Impact of copula selection on geotechnical reliability under incomplete probability information

This paper aims to investigate the impact of copula selection on geotechnical reliability under incomplete probability information. The copula theory is introduced briefly. Thereafter, four copulas, namely Gaussian, Plackett, Frank, and No. 16 copulas, are selected to model the dependence structure between cohesion and friction angle. A copula-based approach is used to construct the joint probability density function of cohesion and friction angle with given marginal distributions and correlation coefficient. The reliability of an infinite slope and a retaining wall is presented to demonstrate the impact of copula selection on reliability. The results indicate that the probabilities of failure of geotechnical structures with given marginal distributions and correlation coefficient of shear strength parameters cannot be determined uniquely. The resulting probabilities of failure associated with different copulas can differ considerably. Such a difference increases with decreasing probability of failure. Significant difference in probabilities of failure could be observed for relatively small coefficients of variation of the shear strength parameters or a strong negative correlation between cohesion and friction angle. The Gaussian copula, often adopted out of expedience without proper validation, may not capture the dependence structure between cohesion and friction angle properly. Furthermore, the Gaussian copula may greatly underestimate the probability of failure for geotechnical structures.     

Estimating storm surge intensity with Poisson bivariate maximum entropy distributions based on copulas

This paper introduces four kinds of novel bivariate maximum entropy distributions based on bivariate normal copula, Gumbel–Hougaard copula, Clayton copula and Frank copula. These joint distributions consist of two marginal univariate maximum entropy distributions. Four types of Poisson bivariate compound maximum entropy distributions are developed, based on the occurrence frequency of typhoons, on these novel bivariate maximum entropy distributions and on bivariate compound extreme value theory. Groups of disaster-induced typhoon processes since 1949–2001 in Qingdao area are selected, and the joint distribution of extreme water level and corresponding significant wave height in the same typhoon processes are established using the above Poisson bivariate compound maximum entropy distributions. The results show that all these four distributions are good enough to fit the original data. A novel grade of disaster-induced typhoon surges intensity is established based on the joint return period of extreme water level and corresponding significant wave height, and the disaster-induced typhoons in Qingdao verify this grade criterion.     

Application of Archimedean copulas to the impact assessment of hydro-climatic variables in semi-arid aquifers of western India

The effects of rainfall and the El Niño Southern Oscillation (ENSO) on groundwater in a semi-arid basin of India were analyzed using Archimedean copulas considering 17 years of data for monsoon rainfall, post-monsoon groundwater level (PMGL) and ENSO Index. The evaluated dependence among these hydro-climatic variables revealed that PMGL-Rainfall and PMGL-ENSO Index pairs have significant dependence. Hence, these pairs were used for modeling dependence by employing four types of Archimedean copulas: Ali-Mikhail-Haq, Clayton, Gumbel-Hougaard, and Frank. For the copula modeling, the results of probability distributions fitting to these hydro-climatic variables indicated that the PMGL and rainfall time series are best represented by Weibull and lognormal distributions, respectively, while the non-parametric kernel-based normal distribution is the most suitable for the ENSO Index. Further, the PMGL-Rainfall pair is best modeled by the Clayton copula, and the PMGL-ENSO Index pair is best modeled by the Frank copula. The Clayton copula-based conditional probability of PMGL being less than or equal to its average value at a given mean rainfall is above 70% for 33% of the study area. In contrast, the spatial variation of the Frank copula-based probability of PMGL being less than or equal to its average value is 35–40% in 23% of the study area during El Niño phase, while it is below 15% in 35% of the area during the La Niña phase. This copula-based methodology can be applied under data-scarce conditions for exploring the impacts of rainfall and ENSO on groundwater at basin scales.     

两变量水文频率分布模型研究述评

水文变量多特征属性的频率分析,以及各种水文事件的遭遇及联合概率分布问题需要采用多变量概率分布模型解决。总结了当前应用最广泛的几种两变量概率分布模型,对各种模型的适用性和局限性做了详细分析,并介绍了一种新的两变量概率模型——Copula函数。现有模型大都基于变量之间的线性相关关系而建立,对于非线性、非对称的随机变量难以很好地描述;大部分模型假定各变量服从相同的边际分布或对变量间的相关性有严格的限定,从而限制了其应用。Copula函数所构造的两变量概率分布模型克服了现有模型的不足,它具有任意的边际分布,可以描述变量间非线性、非对称的相关关系。作为一种用于构造灵活的多变量联合分布的工具,Copula函数在水科学领域具有广阔的应用前景。     

Bivariate simulation using copula and its application to probabilistic pile settlement analysis

This paper aims to propose a procedure for modeling the joint probability distribution of bivariate uncertain data with a nonlinear dependence structure. First, the concept of dependence measures is briefly introduced. Then, both the Akaike Information Criterion and the Bayesian Information Criterion are adopted for identifying the best‐fit copula. Thereafter, simulation of copulas and bivariate distributions based on Monte Carlo simulation are presented. Practical application for serviceability limit state reliability analysis of piles is conducted. Finally, four load–test datasets of load–displacement curves of piles are used to illustrate the proposed procedure. The results indicate that the proposed copula‐based procedure can model and simulate the bivariate probability distribution of two curve‐fitting parameters underlying the load–displacement models of piles in a more general way. The simulated load–displacement curves using the proposed procedure are found to be in good agreement with the measured results. In most cases, the Gaussian copula, often adopted out of expedience without proper validation, is not the best‐fit copula for modeling the dependence structure underlying two curve‐fitting parameters. The conditional probability density functions obtained from the Gaussian copula differ considerably from those obtained from the best‐fit copula. The probabilities of failure associated with the Gaussian copula are significantly smaller than the reference solutions, which are very unconservative for pile safety assessment. If the strong negative correlation between the two curve‐fitting parameters is ignored, the scatter in the measured load–displacement curves cannot be simulated properly, and the probabilities of failure will be highly overestimated. Copyright © 2011 John Wiley & Sons, Ltd.     

Stochastic response surface method for reliability problems involving correlated multivariates with non-Gaussian dependence structure: Analysis under incomplete probability information

This paper aims to provide a stochastic response surface method (SRSM) that can consider non-Gaussian dependent random variables under incomplete probability information. The Rosenblatt transformation is adopted to map the random variables from the original space into the mutually independent standard normal space for the stochastic surrogate model development. The multivariate joint distribution is reconstructed by the pair-copula decomposition approach, in which the pair-copula parameters are retrieved from the incomplete probability information. The proposed method is illustrated in a tunnel excavation example. Three different dependence structures characterized by normal copulas, Frank copulas, and hybrid copulas are respectively investigated to demonstrate the effect of dependence structure on the reliability results. The results show that the widely used Nataf transformation is actually a special case of the proposed method if all pair-copulas are normal copulas. The effect of conditioning order is also examined. This study provides a new insight into the SRSM-based reliability analysis from the copula viewpoint and extends the application of SRSM under incomplete probability information.     

Drought characterization using a new copula-based trivariate approach

Meteorological drought is a natural climatic phenomenon that occurs over various time scales and may cause significant economic, environmental and social damages. Three drought characteristics, namely duration, average severity and peak intensity, are important variables in water resources planning and decision making. This study presents a new method for construction of three-dimensional copulas to describe the joint distribution function of meteorological drought characteristics. Using the inference function for margins, the parameters for six types of copulas were tested to select the best-fitted copulas. According to the values of the log-likelihood function, Galambos, Frank and Clayton were the selected copula models to describe the dependence structure for pairs of duration–severity, severity–peak and duration–peak, respectively. Trivariate cumulative probability, conditional probability and drought return period were also investigated based on the derived copula-based joint distributions. The proposed model was evaluated over the observed data of a Qazvin synoptic station, and the results were compared with the empirical probabilities. For measuring the model accuracy, R ², root mean square error (RMSE) and the Nash–Sutcliffe efficiency (NSE) criteria were used. Results indicated that R ², RMSE and NSE were equal to 0.91, 0.098 and 0.668, respectively, which demonstrate sufficient accuracy of the proposed model. Drought probabilistic characteristics can provide useful information for water resource planning and management.     

Quantile regression C-vine copula model for spatial extremes

A spatial quantile regression model is proposed to estimate the quantile curve for a given probability of non-exceedance, as function of locations and covariates. Canonical vines copulas are considered to represent the spatial dependence structure. The marginal at each location is an asymmetric Laplace distribution where the parameters are functions of the covariates. The full conditional quantile distribution is given using the Joe–Clayton copula. Simulations show the flexibility of the proposed model to estimate the quantiles with special dependence structures. A case study illustrates its applicability to estimate quantiles for spatial temperature anomalies.     

统计相关条件下降水及洪水预报误差相关分析

以三峡水库上游寸滩至万县区间降水预报误差和入库洪水预报误差相应数据为例,在探讨两者统计相关性的基础上,采用Frank、Gumbel、Clayton三种二元Copula连接函数分析了两预报误差的相关结构,以离差平方和最小为准则进行了Copula函数的选择,并与两预报误差独立情况下联合频率分布进行比较和分析。研究结果表明,降水预报误差和洪水预报误差的相关性对其二元联合分布有一定影响,同时,在两预报误差负相关条件下,其联合分布可做简化处理。本文研究结果可为水库预报调度风险管理提供决策参考。     

Impact of sample size on geotechnical probabilistic model identification

This paper aims to investigate the impact of sample size on geotechnical probabilistic model identification. First, the copula approach is presented to model the bivariate distribution of geotechnical parameters. Thereafter, the AIC scores are adopted to identify the best-fit marginal distribution and copula. Second, the variation of AIC scores because of small sample size is investigated using simulated data. Finally, the impact of the variation of AIC scores on identification of the best-fit marginal distribution and copula is examined. The minimum sample sizes for geotechnical data are also suggested to obtain a correct identification of the probabilistic models. The results indicate that the AIC scores estimated from a small sample exhibit large variation. The variation of the AIC scores has a significant impact on probabilistic model identification. The marginal distributions and copulas have a low percentage of correct identification when sample size is small. The percentages of correct identification for the marginal distributions and copulas increase with increasing sample size. The correlation coefficient between geotechnical parameters has a much larger impact on probabilistic model identification than the COV of geotechnical parameters. The suggested minimum sample sizes for geotechnical data are useful for guiding practical geotechnical site investigation.     

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.     

Conditional Simulation of Random Fields with Bivariate Gamma Isofactorial Distributions

This work focuses on a random function model with gamma marginal and bivariate isofactorial distributions, which has been applied in mining geostatistics for estimating recoverable reserves by disjunctive kriging. The objective is to widen its use to conditional simulation and further its application to the modeling of continuous attributes in geosciences. First, the main properties of the bivariate gamma isofactorial distributions are analyzed, with emphasis in the destructuring of the extreme values, the presence of a proportional effect (higher variability in high-valued areas), and the asymmetry in the spatial correlation of the indicator variables with respect to the median threshold. Then, we provide examples of stationary random functions with such bivariate distributions, for which the shape parameter of the marginal distribution is half an integer. These are defined as the sum of squared independent Gaussian random fields. An iterative algorithm based on the Gibbs sampler is proposed to perform the simulation conditional to a set of existing data. Such ‘multivariate chi-square’ model generalizes the well-known multigaussian model and is more flexible, since it allows defining a shape parameter which controls the asymmetry of the marginal and bivariate distributions.     

双参数Copula函数在洪水联合分布中的应用研究

采用金沙江屏山站和岷江高场站的日径流资料,分别构造了两江洪峰流量以及金沙江峰、量之间的联合分布,比较了单参数与双参数Archimedean Copula函数的拟合效果,估计了两江洪峰遭遇风险的条件概率,探讨了双参数Archimedean Copula函数在洪水联合分布中的应用。结果表明:双参数Copula函数拟合效果明显优于单参数Copula函数,且同时具有上尾相关性和下尾相关性,在水文极值分析中是有广阔的应用前景;当金沙江发生千年一遇洪水时,岷江发生千年一遇洪水的概率为3.458%。     

Spatial hydrological drought characteristics in Karkheh River basin,southwest Iran using copulas

Investigation on drought characteristics such as severity, duration, and frequency is crucial for water resources planning and management in a river basin. While the methodology for multivariate drought frequency analysis is well established by applying the copulas, the estimation on the associated parameters by various parameter estimation methods and the effects on the obtained results have not yet been investigated. This research aims at conducting a comparative analysis between the maximum likelihood parametric and non-parametric method of the Kendall \(\tau \) estimation method for copulas parameter estimation. The methods were employed to study joint severity–duration probability and recurrence intervals in Karkheh River basin (southwest Iran) which is facing severe water-deficit problems. Daily streamflow data at three hydrological gauging stations (Tang Sazbon, Huleilan and Polchehr) near the Karkheh dam were used to draw flow duration curves (FDC) of these three stations. The \(Q_{75}\) index extracted from the FDC were set as threshold level to abstract drought characteristics such as drought duration and severity on the basis of the run theory. Drought duration and severity were separately modeled using the univariate probabilistic distributions and gamma–GEV, LN2–exponential, and LN2–gamma were selected as the best paired drought severity–duration inputs for copulas according to the Akaike Information Criteria (AIC), Kolmogorov–Smirnov and chi-square tests. Archimedean Clayton, Frank, and extreme value Gumbel copulas were employed to construct joint cumulative distribution functions (JCDF) of droughts for each station. Frank copula at Tang Sazbon and Gumbel at Huleilan and Polchehr stations were identified as the best copulas based on the performance evaluation criteria including AIC, BIC, log-likelihood and root mean square error (RMSE) values. Based on the RMSE values, nonparametric Kendall-\(\tau \) is preferred to the parametric maximum likelihood estimation method. The results showed greater drought return periods by the parametric ML method in comparison to the nonparametric Kendall \(\tau \) estimation method. The results also showed that stations located in tributaries (Huleilan and Polchehr) have close return periods, while the station along the main river (Tang Sazbon) has the smaller return periods for the drought events with identical drought duration and severity.     

基于Copulas函数的二维干旱变量联合分布

通过构建干旱变量的联合分布揭示干旱演变规律,可作为干旱分析的重要手段。基于8种单参数族的Copulas函数进行新疆乌鲁木齐和石河子气象站二维干旱变量的联合分布。经拟合优度评价:Frank Copula对干旱历时和干旱烈度、干旱历时和烈度峰值的拟合度最好;Clayton Copula对于干旱烈度和烈度峰值的拟合效果最好。二维变量联合超越概率值随单变量值的减小而增大;单变量的重现期介于二维变量联合重现期与同现重现期之间。表明Copulas函数能够描述二维干旱特征变量的联合分布。     

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.     

水文双变量重现期分析及在干旱中应用

单变量水文统计中一些广为接受的概念在多变量环境下尚缺乏深入分析,也易被误解,如N年内重现期大于等于T的多变量事件发生的次数与N/T的关系。实践中,多变量联合重现期与其边缘分布变量重现期的一些经验关系被发现并通过了案例验证分析,但缺乏解释和推导。基于GH Copula推导了双变量联合重现期与边缘分布变量重现期的关系以及双变量事件发生次数与其重现期、变量相关程度间的定量关系。以昆明56年的逐月SPI(Standardized Precipitation Index)和SRI(Standardized Runoff Index)识别了干旱事件,采用GH Copula构建了干旱历时和烈度的联合分布函数,验证了双变量联合重现期与边缘分布变量重现期的关系以及多变量事件发生次数与其重现期的定量关系。表明不宜以“and”第1重现期是否接近于比该干旱事件的旱情更重的干旱发生的平均时间间隔来说明干旱特征值重现期分析的合理性。变量的相关性不强时,需谨慎采用边缘分布变量重现期的较大值近似代替“and”事件的第1重现期。     

Generation of multivariate cross-correlated geotechnical random fields

A geotechnical problem that involves several spatially correlated parameters can be best described using multivariate cross-correlated random fields. The joint distribution of these random variables cannot be uniquely defined using their marginal distributions and correlation coefficients alone. This paper presents a generic methodology for generating multivariate cross-correlated random fields. The joint distribution is rigorously established using a copula function that describes the dependence structure among the individual variables. The cross-correlated random fields are generated through Cholesky decomposition and conditional sampling based on the joint distribution. The random fields are verified regarding the anisotropic scales of fluctuation and copula parameters.     

基于三维copula函数的多水文区丰枯遭遇分析

不同水文区的丰枯遭遇概率分析属于多变量概率分布问题,涉及的水文区越多,变量的维数就越高,问题就越复杂.为找到一种简单通用的多变量(n≥3)水文概率问题的求解方法,以不同水文区丰枯遭遇概率分析为例,引入三维copula函数构建多变量联合概率模型,将其用于分析长江、淮河及黄河流域的径流量的联合概率和条件概率问题。研究结果表明,当变量维数n≥3时,由copula函数可以很容易地构建多变量概率分布模型;对一组水文数据系列,有多个不同copula函数可以选择,可采用拟合优度检验方法择优;copula函数构建的多变量概率模型,可以计算各种条件下的联合概率分布,可以分析各种不同量级水文变量的遭遇概率和条件概率;通过与多维转换为一维方法的比较,三维Frank copula函数具有更优良的拟合优度、无偏性及有效性,且计算更简便。     

Risk variation of reservoir regulation during flood season based on bivariate statistical approach under climate change: a case study in the Chengbihe reservoir,China

The risk analysis of reservoir regulation in the flood season is crucial and provides the valuable information for reservoir flood control, safety operation, and decision making, especially under climate change. The purpose of this study is to propose a framework for reasonably estimating the variation of reservoir regulation risk including the dam extreme risk and the overtopping risk during the flood season under climate change. The framework consists of an integrated diagnostic system for detecting the climate abrupt change time, a copula function-based bivariate statistical approach for modeling the dependence between the flood peak and flood volume, a Monte Carlo simulation for generating numerous random flood peak–volume pairs, and a risk calculation model for routing the design flood hydrographs to obtain the frequency curve of the maximum water level reached in front of dam and evaluating the reservoir regulation risk. The methodology was implemented in the Chengbihe reservoir in south China by using the 55-year (1963–2017) hydrometeorological data, including temperature, evaporation, precipitation, and streamflow, in the flood season. Results show that the hydrometeorological series during the flood season changed abruptly in 1992 and the entire data can be divided into two periods (1963–1992 and 1993–2017). The dam extreme risk and overtopping risk during the two periods are assessed, respectively, and a comparison analysis is made based on different flood limit water-level schemes (185.00–188.50 m). It demonstrates that both the dam extreme risk and the dam overtopping risk increase under the influence of climate change. The dam extreme risk increases by 22.91–95.03%, while the climate change-induced increase in the dam overtopping risk is between 38.62 and 123.59%, which indicates that the dam overtopping risk is more sensitive to climate change than the dam extreme risk. The risk evaluations in the study are of great significance in the safety operation and risk management of reservoirs under future climate change.