Characterisation of Drought Properties with Bivariate Copula Analysis

Drought severity and duration are usually modelled independently. However, these two characteristics are known to be related. To model this relationship, a joint distribution of drought severity and duration using a bivariate copula model is proposed and applied to daily rainfall data (1976–2007) of 30 rain gauge stations in Peninsular Malaysia. The drought characteristics are classified using the standardized precipitation index (SPI) and their univariate marginal distributions are further identified by fitting exponential, gamma, generalized extreme value, generalized gamma, generalized logistics, generalized pareto, gumbel max, gumbel min, log-logistic, log-pearson3, log-normal, normal, pearson 5, pearson 6 and weibull distributions. The three-parameter log-normal distribution is identified as the best fitting distribution for drought severity while the generalized pareto distribution is determined as the most appropriate distribution for drought duration with respect to the application of the Anderson-Darling procedure. The dependency among the drought properties is analysed using Kendall’s τ method. The maximum likelihood estimation of the univariate marginal distributions and the maximisation of the bivariate likelihood are employed to compute the Akaike Information Criterion (AIC) values in verifying the best fitting copula distribution. The Galambos distribution is recognised as the most appropriate copula distribution for describing the relationship between drought severity and duration. The conditional drought probability and drought return period are further described to explain the drought properties comprehensively. The probabilities of drought occurrences under certain circumstances with a specific seriousness or duration can be determined in order to verify the possibility of drought episodes. The return period of a recurrent drought has also been investigated to identify the time-interval for repeated drought occurrences under similar situation.     

Risk Assessment of Droughts in Gujarat Using Bivariate Copulas

This study presents risk assessment of hydrologic extreme events droughts in Saurashtra and Kutch region of Gujarat state, India. Drought is a recurrent phenomenon and risk assessment of droughts can play an important role in proper planning and management of water resources in the study region. In the study, drought events are characterized by severity and duration, and drought occurrences are modeled by Standardized Precipitation Index (SPI) computed on mean areal precipitation, aggregated at a time scale of 6?months for the period 1900?C2008. After evaluating several distribution functions, drought variable??severity is best described by non-parametric kernel density, whereas duration is best fitted by exponential distribution. Considering the extreme nature of drought variables, the upper tail dependence copula families including two Archimedean??Gumbel-Hougaard, BB1 and one elliptical??Student??s t copulas are evaluated for modeling joint distribution of drought variables. On evaluating their performance using various goodness-of-fit measures, Gumbel-Hougaard copula is found to be the best performing copula in modeling the joint dependence structure of drought variables. Also, while comparing with traditional bivariate distributions, the copula based distributions are resulted in better performance as compared to bivariate log-normal and the logistic model for bivariate extreme value distributions. Then joint and conditional return periods of drought characteristics are derived, which can be helpful for risk based planning and management of water resources systems in the study region.     

Bivariate Flood Frequency Analysis of Upper Godavari River Flows Using Archimedean Copulas

In this paper, a copula based methodology is presented for flood frequency analysis of Upper Godavari River flows in India. By using the specific advantages of copula method in modeling the joint dependence structure of uncertain variables, this study applies Archimedean copulas for frequency analysis of flood characteristics annual peak flow, flood volume and flood duration. To determine the best fit marginal distributions for flood variables, few parametric and nonparametric probability distributions are examined and the best fit model is adopted for copula modeling. Four Archimedean family of copulas, namely Ali-Mikhail-Haq, Clayton, Gumbel-Hougaard and Frank copulas are evaluated for modeling the joint dependence of annual peak flow-volume, and flood volume-duration pairs. The performance of two parameter estimation methods, namely method-of-moments-like estimator based on inversion of Kendall??s tau and maximum pseudo-likelihood estimator for copulas are investigated. On performing Monte Carlo simulation to assess the performance of copula distributions in modeling the joint dependence structure of flood variables, it is found that the developed copula models are well representing the observed flood characteristics. From standard statistical tests, Frank copula has been identified as the best fitted copula for both bivariate models. The Frank copula function is used for obtaining joint and conditional return periods of flood characteristics, which can be useful for risk based design of water resources projects.     

Multivariate Frequency Analysis of Meteorological Drought Using Copula

The multivariate frequency analysis of droughts for Agartala (India) was carried out in the present study. The meteorological drought was modelled using Standardised Precipitation Index(SPI) at the time scale of 1, 3, 6 and 12 months. Three droughts variables i.e., duration, severity, interval were determined for SPI at the time scale of 1, 3, 6 and 12 months. For the construction of bivariate and trivariate joint distributions Archimedean and metaelliptical copulas were used. Upper tail dependence test was also carried out. The best copula was selected based on minimum value Akaike’s information criteria (AIC)) and Schwarz information criterion(SIC). The drought risk was estimated using joint probabilities and return period for the study area.     

Multivariate Drought Frequency Analysis using Four-Variate Symmetric and Asymmetric Archimedean Copula Functions

In drought frequency analysis, as the number of drought variables increases, the joint behavior between these variables needs to be studied. Therefore, this study aims to develop a flexible four-variate joint distribution function of the regional stochastic nature of drought. Using run theory, drought duration, severity, peak, and inter-arrival time were abstracted from the Standardized Precipitation Evapotranspiration Index (SPEI) aggregated at six months, observed in mainland China between 1961 and 2013. As these drought variables showed significant dependence properties and followed different marginal distributions, we employed and compared six four-variate symmetric and asymmetric Archimedean copulas (i.e., Frank, Clayton, Gumbel–Hougaard). The best-fitting model for each region was carefully selected using RMSE, AIC, and BIAS goodness-of-fit tests. Results revealed that the empirical and theoretical probabilities of the symmetric Clayton in regions NE (Northeast), CS (Central and Southern China), EMC (Entire China), and symmetric Frank in regions NC (North China), SC (South China), IM (Inner Mongolia), NW (Northwest), TP (Tibet Plateau) agreed well. Symmetric Frank copula was considered the best-fit for station-based drought analysis in EMC. Based on these copulas, the drought probabilities and return periods for the occurrence of drought events over the next 5, 10, 20, 50, and 100 years in each region were hereby comprehensively explained, and the results shown here could be helpful in the appraisal of the adequacies of water supply systems under drought conditions in all regions. This study showed that a four-variate copula approach is a vital tool for probabilistic interpretation of hydrological and meteorological data in the different climatic region of mainland China.

     

Analysis of Rainfall Severity and Duration in Victoria,Australia using Non-parametric Copulas and Marginal Distributions

The analysis of joint probability distributions of rainfall characteristics such as severity and duration is important in water resources management. Deriving their distributions using standard statistical techniques are often problematical due to its complexity. Standard methods usually assume that the rainfall characteristics are independent or that their marginal distributions belong to the same family of distributions. The use of copulas based methodologies can circumvent these restrictions and are therefore increasingly popular. However, the copulas and marginal distributions that are commonly used belong to specific parametric families and their adoption could lead to spurious inferences if the underlying assumptions are violated. For this reason, we recommend a nonparametric or semiparametric approach to estimate the joint distribution of rainfall characteristics. In this paper, we introduce and compare several copula–based approaches, each involving a combination of parametric or nonparametric marginal distributions conjoined by a parametric or nonparametric copula. An empirical illustration of the different approaches using rainfall data collected from six stations in the state of Victoria, Australia, demonstrated that a nonparametric approach can often give better results than a purely parametric approach.     

Analysing Drought Severity and Areal Extent by 2D Archimedean Copulas

Droughts can be considered as multidimensional hazardous phenomena characterised by three attributes: severity, duration and areal extent. Conventionally, drought events are assessed for their severity, using drought indices such as SPI (Standardised Precipitation Index), RDI (Reconnaissance Drought Index), PDSI (Palmer Drought Severity Index) and many others. This approach may be extended to incorporate the modelling of an additional dimension, the duration or the areal extent. Since the marginal distributions describing these dimensions of drought are often different, no simple mixed probability distribution can be used for the bivariate frequency analysis. The copula approach seems to be sufficiently general and suitable for this type of analysis. It is the aim of this paper to analyse droughts as two-dimensional phenomena, including drought severity and areal extent. In this paper, the Gumbel-Hougaard copula from the Archimedean family is used for this two-dimensional frequency analysis. Annual data on historical droughts from Eastern Crete are analysed for their severity and areal extent, producing copula-based probability distributions, incorporating Gumbel marginal probability functions. Useful conclusions are derived for estimating the «OR» return period of drought events related to both severity and areal extent.     

A New Method for Joint Frequency Analysis of Modified Precipitation Anomaly Percentage and Streamflow Drought Index Based on the Conditional Density of Copula Functions

In this study, a new method was proposed to model the occurrence of related variables based on the conditional density of copula functions. The proposed method was adopted to investigate the dynamics of meteorological and hydrological droughts in the Zarinehroud basin, southeast of Lake Urmia, during the period 1994–2015. For this purpose, the modified precipitation anomaly percentage and streamflow drought indices were extracted. Finally, the joint frequency analysis of duration-duration and severity-severity characteristics of meteorological and hydrological droughts was analyzed. Analysis of 7 different copulas used to create the joint distribution in the Zarinehroud basin indicated that the Frank copula had the best performance in describing the relationship between the meteorological and hydrological drought severities and durations. By examining the results of the bivariate analysis of duration-duration of meteorological and hydrological droughts at different stations, the expected meteorological and hydrological drought durations were estimated in the years ahead. For example, at Chalkhmaz station, 4- to 7-month duration for the hydrological drought and 9- to 12-month duration for the meteorological drought is expected in the years ahead. The joint frequency analysis of drought characteristics allows to determine the meteorological and hydrological drought characteristics at a single station at the same time using joint probabilities. Also, the results indicated that by knowing the conditional density, the hydrological drought characteristics can be easily estimated for the given meteorological drought characteristics. This could provide users and researchers useful information about the probabilistic behavior of drought characteristics for optimal operation of surface water.

     

基于Copula函数的水文干旱联合概率分布研究

双变量联合分布及重现期计算是水文干旱研究的重要内容。根据游程理论确定水文干旱事件,利用Copula函数建立干旱历时和烈度的联合分布,计算不同干旱事件的重现期。实例分析结果表明,联合分布可以考虑干旱历时与烈度不同组合情况,计算的联合重现期大于边际分布计算结果,泾河和北洛河最长历时干旱事件联合重现期为480 a和342 a。     

Goodness-of-fit tests for multi-dimensional copulas:Expanding application to historical drought data

The question of how to choose a copula model that best fits a given dataset is a predominant limitation of the copula approach, and the present study aims to investigate the techniques of goodness-of-fit tests for multi-dimensional copulas. A goodness-of-fit test based on Rosenblatt’s transformation was mathematically expanded from two dimensions to three dimensions and procedures of a bootstrap version of the test were provided. Through stochastic copula simulation, an empirical application of historical drought data at the Lintong Gauge Station shows that the goodness-of-fit tests perform well, revealing that both trivariate Gaussian and Student t copulas are acceptable for modeling the dependence structures of the observed drought duration, severity, and peak. The goodness-of-fit tests for multi-dimensional copulas can provide further support and help a lot in the potential applications of a wider range of copulas to describe the associations of correlated hydrological variables. However, for the application of copulas with the number of dimensions larger than three, more complicated computational efforts as well as exploration and parameterization of corresponding copulas are required.     

基于Copula函数的鄱阳湖都昌站枯水多变量频率分析

以鄱阳湖湖区都昌水文站1958—2007年长系列日平均水位资料为基础,采用干旱分析中应用广泛的游程理论进行湖泊枯水的识别,提取枯水历时、平均枯水强度和最大枯水强度作为湖泊枯水特征变量;采用Archimedean Copula函数构造了变量的联合分布,分析了枯水事件出现的概率,计算了枯水事件的联合重现期和条件重现期。结果表明Weibull分布、Logistic分布和GEV分布分别适于作为干旱历时、平均枯水强度和最大枯水程度的边缘分布函数;Clayton Copula函数适于都昌站枯水多变量联合分布的拟合;联合分布考虑了枯水事件多个变量之间的相关性,能提供更多的湖泊枯水特征信息,更能全面揭示湖泊枯水的概率统计特征。     

干旱事件重现期计算问题研究

干旱重现期大小是用于评价干旱事件严重程度的重要指标。干旱重现期的计算涉及给定阈值下干旱过程划分(识别)、样本系列分布函数拟合等关键环节,其中干旱阈值的确定是前提。提出以干旱事件的最长调查期为约束条件确定干旱阈值的思路,即根据样本计算的干旱事件最大重现期不应超过最长调查期,以此为据确定干旱阈值并从样本序列中识别干旱事件。同时,针对因干旱历时样本经验点据"平台式"过度集中而导致的频率曲线适线困难问题,建议采用基于游程理论的游程长度分布函数估计干旱历时概率分布。以青海民和县1932年-2010年的月降雨资料为例,对上述方法进行了应用研究,结合Copula函数计算了干旱事件的重现期。     

Analysis of Irrigation Needs Using an Approach Based on a Bivariate Copula Methodology

The problem of drought probability has been investigated by several authors, who have usually analysed droughts using various drought indices such as the Standard Precipitation Index. Various aspects of time series of such indices (intensity, severity and duration) were investigated by several authors using a copula method. Because such analysis is based on only one basic climatic variable, this paper addresses a different approach, i.e., joint analysis of the severity and duration of the most demanding potential annual irrigation periods by a bivariate copula method. Characteristics of these periods are derived from both temperature and precipitation. Maximum annual duration of the potential irrigation period and corresponding rainfall deficit were inferred from these basic variables as inputs to two-dimensional probability analysis by the copula method, because this offers more direct answers to questions of irrigation needs. Results indicate the suitability of the proposed method for analysis of irrigation needs, with greater benefits than the typical one-dimensional analysis of individual climatic variables. A case study for testing the method was done for southwestern Slovakia, for which the frequency of irrigation needs was estimated. Example results indicate that every second year, a one-month period can be expected in which temperatures are >25^°C and there is a moisture deficit of ～30 mm. Even more significant periods of drought can be expected, for example, with a 5 or 10-year return period. These phenomena significantly damage agriculture yields, so requirements for irrigation structures in the study area are indicated by the proposed method.     

Stochastic Simulation of Daily Suspended Sediment Concentration Using Multivariate Copulas

An estimation of daily suspended sediment concentration (SSC) is required for water resource and environmental management. The traditional methods for simulating daily SSC focus on modeling the SSCs themselves, whereas the cross-correlation structure between SSC and streamflow has received only minor attention. To address this issue, we propose a stochastic method to generate long-term daily SSC using multivariate copula functions that account for temporal and cross dependences in daily SSCs. We use the conditional copula method to construct daily multivariate distributions to alleviate the complications and workload of parameter estimations using high-dimensional copulas. The observed daily streamflow and SSC data are normalized using the normal quantile transform method to relax the computationally intensive model of building daily marginal distributions. Daily SSCs can thus be simulated through the multivariate conditional distribution using previous daily SSC and concurrent daily streamflow values. The proposed method is rigorously examined by application to a case study at the Pingshan station in the Jinsha River Basin, China, and compared with the bivariate copula method. The results show that the proposed method has a high degree of accuracy, in preserving the statistics and temporal correlation of daily SSC observations, and better preserves the lag-0 cross correlation compared with the bivariate copula method. The multivariate copula framework proposed here can accurately and efficiently generate long-term daily SSC data for water resource and environmental management, which play a critical role in accurately estimating the frequency and magnitude of extreme SSC events.

     

Applying Bivariate Normal Distribution to Flood Frequency Analysis

Abstract

This article provides a procedure for using the bivariate normal distribution to describe the joint distributions of correlated flood peaks and volumes as well as correlated flood volumes and durations. The Box-Cox transformation (power-transformation) method is used to normalize original marginal distributions of flood peaks, volumes, and durations regardless of the original forms of these distributions. The power-transformation parameter is estimated using the maximum likelihood method. The joint cumulative distribution function, the conditional cumulative distribution function, and the associated return periods can be readily obtained based on the bivariate normal distribution. The method is tested and validated using observed flood data from the Batiscan River basin in the province of Quebec, Canada. The resulting theoretical distributions show a good fit to observed ones.     

The Impact of Climate Change on Hydro-Meteorological Droughts Using Copula Functions

Climate change has made many alterations to the climate of earth, including hydro-climatic extreme events. To investigate the impact of climate change on hydro-meteorological droughts in the Kamal-Saleh dam basin in Markazi province, Iran, proportional to future climate conditions, a new and comprehensive index was developed with the aim of accurately estimating drought in a more realistic condition. This aggregate drought index (ADI) represented the main meteorological and hydrological characteristics of drought. Temperature and precipitation projections for future climates were simulated by five CMIP5 models and downscaled over the study area during 2050s (2040–2069) and 2080s (2070–2099) relative to the baseline period (1976–2005). By fitting five univariate distribution functions on drought severity and duration, proper marginal distributions were selected. The joint distribution of drought severity and duration was chosen from five types of copula functions. The results revealed that in future, severe droughts are expected to frequently occur in a shorter period.

     

Fuzzy Stress-based Modeling for Probabilistic Irrigation Planning Using Copula-NSPSO

Hydrological uncertainties are the main components of a sustainable framework in agricultural water management. Prediction of drought as a meteorological phenomenon should be considered to define the groundwater exploitation strategies. This study was conducted to develop a multiobjective-bivariate structure for reducing the soil moisture deficit and groundwater withdrawal in the Qazvin Irrigation District, Qazvin province, Iran. Therefore, non-dominated sorting theory, self-organizing particle swarm optimization and bivariate copula functions were incorporated under fuzzy uncertainty analysis. The results showed that the generalized extreme values and log-normal distribution functions had the best fitness on the drought peak and severity with Kolmogorov Smirnov amounts of 0.08 and 0.17, respectively. Furthermore, the goodness-of-fit tests were indicated the Joe joint function (MLE = 11) is the appropriate function for estimating the probabilistic values of drought characteristics. Proposed plans were to increase the water use efficiency for improving the expected yield production by an average of 20%. Furthermore, the standardized groundwater index was decreased from 1.1 to –4.3 for winter crops.

     

基于地下水埋深的区域干旱频率分析研究

以地下水埋深为水文干旱指标，在分析研究区实际旱情发生频率的基础上，采用相邻时段地下水埋深变化的累积频率法，识别由地下水干旱历时和干旱烈度组成的干旱特征变量值，并从降水的角度，分析了用相邻时段地下水埋深变化表征干旱的合理性。在采用适线法确定单个干旱特征变量累积分布的基础上，利用Copula函数构建了干旱历时与干旱烈度间的联合分布，并计算了相应的干旱重现期。对淮北平原砀山县的地下水干旱频率分析结果表明：基于相邻时段地下水埋深变化的累积频率法，识别的干旱历时和干旱烈度及其对应的干旱重现期与砀山县实际受旱情况相符。提出的基于地下水埋深的区域干旱频率分析法，概念清晰，在其它类似的平原区域具有推广价值。     

The Gumbel Mixed Model Applied to Storm Frequency Analysis

A bivariate extreme value distribution, namely theGumbel mixed model constructed from Gumbel marginaldistributions is employed to analyze the joint distributionof correlated storm peak (maximum rainfall intensity) andamount. Based on its marginal distributions, the jointdistribution, the conditional probability distribution, andthe associated return periods can be deduced. Parameters ofthe bivariate distribution model are estimated based on itsmarginal distributions by the method of moments (MM). Theusefulness of the model is demonstrated by using it torepresent multivariate storm events at the Niigatameteorological station in Japan.