Bivariate frequency analysis of low flow using copula functions (case study: Dez River Basin,Iran)

Accurate estimation of low flow as a criterion for different objectives in water resource management, including drought is of crucial importance. Despite the complex nature of water deficits, univariate methods have often been used to analyze the frequency of low flows. In this study, low flows of Dez River basin were examined during period of 1956–2012 using copula functions at the upstream of headbranches’ junction. For this purpose, at first 7-day series of low flow was extracted at the studied stations, then their homogeneity was examined by Mann–Kendall test. The results indicated that 7-day low flow series of Dez basin were homogenous. In the next stage, 12 different distribution functions were fitted onto the low flow data. Finally, for Sepid Dasht Sezar (SDS), Sepid Dasht Zaz (SDZ), and Tang Panj Bakhtiyari (TPB) stations, logistic distribution had the best fit, while for Tang Panj Sezar (TPS) station, GEV distribution enjoyed the best fit. After specifying the best fitted marginal distributions, seven different copula functions including Ali–Mikhail–Haq (AMH), Frank, Clayton, Galambos, Farlie–Gumbel–Morgenstern (FGM), Gumbel–Hougaard (GH), and Plackett were used for bivariate frequency analysis of the 7-day low flow series. The results revealed that the GH copula had the best fitness on paired data of SDS and SDZ stations. For TPS and TPB stations, Frank copula has had the best correspondence with empirical copula values. Next, joint and conditional return periods were calculated for the low flow series at the upstream of branches’ junction. The results of this study indicated that the risk of incidence of severe drought is higher in upstream stations (SDZ and SDS) when compared with downstream stations (TPB and TPS) in Dez basin. Generally, application of multivariate analysis allows researchers to investigate hydrological events with a more comprehensive view by considering the simultaneous effect of the influencing factors on the phenomenon of interest. It also enables them to evaluate different combinations of required scenarios for integrated management of basin and planning to cope with the damages caused by natural phenomena.     

应用超阈值抽样及Copula函数推求水库设计洪水

按照超阈值理论对洪水资料进行抽样,应用Copula函数分别建立了超阈值洪水洪峰、3d洪量和7d洪量与其发生时间的联合分布,得到了不同量级洪水在发生时间上的概率分布图,对水库汛期防洪调度运用具有一定的指导意义。同时,应用Copula函数建立了年最大超阈值洪水洪峰、3d洪量和7d洪量的联合分布,以此得到设计标准和校核标准的设计洪水,并通过同频率放大法得到了设计洪水过程线,该洪水过程线更符合水文现象的内在规律,展现了超阈值理论和Copula函数在水文洪水频率分析计算领域的良好应用前景。     

拓扑克里格法与普通克里格法在区域洪水频率分析中的比较研究

依据北江(珠江流域支流)流域6个水文测站年最大洪峰流量资料,分别用Top-kriging(拓扑克里格法)和普通克里格法进行区域洪水频率估计。采用均方根误差作为频率分布线型拟合优度指标。运用线性矩法进行单站洪水频率分析,确定10、50、100、1000年一遇设计洪水值。在此基础上,从Topkriging和普通克里格法设计洪水估计不确定性和相对线性矩法单站洪水频率的估计误差两个方面比较Top-kriging和普通克里格法。结果表明:(1)Top-kriging法是更好的线性无偏估计,相比普通克里格法更适合于区域洪水频率估计;(2)Top-kriging法设计洪水估计不确定性明显小于普通克里格法;(3)Top-kriging法设计洪水估计结果更接近线性矩法单站洪水频率分析结果。     

Flood estimation in Mahanadi river system,India using partial duration series

Flood frequency analysis is a pre-requisite for setting up and safeguarding of many hydraulic structures, such as dams, barrages, check-dams, culverts and urban drainage systems. In the flood frequency analysis, partial duration series (PDS) may be considered when dealing with values exceeding certain limits causing floods. In fact, the PDS is capable of getting more information about extreme events than the annual maximum series (AMS). Additionally, an assumption that, the magnitude of the extreme events of a PDS is best described by a generalized Pareto (GP) distribution. The present work investigates the at-site flood frequency analysis to find the average number of peaks (λ) for modelling the PDS on the basis of the PDS/GP assumption and variability in the GP parameters coupled with the quantile estimation with an increase in the value of average number of peaks (λ) each year in the Mahanadi river system, Odisha, India. Also, to verify the PDS/GP assumption we tested seven different frequency distributions (Exponential, Gumbel, logistics, generalized extreme value (GEV), Lognormal (LN), generalized logistics (GL) and Pearson Type 3). Extensive daily discharge data collected from 23 gauging sites were used for the analysis. The results indicate precision and stability of GP distribution parameters for λ?=?4 for almost all the discharge sites. The peak flood estimated for various return periods in the Mahanadi river system using GP distribution is endowed with high correlation statistics for this λ value.     

A study on selection of probability distributions for at-site flood frequency analysis in Australia

The most direct method of design flood estimation is at-site flood frequency analysis, which relies on a relatively long period of recorded streamflow data at a given site. Selection of an appropriate probability distribution and associated parameter estimation procedure is of prime importance in at-site flood frequency analysis. The choice of the probability distribution for a given application is generally made arbitrarily as there is no sound physical basis to justify the selection. In this study, an attempt is made to investigate the suitability of as many as fifteen different probability distributions and three parameter estimation methods based on a large Australian annual maximum flood data set. A total of four goodness-of-fit tests are adopted, i.e., the Akaike information criterion, the Bayesian information criterion, Anderson–Darling test, and Kolmogorov–Smirnov test, to identify the best-fit probability distributions. Furthermore, the L-moments ratio diagram is used to make a visual assessment of the alternative distributions. It has been found that a single distribution cannot be specified as the best-fit distribution for all the Australian states as it was recommended in the Australian rainfall and runoff 1987. The log-Pearson 3, generalized extreme value, and generalized Pareto distributions have been identified as the top three best-fit distributions. It is thus recommended that these three distributions should be compared as a minimum in practical applications when making the final selection of the best-fit probability distribution in a given application in Australia.     

On the frequency distribution of turbidite thickness

The frequency distribution of turbidite thickness records information on flow hydrodynamics, initial sediment volumes and source migration and is an important component of petroleum reservoir models. However, the nature of this thickness distribution is currently uncertain, with log‐normal or negative‐exponential frequency distributions and power‐law cumulative frequency distributions having been proposed by different authors. A detailed analysis of the Miocene Marnoso Arenacea Formation of the Italian Apennines shows that turbidite bed thickness and sand‐interval thickness within each bed have a frequency distribution comprising the sum of a series of log‐normal frequency distributions. These strata were deposited predominantly in a basin‐plain setting, and bed amalgamation is relatively rare. Beds or sand intervals truncated by erosion were excluded from this analysis. Each log‐normal frequency distribution characterizes bed or sand‐interval thickness for a given basal grain‐size or basal Bouma division. Measurements from the Silurian Aberystwyth Grits in Wales, the Cretaceous Great Valley Sequence in California and the Permian Karoo Basin in South Africa show that this conclusion holds for sequences of disparate age and variable location. The median thickness of these log‐normal distributions is positively correlated with basal grain‐size. The power‐law exponent relating the basal grain‐size and median thickness is different for turbidites with a basal A or B division and those with only C, D and E divisions. These two types of turbidite have been termed ‘thin bedded’ and ‘thick bedded’ by previous workers. A change in the power‐law exponent is proposed to be related to: (i) a transition from viscous to inertial settling of sediment grains; and (ii) hindered settling at high sediment concentrations. The bimodal thickness distribution of ‘thin‐bedded’ and ‘thick‐bedded’ turbidites noted by previous workers is explained as the result of a change in the power‐law exponent. This analysis supports the view that A and B divisions were deposited from high‐concentration flow components and that distinct grain‐size modes undergo different depositional processes. Summation of log‐normal frequency distributions for thin‐ and thick‐bedded turbidites produces a cumulative frequency distribution of thickness with a segmented power‐law trend. Thus, the occurrence of both log‐normal and segmented power‐law frequency distributions can be explained in a holistic fashion. Power‐law frequency distributions of turbidite thickness have previously been linked to power‐law distributions of earthquake magnitude or volumes of submarine slope failure. The log‐normal distribution for a given grain‐size class observed in this study suggests an alternative view, that turbidite thickness is determined by the multiplicative addition of several randomly distributed parameters, in addition to the settling velocity of the grain‐sizes present.     

梯级水电开发对大通河流域洪水过程的影响分析

应用大通河流域实测洪水资料和水利普查数据,定性和定量分析了梯级水电开发对流域洪水过程和洪水特征的影响,建立了梯级水电站数量与洪水涨落率、涨落频次的数学关系模型。结果表明:随着流域水电站数量的增多,洪水过程变化频繁,洪水涨落率增大,涨落频次增多;梯级水电开发使流域最大洪峰流量增大了19.7%~21.5%,最大洪峰传播时间减少了6.1%,而对次洪水总量的影响较小。研究水文过程受人类活动的影响规律,对加强流域水电站统一调度、水能水资源可持续利用和水生态文明建设具有十分重要的意义。     

Joint pattern of seasonal hydrological droughts and floods alternation in China’s Huai River Basin using the multivariate L-moments

Under the current condition of climate change, droughts and floods occur more frequently, and events in which flooding occurs after a prolonged drought or a drought occurs after an extreme flood may have a more severe impact on natural systems and human lives. This challenges the traditional approach wherein droughts and floods are considered separately, which may largely underestimate the risk of the disasters. In our study, the sudden alternation of droughts and flood events (ADFEs) between adjacent seasons is studied using the multivariate L-moments theory and the bivariate copula functions in the Huai River Basin (HRB) of China with monthly streamflow data at 32 hydrological stations from 1956 to 2012. The dry and wet conditions are characterized by the standardized streamflow index (SSI) at a 3-month time scale. The results show that: (1) The summer streamflow makes the largest contribution to the annual streamflow, followed by the autumn streamflow and spring streamflow. (2) The entire study area can be divided into five homogeneous sub-regions using the multivariate regional homogeneity test. The generalized logistic distribution (GLO) and log-normal distribution (LN3) are acceptable to be the optimal marginal distributions under most conditions, and the Frank copula is more appropriate for spring-summer and summer-autumn SSI series. Continuous flood events dominate at most sites both in spring-summer and summer-autumn (with an average frequency of 13.78% and 17.06%, respectively), while continuous drought events come second (with an average frequency of 11.27% and 13.79%, respectively). Moreover, seasonal ADFEs most probably occurred near the mainstream of HRB, and drought and flood events are more likely to occur in summer-autumn than in spring-summer.     

甘肃省境内典型小流域暴雨特性及洪水过程模拟研究

选取甘肃省境内11个典型小流域的水文站实测资料,在分析资料可靠性、一致性、代表性及暴雨洪水特性的基础上,综合利用水文学、数理统计学等理论和方法,建立了小流域最大暴雨量-暴雨历时关系模型、最大洪峰流量-流域特征关系模型。结果表明:各时段最大暴雨量较大值主要分布在泾河流域和嘉陵江流域东南部,较小值主要分布在渭河流域、洮河流域、嘉陵江流域的北峪河;绘制了各代表站最大暴雨量-暴雨历时关系曲线,嘉陵江流域以90min为转折点,其他流域均以180min为转折点,小于转折点时,暴雨量随暴雨历时增长而急剧增加,大于转折点后,变化趋于平缓稳定。重点选取危害较大、难以防范的330场次短历时暴雨洪水,建立各小流域暴雨洪水概化模型,短历时洪水发生时间很短,洪水起涨时间约2～5h,洪峰出现时间约5～7h。研究典型小流域水文要素变化规律,分析局地短历时暴雨洪水特性,并模拟其变化过程,对区域抗旱防汛减灾、水资源管理、小流域治理及生态环境保护具有重要意义。     

洪水频率分析中PDS模型研究进展

为了从实测洪水资料中获取更多的信息,提出PDS(Partial duration series)洪水频率分析模型并得到发展。介绍了PDS模型及其导出的AMS(Annual Maximal Series)模型的基本理论,综述了两种模型之间的比较研究。总结了PDS模型应用的关键技术,包括独立性判别、阈值的选取、超定量次数概率分布的选择。描述了PDS模型在区域洪水频率分析、PDS模型与贝叶斯理论结合研究的新进展,展望了PDS模型进一步研究的前景。     

基于Copula函数的组合变量联合概率分布研究及应用

基于Copula函数原理,利用武江流域实测水文资料,以广义GDP为洪峰洪量边缘分布,构建了流域组合变量Copula概率分布模型,分析了洪峰与洪量、洪量与洪水历时、洪峰与洪水历时的联合概率分布,绘制各种变量组合下的联合分布图及重现期等值线图,并比较了同重现期条件下,洪水单变量设计值与多维联合设计值的区别。结果表明:广义GDP分布能很好的描述洪峰、洪量边缘分布,而基于广义GDP分布和指数分布构建的两变量Copula联合概率分布模型不限定变量的边缘分布,对各种类型的水文变量联合分布拟合效果较好;能全面反映洪水各特征属性不同等级下的联合发生频率,对同一频率下联合分布推求的洪水设计值比单变量设计值偏于安全。基于Copula函数的组合变量概率分布模型描述洪峰流量、洪量、洪水历时等特征的联合分布,较为全面地反映组合特征的洪水发生的概率和重现期,进一步反映洪水风险。     

群智能优化算法在水文频率曲线适线中的应用

根据不同适线准则,研究群智能优化算法在水文频率曲线适线中的计算问题,为水利工程规划、水资源优化配置等提供依据。以陕北地区12个主要测站的年径流系列为例,选取五大分布类共12种分布线型,根据我国现行水利水电工程设计洪水计算规范,按照离(残)差平方和最小准则(OLS)、离(残)差绝对值和最小准则(ABS)、相对离差平方和最小准则(WLS),以MATLAB7.6为计算平台,研究模拟退火算法、遗传算法、粒子群算法和蚁群算法进行水文频率参数的估计。在不同适线准则下,陕北地区年径流最优频率分布模型为广义Logistic分布(Generalized Logistic Distribution,GLO),粒子群算法进行参数估算偏差最小。与传统优化方法相比,群智能优化算法对优化目标函数要求低,是一种推求年径流频率曲线统计参数的新途径。     

Assessing the copula selection for bivariate frequency analysis based on the tail dependence test

The flood characteristics, namely, peak, duration and volume provide important information for the design of hydraulic structures, water resources planning, reservoir management and flood hazard mapping. Flood is a complex phenomenon defined by strongly correlated characteristics such as peak, duration and volume. Therefore, it is necessary to study the simultaneous, multivariate, probabilistic behaviour of flood characteristics. Traditional multivariate parametric distributions have widely been applied for hydrological applications. However, this approach has some drawbacks such as the dependence structure between the variables, which depends on the marginal distributions or the flood variables that have the same type of marginal distributions. Copulas are applied to overcome the restriction of traditional bivariate frequency analysis by choosing the marginals from different families of the probability distribution for flood variables. The most important step in the modelling process using copula is the selection of copula function which is the best fit for the data sample. The choice of copula may significantly impact the bivariate quantiles. Indeed, this study indicates that there is a huge difference in the joint return period estimation using the families of extreme value copulas and no upper tail copulas (Frank, Clayton and Gaussian) if there exists asymptotic dependence in the flood characteristics. This study suggests that the copula function should be selected based on the dependence structure of the variables. From the results, it is observed that the result from tail dependence test is very useful in selecting the appropriate copula for modelling the joint dependence structure of flood variables. The extreme value copulas with upper tail dependence have proved that they are appropriate models for the dependence structure of the flood characteristics and Frank, Clayton and Gaussian copulas are the appropriate copula models in case of variables which are diagnosed as asymptotic independence.     

水文多变量趋势分析的应用研究

水文频率分析需要进行一系列的假设,包括一致性、同分布性和独立性。由于水文事件通常都包含多个水文变量,例如洪水过程包含洪峰和洪量,低径流过程包括年最小月径流和年最小3个月径流等,因此近年来多变量水文分析与计算方法在水文领域中得到广泛应用。然而,多变量频率分析并未对上述假设进行检验,而是直接对频率分布进行函数拟合,不能保证得到的频率分布函数符合实际情况。引入水质研究领域内广泛使用的多变量MK检验方法,分别对长江上游干支流主要控制站点的年最大洪峰、年最大7d洪量和年最低月径流、年最低3个月径流的单变量和联合变量进行变化趋势分析。结果表明,单变量趋势分析仅仅能够检验出单个变量的变化趋势,而多变量趋势分析则能够综合检验出联合变量是否存在变化趋势。因此,为保证水文频率分析结果的准确,有必要在进行多变量水文频率分析前对单变量和联合变量进行趋势分析。     

Long-duration PMP and PMF estimation with SWAT model for the sparsely gauged Upper Nujiang River Basin

For large sparsely gauged basins, it is difficult to estimate long-duration probable maximum precipitation (PMP) and probable maximum flood (PMF) due to insufficient observed data and precipitation spatial distribution uncertainty. In this paper, a framework coupling the China Grid Daily Precipitation Datasets (CGDPDs) with Soil and Water Assessment Tool (SWAT) was proposed to estimate the 15-day PMP and PMF for the sparsely gauged Upper Nujiang River Basin (with a drainage area of 73,484 km²). CGDPD was tested against the observations and further corrected considering the error distribution characteristics. Results showed that 1-, 3-, 7- and 15-day maximum areal precipitations based on the corrected CGDPD were 17, 7, 4 and 18% larger than those calculated only by six observed stations’ precipitation. Then CGDPD was used as the precipitation data to estimate PMP. For the spatial distribution of PMP, the 15-day PMP process on the sub-basin scale (PMP_sub-basin) could be obtained with the following procedure. First, the basin’s 15-day areal PMP was estimated. Among this estimation, the maximum 3-day PMP was estimated by moisture maximization, while the remaining 12-day PMP was estimated with the combined storm obtained by the similar process substitution method. Second, the model storm amplification approach based on water balance principle was used to distribute the areal PMP to each sub-basin to obtain the PMP_sub-basin at all 27 sub-basins. The designed PMF could be finally estimated through inputting PMP_sub-basin into SWAT. In comparison with PMF derived from PMP without spatial distribution, different duration PMFs could increase by 3–15% when considering PMP spatial distribution uncertainty. This study could provide a reasonable procedure to estimate long-duration PMP and PMF for similar basins.     

Flood frequency analysis of river swat using Log Pearson type 3, Generalized Extreme Value,Normal, and Gumbel Max distribution methods

In Pakistan, floods are among the most devastating and recurring natural hazards. Flood hazard assessment requires flood event magnitude and probability of occurrence. Flood frequency analysis is the most common technique used for the at-site estimation of flood recurrence magnitude. This paper evaluates four most commonly used distribution methods, i.e., Generalized Extreme Value (GEV), Log Pearson 3 (LP3), Gumbel Max, and Normal for the flood frequency and estimation of flood recurrence. Different hydrological stations data namely Khwazakhela, Chakdarra, Panjkora, and Munda Headwork located at Swat river was taken from Provincial Irrigation Department, Khyber Pakhtunkhwa. The analysis is done for 5-, 10-, 25-, 50-, and 100-year return periods by using annual maximum discharge data from 1980 to 2016 (37 yr). Three goodness-of-fit tests were applied to the fitted distributions, i.e., Kolmogorov–Smirnov, Anderson–Darling, and Chi-squared at 5% significance level. Results indicate that LP3 and GEV were ranked top two distributions at all locations while Gumbel Max and Normal were the least fitted having rank 3 and 4, respectively. Based on the goodness-of-fit ranking, LP3 was selected for the estimation of flood magnitude and return periods at Khwazakhela. Designed hydrographs based on probabilistic approach and flood 2010 hydrograph are presented for flood simulation.     

基于高阶概率权重矩的广义极值分布参数估计

研究基于高阶概率权重矩的广义极值分布参数估计。根据高阶概率权重矩法原理,建立了广义极值分布高阶概率权重矩估算参数模型。以陕北地区4个水文测站的年最大洪峰流量序列为例,结果表明:高阶概率权重矩法能赋予大洪水值更多的权重。蒙特卡洛试验表明:适当提高阶数可以减小误差,但阶数过高反而会增大误差。     

变化环境下石羊河流域年径流频率计算

受气候变化与人类活动的影响,水文序列一致性遭到破坏,传统的频率计算方法受到挑战。针对石羊河流域8条河流年径流量序列,在基于水文变异诊断系统进行变异分析的基础上,采用条件概率分布法和混合分布法进行了变化环境下的年径流频率计算。结果表明,石羊河流域年径流量序列以向下跳跃变异为主,变异年份多集中在1960年附近;变异序列的两种分布均通过K-S检验,最优分布的拟合效率系数均大于0.97;对比由考虑变异的理论分布与不考虑变异的一致性P-III型分布确定的设计年径流量,当设计频率高于90%时,大靖河与西营河的设计年径流量值相差较大。因此,石羊河流域新建水利工程的规划设计,应适当考虑水文资料的变异特性,选择合理的频率分析方法确定设计年径流量。     

Quantifying uncertainties associated with depth duration frequency curves

Uncertainty in depth–duration–frequency (DDF) curves is usually disregarded in the view of difficulties associated in assigning a value to it. In central Iran, precipitation duration is often long and characterized with low intensity leading to a considerable uncertainty in the parameters of the probabilistic distributions describing rainfall depth. In this paper, the daily rainfall depths from 4 stations in the Zayanderood basin, Iran, were analysed, and a generalized extreme value distribution was fitted to the maximum yearly rainfall for durations of 1, 2, 3, 4 and 5 days. DDF curves were described as a function of rainfall duration (D) and return period (T). Uncertainties of the rainfall depth in the DDF curves were estimated with the bootstrap sampling method and were described by a normal probability density function. Standard deviations were modeled as a function of rainfall duration and rainfall depth using 10⁴ bootstrap samples for all the durations and return periods considered for each rainfall station.     

Singularity analysis of global zircon U-Pb age series and implication of continental crust evolution

Recently compiled global databases of igneous and detrital zircon U-Pb ages have been integrated with other types of isotope data (e.g., neodymium, hafnium, and oxygen) to characterize the episodic growth of the continental crust and the development of supercontinents. However, to what extent do the age peaks reflect the rate of continent crust growth or the differential preservation due to supercontinent settings is a long-standing question. Instead of analyzing amplitudes and periodicity of the age series, here, the analysis focuses on shapes of the individual age peaks described by a power law model for measuring the scale invariant fractality and singularity of time-series records. The results indicate that zircon age distributions around peaks follow power law distribution, regardless of the bin-size used to measure the age distribution. Based on the commonly accepted mechanisms (phase transition, self-organized criticality and multiplicative cascade processes) for generation of power law distributions one can relate the nonlinearity of the age peaks to short spurts of accelerated magmatic activities due to “avalanches” (superplumes, slab breakoff etc.) occurred during episodic convection of the mantle. The exponents of the power law age distributions estimated from the age peaks can be used as an index to quantify the intensity of a singularity. The values of exponents calculated at all major age peaks from five global databases exhibit an episodic nature, with a mean duration of approximately 600–800 Myr. Both intensity of the zircon episodes and their duration for the interval from 3 Ga to 0.5 Ga depict a descending trend, which may signify mantle cooling.