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.