Asymmetric copula in multivariate flood frequency analysis

The univariate flood frequency analysis is widely used in hydrological studies. Often only flood peak or flood volume is statistically analyzed. For a more complete analysis the three main characteristics of a flood event i.e. peak, volume and duration are required. To fully understand these variables and their relationships, a multivariate statistical approach is necessary. The main aim of this paper is to define the trivariate probability density and cumulative distribution functions. When the joint distribution is known, it is possible to define the bivariate distribution of volume and duration conditioned on the peak discharge. Consequently volume–duration pairs, statistically linked to peak values, become available. The authors build trivariate joint distribution of flood event variables using the fully nested or asymmetric Archimedean copula functions. They describe properties of this copula class and perform extensive simulations to highlight differences with the well-known symmetric Archimedean copulas. They apply asymmetric distributions to observed flood data and compare the results those obtained using distributions built with symmetric copula and the standard Gumbel Logistic model.     

Assessment of hydrological droughts for the Yellow River,China, using copulas

Droughts are one of the normal and recurrent climatic phenomena on Earth. However, recurring prolonged droughts have caused far‐reaching and diverse impacts because of water deficits. This study aims to investigate the hydrological droughts of the Yellow River in northern China. Since drought duration and drought severity exhibit significant correlation, a bivariate distribution is used to model the drought duration and severity jointly. However, drought duration and drought severity are often modelled by different distributions; the commonly used bivariate distributions cannot be applied. In this study, a copula is employed to construct the bivariate drought distribution. The copula is a function that links the univariate marginal distributions to form the bivariate distribution. The bivariate return periods are also established to explore the drought characteristics of the historically noticeable droughts. The results show that the return period of the drought that occurred in late 1920s to early 1930s is 105 years. The significant 1997 dry‐up phenomenon that occurred in the downstream Yellow River (resulting from the 1997–1998 drought) only has a return period of 4·4 years and is probably induced by two successive droughts and deteriorated by other factors, such as human activities. Copyright © 2007 John Wiley & Sons, Ltd.     

基于 Copula函数的韩江流域旱涝组合事件概率特征研究

以韩江流域潮安、横山和溪口3个代表性站点1959—2000年日流量数据为基础资料,选取4种边缘分布函数拟合各站点春、夏、秋、冬四季的径流距平百分率序列,并利用线性矩法估计参数,在此基础上采用4种Copula函数拟合春－夏、夏－秋、秋－冬、冬－春径流距平百分率序列,然后计算出各站点连旱连涝和旱涝交替这2类旱涝组合事件发生的概率,结果表明：1）基于Copula函数建立的联合分布模型可以较好地描述季节间连旱连涝和旱涝交替这2类旱涝组合事件发生的概率;2）潮安与溪口站夏－秋、秋－冬时段平均每4年发生一次连旱连涝的事件,春－夏、冬－春时段发生连旱和先旱后涝的概率均在10％左右;横山站秋冬季节平均每1.28年就发生一次连续干旱的事件。     

基于多变量概率分析的珠江流域干旱特征研究

根据珠江流域42 个站点1960-2005 年的46 年日降水资料,以6 个月尺度的SPI 值表征珠江流域干旱情况。通过Mann-Kendall 趋势分析研究了珠江流域干旱时间演变特征,同时基于多变量Copula 函数,在定义的两种不同干旱情景下,根据两变量联合重现期及其对应的第二重现期,比较性地研究了珠江流域的干旱风险。研究结果表明：(1) 珠江流域西部有变旱的趋势而东部有变湿润的趋势,其中显著的变旱趋势集中在11、12、1 月,显著的变湿趋势集中在6、7 月。同时除个别站点外,干旱历时和干旱严重程度趋势不显著;(2) 珠江流域整体上干旱风险较大,东部要比西部干旱风险高。珠江流域内发生长历时干旱时,干旱严重程度也往往很大,对干旱风险管理很不利。在发生严重程度的干旱时,珠江流域东部的干旱高风险区域增加,威胁珠江三角洲地区水资源安全;(3) 由于第二重现期综合考虑了各种情况,可能第二重现期对干旱风险分析的结果更稳健一些。     

Copula joint function and its application in probability seismic hazard analysis

A mature mathematical technique called copula joint function is introduced in this paper, which is commonly used in the financial risk analysis to estimate uncertainty. The joint function is generalized to the n-dimensional Frank’s copula. In addition, we adopt two attenuation models proposed by YU and Boore et al, respectively, and construct a two-dimensional copula joint probabilistic function as an example to illustrate the uncertainty treatment at low probability. The results show that copula joint func...     

基于Copula函数的珠江流域河川径流丰枯遭遇

面对珠江水量统一调度的新要求,采用Copula函数对珠江流域内3个控制性水文站的径流量进行研究,分析其丰枯遭遇概率。结果显示：1）三大水系两两间丰枯同步的概率大于丰枯异步的概率（汛期西江和东江除外）,其中北江与东江径流量一致性较好,丰枯同步的概率最大,其中全年丰枯同步的概率高达69.88%,汛期丰枯同步的概率为63.36%;西江与东江丰枯同步的概率最小,汛期只有47.88%;西北江丰枯同步的概率居中,全年、汛期和非汛期丰枯同步的概率分别为55.10%、56.02%和50.12%。2）三大水系三维联合分布研究表明,三大水系丰枯同步的概率在全年、汛期和非汛期均较大,依次为40.16%、38.11%和36.49%,其中同丰和同枯的概率远大于同平的概率。3）利用三大水文站的联合分布,可获得各水文站不同径流量遭遇的概率,以及特定概率下各水文站径流量的可能组合,可对珠江水量统一调度提供参考。     

Modelling bivariate extreme precipitation distribution for data‐scarce regions using Gumbel–Hougaard copula with maximum entropy estimation

A new method of parameter estimation in data scarce regions is valuable for bivariate hydrological extreme frequency analysis. This paper proposes a new method of parameter estimation (maximum entropy estimation, MEE) for both Gumbel and Gumbel–Hougaard copula in situations when insufficient data are available. MEE requires only the lower and upper bounds of two hydrological variables. To test our new method, two experiments to model the joint distribution of the maximum daily precipitation at two pairs of stations on the tributaries of Heihe and Jinghe River, respectively, were performed and compared with the method of moments, correlation index estimation, and maximum likelihood estimation, which require a large amount of data. Both experiments show that for the Ye Niugou and Qilian stations, the performance of MEE is nearly identical to those of the conventional methods. For the Xifeng and Huanxian stations, MEE can capture information indicating that the maximum daily precipitation at the Xifeng and Huanxian stations has an upper tail dependence, whereas the results generated by correlation index estimation and maximum likelihood estimation are unreasonable. Moreover, MEE is proved to be generally reliable and robust by many simulations under three different situations. The Gumbel–Hougaard copula with MEE can also be applied to the bivariate frequency analysis of other extreme events in data‐scarce regions.     

Hydrological uncertainty processor based on a copula function

Quantifying the uncertainty in hydrological forecasting is valuable for water resources management and decision-making processes. The hydrological uncertainty processor (HUP) can quantify hydrological uncertainty and produce probabilistic forecasts under the hypothesis that there is no input uncertainty. This study proposes a HUP based on a copula function, in which the prior density and likelihood function are explicitly expressed, and the posterior density and distribution obtained using Monte Carlo sampling. The copula-based HUP was applied to the Three Gorges Reservoir, and compared with the meta-Gaussian HUP. The Nash-Sutcliffe efficiency and relative error were used as evaluation criteria for deterministic forecasts, while predictive QQ plot, reliability, resolution and continuous rank probability score (CRPS) were used for probabilistic forecasts. The results show that the proposed copula-based HUP is comparable to the meta-Gaussian HUP in terms of the posterior median forecasts, and that its probabilistic forecasts have slightly higher reliability and lower resolution compared to the meta-Gaussian HUP. Based on the CRPS, both HUPs were found superior to deterministic forecasts, highlighting the effectiveness of probabilistic forecasts, with the copula-based HUP marginally better than the meta-Gaussian HUP.     

Check dam layout optimization on the stream network for flood mitigation: surrogate modelling with uncertainty handling

The configuration of check dams and their numbers throughout a basin are important factors for reducing floods in downstream reaches of rivers. In this paper, a stochastic model based on surrogate modelling and Monte Carlo simulation, linked to an evolutionary optimization tool, is developed to assign the optimal sites and number of check dams on a stream network. To handle uncertainty of rainfall variables and their correlation structures, the copula method is employed and an artificial neural network (ANN) is used to emulate the computationally expensive hydrological model, HEC-HMS, within the optimization routines. The prepared modelling framework is applied to a mountainous basin to determine the arrangement of check dams in its sub-basins. The experimental results show that optimal strategies can reduce the expected value of peak flood discharges by up to 50%, with significantly lower costs or number of check dams, relative to a traditional approach with a large number of check dams in sub-basins, presenting a maximum of 21% efficiency.