The main contents of this dissertation are as follows.
1. Research on the application of tail-index estimation theory in hydrological events.Whether the monthly precipitation series obey the heavy-tailed distribution was judged. Threekinds of estimators were selected to estimate the extreme index and tail index of the data.
2. Research on the principle of nonidentically distributed hydrologic data and itsapplication in annual runoff series. By dividing the nonidentically distributed annual runoffseries into several sub-series according to the change points, the formulas of the theoreticalprobability, the empirical probability, the moments and the quantile were deduced. P-IIIdistribution was chosen for application and assessment.
3. Research on the method of partial probability weighted moments, the method ofhigher-order probability weighted moments and their application in annual runoff series andannual maximum flows. Giving different censoring levels, the parameters of GEV distributionwere estimated using partial probability weighted moments. The effects of fitting the largervalue of data were evaluated. Monte Carlo simulation was carried to analyze the statisticalproperties of partial probability weighted moments. The expressions of higher-orderprobability weighted moments for P-III distribution were derived. The parameters of GEVdistribution and P-III distribution were estimated separately. The effects of fitting the largervalue of data were evaluated. Monte Carlo simulation was carried to analyze the statistical properties of higher-order probability weighted moments.
4. Research on the application of Bayesian theory in parameter estimation for P-IIIdistribution. The estimators of parameters for P-III distribution based on Bayesian theorywere presented. Three kinds of MCMC sampling methods were used to obtain the parametersamples of annual maximum flows, and their properties were evaluated in application.
5. Research on the application of hierarchical Archimedean copulas in multivariatedraught analysis. Commonly-used Archimedean copulas are distribution functions of severalexchangeable uniform random variates, and take no account of dependence differencebetween variates. For this reason, these copulas suffer from a very limited dependencestructure. Contrapose this disadvantages of ordinary multivariate Archimedean copulas, weconstructed trivariate hierarchical Archimedean copulas (HAC) that combines Claytoncopulas with Frank copulas to model multivariate joint distributions according to thedependence of draught duration, draught severity and severity peak. The expressions ofdensity, cumulative distribution functions and conditional copulas of the trivariate HAC werederived. A goodness-of-fit test for trivariate copulas was conducted. This kind of trivariateHAC was used for draught analysis.
The main conclusions are shown as follows.
1. The distribution of monthly precipitation series belongs to heavy-tailed distribution,and the tail-index estimation theory is suitable to measure its tail behavior. All of thecommonly-used Hill estimator, Pickands estimator and moment estimator can estimate theextreme index and the tail index conveniently. Sum-plot was used for selecting the best k, andwas justified its good effect and high accuracy.
2. The formulas of nonidentically distributed hydrologic data have definite physicalsignificance and strict mathematical foundation. Case studies show that this method can befitted by arbitrary common distributions and is convenient to use.
3. Both method of partial probability weighted moments and method of higher-orderprobability weighted moments can improve the fitting effect of larger value of flood dataeffectively, and can significantly reduce the error of design values. The errors of design valuesthat calculated using PPWM for GEV distribution and higher-order PWMs for P-IIIdistribution are relatively smaller. The errors of design values that calculated usinghigher-order PWMs for GEV distribution are relatively larger. However, the calculation ofhigher-order PWMs for GEV distribution is extremely simple.
4. Regarding annual runoff series, neither method of partial probability weightedmoments nor method of higher-order probability weighted moments can improve the fittingeffect of larger value of runoff data. Method of PWMs for P-III distribution can fit the annual runoff data quite well.
5. In estimating the parameters of P-III distribution based on Bayesian method, differentsampling method has some influence on the performance of the MCMC method. Fast inconvergence rate, high in computational efficiency and good effects in parameter estimation,AM algorithm was much better than MH and DR algorithm.
6. Trivariate hierarchical Archimedean copulas that combine Clayton copulas with Frankcopulas can describe the joint distribution of drought duration, drought severity and severitypeak. Comparing with the symmetrical trivariate Archimedean copulas, trivariate HACreflected the different dependence structures among draught variates more reasonably, andhad more flexible structure. This kind of copulas describes the joint distributioncharacteristics of drought variates comprehensively.
引文
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