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.     

Fitting Drought Duration and Severity with Two-Dimensional Copulas

This study aims to model the joint drought duration and severity distribution using two-dimensional copulas. The method of inference function for margins (IFM method) is employed to construct copulas. Two separate maximum likelihood estimations of univariate marginal distributions are performed first, then followed by a maximization of the bivariate likelihood as a function of the dependence parameters. The drought duration and severity are assumed to be exponential and gamma distributions, respectively. Several copulas are tested to determine the best data fitted copula. Droughts, defined using the Standardized Precipitation Index (SPI), of Wushantou (Taiwan) are employed as an example to illustrate the proposed methodology. The copula fitting results for drought duration and severity are quite satisfactory. The bivariate drought analyses, including the joint probabilities and bivariate return periods, based on the derived copula-based joint distribution are also investigated to demonstrate the advantages of bivariate modeling of droughts.     

Trivariate Frequency Analyses of Peak Discharge,Hydrograph Volume and Suspended Sediment Concentration Data Using Copulas

Copula functions are often used for multivariate frequency analyses, but discharge and suspended sediment concentrations have not yet been modelled together with the use of 3-dimensional copula functions. One hydrological station from Slovenia and five stations from USA with watershed areas from 920 km² to 24,996 km² were used for trivariate frequency analyses of peak discharges, hydrograph volumes and suspended sediment concentrations. Different parametric marginal distributions were applied and parameters were estimated with the method of L-moments. Maximum pseudo-likelihood method was used for copula parameters estimation. With the use of statistical and graphical tests we selected the most appropriate copula model. Symmetric and asymmetric versions of Archimedean copulas were applied according to the dependence characteristics of the individual stations. We selected Gumbel-Hougaard copula as the most appropriate model for all discussed stations. Primary joint return periods OR and secondary Kendall’s return periods were calculated and comparison between selected copula functions was made. We can conclude that copula functions are useful mathematical tool, which can also be used for modelling variables that are presented in this paper.     

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.     

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.     

Spatial Assessment of Climate Risk for Investigating Climate Adaptation Strategies by Evaluating Spatial-Temporal Variability of Extreme Precipitation

In response to the impacts of extreme precipitation on human or natural systems under climate change, the development of climate risk assessment approach is a crucial task. In this paper, a novel risk assessing approach based on a climate risk assessment framework with copula-based approaches is proposed. Firstly, extreme precipitation indices (EPIs) and their marginal distributions are estimated for historical and future periods. Next, the joint probability distributions of extreme precipitation are constructed by copula methods and tested by goodness-of-fit indices. The future joint probabilities and joint return periods (JRPs) of the EPIs are then evaluated. Finally, change rates of JRPs for future periods are estimated to assess climate risk with the quantitative data of exposure and vulnerability of a protected target. An actual application in Taiwan Island is successfully conducted for climate risk assessment with the impacts of extreme precipitation. The results indicate that most of regions in Taiwan Island might have higher potential climate risk under different scenarios in the future. The future joint probabilities of precipitation extremes might cause the high risk of landslide and flood disasters in the mountainous area, and of inundation in the plain area. In sum, the proposed climate risk assessing approach is expected to be useful for assisting decision makers to draft adaptation strategies and face high risk of the possible occurrence of natural disasters.

     

Uncertainty evaluation of design rainfall for urban flood risk analysis

A reliable and long dataset describing urban flood locations, volumes and depths would be an ideal prerequisite for assessing flood frequency distributions. However, data are often piecemeal and long-term hydraulic modelling is often adopted to estimate floods from historical rainfall series. Long-term modelling approaches are time- and resource-consuming, and synthetically designed rainfalls are often used to estimate flood frequencies. The present paper aims to assess the uncertainty of such an approach and for suggesting improvements in the definition of synthetic rainfall data for flooding frequency analysis. According to this aim, a multivariate statistical analysis based on a copula method was applied to rainfall features (total depth, duration and maximum intensity) to generate synthetic rainfalls that are more consistent with historical events. The procedure was applied to a real case study, and the results were compared with those obtained by simulating other typical synthetic rainfall events linked to intensity-duration-frequency (IDF) curves. The copula-based multi-variate analysis is more robust and adapts well to experimental flood locations even if it is more complex and time-consuming. This study demonstrates that statistical correlations amongst rainfall frequency, duration, volume and peak intensity can partially explain the weak reliability of flood-frequency analyses based on synthetic rainfall events.     

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.

     

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.     

Joint probability distribution of winds and waves from wave simulation of 20 years (1989-2008) in Bohai Bay

The joint probability distribution of wind speed and significant wave height in the Bohai Bay was investigated by comparing the Gumbel logistic model, the Gumbel-Hougaard (GH) copula function, and the Clayton copula function. Twenty years of wind data from 1989 to 2008 were collected from the European Centre for Medium-Range Weather Forecasts (ECMWF) database and the blended wind data of the Quick Scatterometer (QSCAT) satellite data set and re-analysis data from the United States National Centers for Environmental Prediction (NCEP). Several typhoons were taken into account and merged with the background wind fields from the ECMWF or QSCAT/NCEP database. The 20-year data of significant wave height were calculated with the unstructured-grid version of the third-generation wind wave model Simulating WAves Nearshore (SWAN) under extreme wind process conditions. The Gumbel distribution was used for univariate and marginal distributions. The distribution parameters were estimated with the method of L-moments. Based on the marginal distributions, the joint probability distributions, the associated return periods, and the conditional probability distributions were obtained. The GH copula function was found to be optimal according to the ordinary least squares (OLS) test. The results show that wind waves are the prevailing type of wave in the Bohai Bay.     

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。     

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.     

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.

     

Generating Synthetic Rainfall Total Using Multivariate Skew-<Emphasis Type="Italic">t</Emphasis> and Checkerboard Copula of Maximum Entropy

This study aims to test the appropriateness of multivariate skew-t copula and checkerboard copula of maximum entropy in generating monthly rainfall total data. The generation of synthetic data is important, as it provides hypothetical data in areas for which data availability remains limited. Three selected meteorological stations in Kelantan, Malaysia, Stesen Pertanian Melor, Rumah Pam Salor, and Ladang Lepan Kabu, are considered in this study. Monthly rainfall total data for the driest and wettest months in the year are tested in this study. For these three stations, the identified month with the least total of rainfall received (driest) is May, while the month with the highest total of rainfall received (wettest) is November. The data is fitted to gamma distribution with the corresponding parameters estimated. The observed data will be transformed to be in unit uniform using the gamma marginal. The resulting data is compared to simulated uniform data generated using multivariate skew-t copula and checkerboard copula of maximum entropy models based on the correlation values of the observed and simulated data. Next, the Kolmogorov-Smirnov test is used to assess the fit between the observed and generated data. The results show that the values of simulated correlation coefficients do not differ much for gamma distribution, multivariate skew-t, and maximum entropy approaches. This implies that the multivariate skew-t and maximum entropy may be used to generate monthly rainfall total for cases in which actual data is unavailable.     

Research on the Relationships Between Rainfall and Meteorological Yield in Irrigation District

Rainfall and grain yield are two closely related random variables to be worthy of studying. The meteorological yield explains the influences of weather changes on grain yield. Based on the data series from 1980 to 2006 in Jinghuiqu irrigation district of Shaanxi Province in China, the meteorological yield is achieved from grain yield. Then, the empirical mode decomposition method is applied to analyze fluctuating periods and local features of rainfall and meteorological yield. Meanwhile, the copula method is introduced into describe the joint probability distribution of rainfall and meteorological yield. The studied results show that rainfall and meteorological yield exist vary fluctuation periods with multi-time scales, including 2 to 4 years of short period level, 4 to 6 (or 7) years of middle period level and 19 (or 10 to 11) years of long period level. Using the frank copula method, the bivariate distribution and return period of rainfall and meteorological yield was successfully developed to reveal the encounter risk of their different magnitudes. Finally, similarly with rainfall and meteorological yield, the complex changes and fluctuation periods are also proven to be existed in their joint probability.     

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.     

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.

     

Spatial Pattern Characterization and Multivariate Hydrological Frequency Analysis of Extreme Precipitation in the Pearl River Basin, China

Consecutive extreme rainfall events, especially those having unfavourable spatio-temporal patterns, always trigger large floods. This paper aims to examine, through the multivariate hydrological frequency analysis, the probability of the synchronous occurrence of rainfall extremes in the Pearl River basin. The copula method together with the stationarity and independence tests, which are crucial to the valid use of statistical methods in regional frequency analyses, were applied in the study. The obtained results indicate that: (1) major precipitation events of the annual maximum 1-, 3-, 5- and 7-day rainfall recorded at 42 stations are the flat looking series and variables are independent, (2) the marginal distribution of all extreme rainfall variables in four homogeneous hydrologic regions fits the log-normal probability distribution and most of their joint distribution fits the Gumbel-Hougaard distribution, (3) on that basis the contour maps of the joint distribution of annual maximum 1-, 3-, 5- and 7-day rainfall between different regions are drawn and the probability of the synchronous occurrence of the extreme rainfalls in different regions are estimated. These findings have great practical value for the regional water resources and flood risk management and are important in exploration of the spatial patterns of rainfall extremes in the Pearl River basin in order to reveal the underlying linkages between precipitation and floods from a broader geographical perspective.     

Probabilistic Event Based Rainfall-Runoff Modeling Using Copula Functions

Multivariate probability analysis of hydrological elements using copula functions can significantly improve the modeling of complex phenomena by considering several dependent variables simultaneously. The main objectives of this study were to: (i) develop a stand-alone and event-based rainfall-runoff (RR) model using the common bivariate copula functions (i.e. the BCRR model); (ii) improve the structure of the developed copula-based RR model by using a trivariate version of fully-nested Archimedean copulas (i.e. the FCRR model); and (iii) compare the performance of the developed copula-based RR models in an Iranian watershed. Results showed that both of the developed models had acceptable performance. However, the FCRR model outperformed the BCRR model and provided more reliable estimations, especially for lower joint probabilities. For example, when joint probabilities were increased from 0.5 to 0.8 for the peak discharge (q_p) variable, the reliability criteria value increased from 0.0039 to 0.8000 in the FCRR model, but only from 0.0010 to 0.6400 in the BCRR model. This is likely because the FCRR considers more than one rainfall predictor, while the BCRR considers only one.

     

Nonstationary Flood Frequency Analysis for Annual Flood Peak and Volume Series in Both Univariate and Bivariate Domain

Flood frequency analysis for practical application is traditionally based on the assumption of stationarity, but this assumption has been open to doubt in recent years. A number of studies have focused on the nonstationary flood frequency analysis, and the associated causes of nonstationarity. In this study, the annual maximum flood peak and flood volume of Wangkuai reservoir watershed were used, and several univariate and bivariate models were established to investigate the nonstationary flood frequency, with the distribution parameters changing over the climate indices (NPO, Niño3) and the check dam indices (CDIp, CDIv). In the univariate models, the Weibull distribution performed best and exhibited an undulate behavior for both flood peak and volume, which tended to describe the nonstationarity reasonably well. The bivariate models were constructed using copulas, of which the optimal Weibull distribution in the univariate flood frequency analysis was considered as marginal distributions within the joint distribution. The results showed that the Gumbel-Hougaard copula offered the best joint distribution, and most of the probability isolines crossed each other, which demonstrated the possibility that the occurrence of combinations of the flood peak and volume may be the same under multiple effects of phase changes in the climate patterns and certain human activities (i.e. soil and water conservation). The most likely events were elaborated in diagrams, and the associated combinations of the flood peak and volume were smaller than that estimated by the fixed parameters (i.e. stationary condition) during most of the study period, while it was the opposite in 1956, 1959 and 1963. The results highlight the necessity of nonstationary flood frequency analysis under various conditions in both univariate and multivariate domains.