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Multivariate quantiles in hydrological frequency analysis
Abstract
Several hydrological phenomena are described by two or more correlated characteristics. These dependent characteristics should be considered jointly to be more representative of the multivariate nature of the phenomenon. Consequently, probabilities of occurrence cannot be estimated on the basis of univariate frequency analysis (FA). The quantile, representing the value of the variable(s) corresponding to a given risk, is one of the most important notions in FA. The estimation of multivariate quantiles has not been specifically treated in the hydrological FA literature. In the present paper, we present a new and general framework for local FA based on a multivariate quantile version. The multivariate quantile offers several combinations of the variable values that lead to the same risk. A simulation study is carried out to evaluate the performance of the proposed estimation procedure and a case study is conducted. Results show that the bivariate estimation procedure has an analogous behaviour to the univariate one with respect to the risk and the sample size. However, the dependence structure between variables is ignored in the univariate case. The univariate estimates are obtained as special combinations by the multivariate procedure and with equivalent accuracy. Copyright © 2009 John Wiley & Sons, Ltd.
... To model the dependence structure between hydrological variables, several studies employed copulas (e.g. Chebana and Ouarda, 2011;Dupuis, 2007;Genest and Chebana, 2017;Grimaldi and Serinaldi, 2006;Li et al., 2019;Shiau et al., 2006;Zhang and Singh, 2006;Zhu et al., 2019). Copulas are the link between the margins and the multivariate distribution function (Sklar, 1959). ...
... In addition, copulas are completely independent of the marginal distributions and the margins need not to be from the same family, which gives more flexibility to the model. Since the hydrological risk is closely related to the quantile notion, Chebana and Ouarda (2011) considered the multivariate quantiles where they showed that taking into account several variables simultaneously to assess multivariate quantiles, is more flexible, more realistic and improves the accuracy of the results. ...
... The reader is referred to Serfling (2002) for a review and a classification of some of these multivariate quantile versions. Recently, Belzunce et al. (2007) defined a bivariate vectorvalued quantile version which is adapted and shown to be appropriate for hydrological application by Chebana and Ouarda (2011). For p in (0, 1), the pth bivariate quantile for the event {X⩽x, Y⩽y} is given by ...
To study hydrological events, such as floods and droughts, frequency analysis (FA) techniques are commonly employed. FA relies on some assumptions, especially, the stationarity of the data series. However, the stationarity assumption is not always fulfilled for a variety of reasons such as climate change and human activities. Thus, it is essential to check the stationarity or we should develop models that take into account the non-stationarity in a new risk assessment framework. On the other hand, a majority of hydrological phenomena are described by a number of correlated characteristics. To model the dependence structure between these hydrological variables, copulas are the most employed tool. Generally in the literature, the multivariate model is assumed to be the same over time even though multivariate stationarity is required. Considering the non-stationarity in the dependence structure is important because when the copula parameter changes, the multivariate quantile curve changes accordingly. Different scenarios can be considered when choosing a multivariate non-stationary model since several variables and a dependence structure are involved. The objective of the present study is to construct a model that integrates simultaneously multivariate and non-stationarity aspects along with hypothesis testing. For the copula part, we consider versions called Dynamic copulas and series of association measures are obtained through rolling windows of the corresponding series. Adapted versions of the AIC criterion are employed to select the final model (margins and copula). The procedure is applied to a flood volume and peak dataset from Iran. The obtained model constitutes of a lognormal distribution for the margins with linear trend in the peak series, stationary for the volume series and a quadratic trend in the logistic Gumbel copula parameter for the dependence structure.
... Equation (17) refers to the "structure-based" return period introduced by Volpi and Fiori (2012). Among these seven types of probabilities, we selected the "AND" and the "COND1" probabilities (see Fig. 4) as these are commonly used in the literature (Chebana and Ouarda, 2011;Tencer et al., 2014;Sadegh et al., 2018) and correspond to the two types of interrelations we are interested in (i.e. compound and cascade). ...
... joint exceedance curves (Hawkes et al., 2002;Hawkes, 2008;Mazas and Hamm, 2017) quantile curves (De Haan and De Ronde, 1998;Chebana and Ouarda, 2011). ...
... Synthetic datasets are often used to compare different statistical models (Chebana and Ouarda, 2011;Zheng et al., 2014;Cooley et al., 2019). Here we generated 60 bivariate synthetic datasets representative of environmental data such as daily rainfall, daily wind gust and daily wildfire occurrences (see Sect. 4). ...
Modelling multiple hazard interrelations remains a challenge for practitioners. This article primarily focuses on the interrelations between pairs of hazards. The efficacy of six distinct bivariate extreme models is evaluated through their fitting capabilities to 60 synthetic datasets. The properties of the synthetic datasets (marginal distributions, tail dependence structure) are chosen to match bivariate time series of environmental variables. The six models are copulas (one non-parametric, one semi-parametric, four parametric). We build 60 distinct synthetic datasets based on different parameters of log-normal margins and two different copulas. The systematic framework developed contrasts the model strengths (model flexibility) and weaknesses (poorer fits to the data). We find that no one model fits our synthetic data for all parameters but rather a range of models depending on the characteristics of the data. To highlight the benefits of the systematic modelling framework developed, we consider the following environmental data: (i) daily precipitation and maximum wind gusts for 1971 to 2018 in London, UK, and (ii) daily mean temperature and wildfire numbers for 1980 to 2005 in Porto District, Portugal. In both cases there is good agreement in the estimation of bivariate return periods between models selected from the systematic framework developed in this study. Within this framework, we have explored a way to model multi-hazard events and identify the most efficient models for a given set of synthetic data and hazard sets.
... Other major factors are high tides and sea level rise. When these factors are positively correlated, analyzing them separately in a univariate framework underestimates the associated risk [4]. Less-or un-correlated factors can also simultaneously reach a high level and thus should be considered jointly. ...
... The covariate added at each step is indicated in bold 1 . These results correspond to the parametric regressions of Equations (2) to(4). Information criteria are not reported in this table because in this case of nested models, using them would be equivalent to having likelihood ratio tests with different ...
Coastal flooding compound events can be caused by climate-driven extremes of storm surges and waves. To assess the risk associated with these events in the context of climate variability, the bivariate extremes of skew surge (S) and significant wave height (HS) are modeled in a nonstationary framework using physical atmospheric/oceanic parameters as covariates (atmospheric pressure, wind speed and sea surface temperature). This bivariate nonstationary distribution is modeled using a threshold-based approach for the margins of S and HS and a dynamic copula for their dependence structure. Among the covariates considered, atmospheric pressure and related wind speed are primary forcings for the margins of S and HS, but temperature is the main positive forcing of their dependence. This latter relation implies an increasing risk of compound events of S and HS for the studied site in the context of increasing global temperature.
... In a multivariate setting, the notions of return period and return values are ambiguous. There are various definitions of a multivariate extreme, as discussed in Serinaldi (2015); Girard and Stupfler (2015); Vanem (2018bVanem ( , 2020, giving rise to different versions of multivariate quantiles, see Chebana and Ouarda (2011). The following discussion considers the bivariate case for simplicity, but may easily be generalized to higher dimensions. ...
... The following discussion considers the bivariate case for simplicity, but may easily be generalized to higher dimensions. Typically, a bivariate quantile will be vector-valued and can be expressed in the following form (see Chebana and Ouarda (2011)): ...
Environmental contours are widely used as basis for design of structures exposed to environmental loads. The basic idea of the method is to decouple the environmental description from the structural response. This is done by establishing an envelope of joint extreme values representing critical environmental conditions, such that any structure tolerating loads on this envelope will have a failure probability smaller than a prescribed value. Specifically, given an n-dimensional random variable \(\mathbf {X}\) and a target probability of failure \(p_{e}\), an environmental contour is the boundary of a set \(\mathcal {B} \subset \mathbb {R}^{n}\) with the following property: For any failure set \(\mathcal {F} \subset \mathbb {R}^{n}\), if \(\mathcal {F}\) does not intersect the interior of \(\mathcal {B}\), then the probability of failure, \(P(\mathbf {X} \in \mathcal {F})\), is bounded above by \(p_{e}\). We work under the assumption that failure sets are convex, which is relevant for many real-world applications. In this paper, we show that such environmental contours may be regarded as boundaries of Voronoi cells. This geometric interpretation leads to new theoretical insights and suggests a simple novel construction algorithm that guarantees the desired probabilistic properties. The method is illustrated with examples in two and three dimensions, but the results extend to environmental contours in arbitrary dimensions. Inspired by the Voronoi-Delaunay duality in the numerical discrete scenario, we are also able to derive an analytical representation where the environmental contour is considered as a differentiable manifold, and a criterion for its existence is established.
... Calculation of distribution quantiles in a multidimensional space enables different combinations of variables that yield the same risk [7]. By observing complex phenomena, the following authors applied the following probability distributions in a multidimensional space: double Gaussian distribution and exponential distribution [8], double Gumbel or mixed Gumbel distribution [9], double normal distribution [10][11][12], double lognormal distribution [13], and double gamma distribution [14]. ...
... Copulas are used for modelling in the following fields: (1) droughts [2,[17][18][19][20], (2) precipitation [21][22][23][24][25][26], (3) groundwater [27], (4) dam overtopping risk determination [28], (5) dependence between flow and concentration of suspended sediment load [29], (6) probabilistic analyses of floods [30][31][32][33][34][35][36], (7) flood wave coincidences [1,37,38], and (8) the impact of climate change on precipitation extremes [39,40]. ...
In confluence areas, where river flow slowdowns can be substantial, the selection of design flows for the purpose of designing a flood control system directly depends on flood waves at the mainstem and its tributaries. The aim of this paper is twofold: to elaborate a procedure that will define the flood exceedance probability in the analyzed area within a multidimensional probability space and to develop a methodology for defining flood coincidence in the mainstem reach with its main tributaries and thus determine design flows in order to design the flood control system in the confluence area. Mathematical models are based on (1) two-dimensional probability distribution (PROIL model) and (2) the copula function (Archimedean class of copulas), and fitted to a practical application in a two-dimensional probability distribution space. In case of random variables, simultaneous quantitative characteristics of flood wave (water flow) hydrographs for the mainstem and a tributary are considered. The paper discusses the specific reach of the Danube, from the Hungarian-Serbian border to the city of Smederevo, which encompasses three significant tributaries—the Drava, the Tisa, and the Sava.
... Par exemple, les pluies intenses sont décrites par leur intensité et leur durée dont la corrélation n'est pas négligeable généralement, ou encore, les crues qui sont décrites par les trois variables débit de pointe, durée et volume dont la corrélation n'est pas nulle. Par ailleurs, de nombreuses études ont démontré que l'analyse fréquentielle univariée ne permet pas d'estimer d'une manière complète la probabilité des valeurs extrêmes dans le cas d'un aléa caractérisé par plusieurs variables [29]. Selon Salvadori et De Mechele [119], la modélisation de la dépendance permet une meilleure compréhension de l'aléa et évite de sous/sur-estimer le risque. ...
... Quelques travaux dans la littérature ont traité cette question de coïncidence dans le cas de dépendance et dans un contexte de sélection des combinaisons (couples) en particulier. Chebana and Ouarda [29] ont proposé la décomposition d'un contour d'isoprobabilité en partie "naïve" (queue de distribution) et partie "propre" (centrale), ils ont ensuite proposé de sélectionner uniquement les couples formant les extrémités de la partie propre. Salvadori et al. [120] ont proposé d'introduire une fonction liée au dimensionnement (i.e., courbe de fragilité d'un ouvrage) et de sélectionner un seul couple du contour bivarié (combinaison) permettant de minimiser cette fonction, identiquement à la méthode appliquée dans le projet "SAO POLO" [125]. ...
La modélisation des combinaisons de phénomènes d’inondation est une problématique d’actualité pour la communauté scientifique qui s’intéresse en priorité aux sites urbains et nucléaires. En effet, il est fort probable que l’approche déterministe explorant un certain nombre de scénarios possède certaines limites car ces scénarios déterministes assurent un conservatisme souvent excessif. Les approches probabilistes apportent une précision supplémentaire en s’appuyant sur les statistiques et les probabilités pour compléter les approches déterministes. Ces approches probabilistes visent à identifier et à combiner plusieurs scénarios d’aléa possibles pour couvrir plusieurs sources possibles du risque. L’approche probabiliste d’évaluation de l’aléa inondation (Probabilistic Flood Hazard Assessment ou PFHA) proposée dans cette thèse permet de caractériser une (des) quantité(s) d’intérêt (niveau d’eau, volume, durée d’immersion, etc.) à différents points d’un site en se basant sur les distributions des différents phénomènes de l’aléa inondation ainsi que les caractéristiques du site. Les principales étapes du PFHA sont : i) identification des phénomènes possibles (pluies, niveau marin, vagues, etc.), ii) identification et probabilisation des paramètres associés aux phénomènes d’inondation sélectionnés, iii) propagation de ces phénomènes depuis les sources jusqu’aux point d’intérêt sur le site, iv) construction de courbes d’aléa en agrégeant les contributions des phénomènes d’inondation. Les incertitudes sont un point important de la thèse dans la mesure où elles seront prises en compte dans toutes les étapes de l’approche probabiliste. Les travaux de cette thèse reposent sur l’étude de la conjonction de la pluie et du niveau marin et apportent une nouvelle méthode de prise en compte du déphasage temporel entre les phénomènes (coïncidence). Un modèle d’agrégation a été développé afin de combiner les contributions des différents phénomènes d’inondation. La question des incertitudes a été étudiée et une méthode reposant sur la théorie des fonctions de croyance a été utilisée car elle présente des avantages divers par rapport aux autres concepts (modélisation fidèle dans les cas d’ignorance totale et de manque d’informations, possibilité de combiner des informations d’origines et de natures différentes, etc.). La méthodologie proposée est appliquée au site du Havre, en France.
... Univariate HFA can provide only limited assessment of extreme events and their probability of occurrence (e.g. Chebana and Ouarda, 2011;Joyce et al., 2018). ...
... This combined effect and relationship between precipitation and storm surge has previously been investigated by Svensson and Jones [28]. These studies emphasize that univariate flood frequency analysis falls short when assessing extreme events caused by interconnected random variables [29][30][31][32]. But until now, to our knowledge, limited research has explored the compound impacts between storm surges and flood variables in coastal areas for distinct pair combinations. ...
In low-lying coastal areas, the interplay of various factors including precipitation, river flow, and storm surge can lead to greater influence on floods when they occur simultaneously. The copula method was used in this study to investigate the bivariate flood risk of compounding storm surge and river discharge events in the Pearl River Delta (PRD). Our results indicate that while the correlation between storm surge and flood peak (S-Q) was weak, there was a strong dependence between the pairs of storm surge–flood volume (S-V) and storm surge–flood duration (S-D). For these three pairs, the Clayton copula was the optimal function for S-Q, while the Frank copula was the optimal function for S-V and S-D, respectively. When the flood volume exceeds 2.0 × 104 m3/s and the flood duration is more than 10 days, the bivariate hydrologic risk for S-V and S-D is observed to decrease rapidly. Furthermore, the failure probability (FP) would be underestimated when the combined impact of river flow and storm surge is ignored in coastal flood risk assessment. Such bivariate hydrologic risk analysis implies that when determining design values in coastal flood risk assessment, the combined impact of river flow and storm surge should be taken into account.
... Therefore, it is necessary to estimate the probabilistic relationship between the fluctuation of variables and the fluctuation of crop yield by determining the probability distribution functions for the variables under study as well as crop yield. A better understanding of hydrological phenomena can be achieved by constructing a joint distribution that accounts for all relevant variables (Chebana and Ouarda, 2011). ...
The trend of population growth and the consequent need to increase agricultural production in order to provide for the necessary food is critical topic nowadays. To account for the risks and damage caused by changes in weather and climate constrains, it is important to know how these changes affect the agricultural productivity. Wheat is the world's most widely grown crop and is directly related to worldwide food security issues. As a result, it has always been of interest to researchers as a strategic crop. In this study, the correlation of the Water Requirement Satisfaction Index in six growing periods of rainfed wheat with its yield was examined and found that the highest correlation is seen over the entire growing season. Based on statistical distributions and copula functions, the sensitivity and rainfed wheat yield to the Water Requirement Satisfaction Index was examined. Clayton Copula was selected based on AIC and RMSE evaluation indices. The probability of having crop losses is around 50%, in general. The results showed that the yield risk of rainfed wheat in the Tabriz region drops to 46% under low climatic risk conditions and reaches to 98% under medium and high climate risk conditions. The results allow farmers and stakeholders to better plan and manage food security by knowing the changes in rainfed wheat yield depending on weather conditions. By using different options under different conditions such as wet events, it is also possible to work towards higher yield amounts of rainfed wheat and other rainfed crops.
... The copula method is a popular approach for estimating return periods of extreme events in hydrology or finance and is com-120 monly used as a tool of risk management as univariate statistical analysis might not be enough to provide reliable probabilities with correlated variables as stated by Chebana and Ouarda (2011). One key advantage of the copula method over other methods is that it allows more flexible modeling of the dependence structure between variables. ...
The study aims to provide a complete analysis framework applied to an earthen dyke located in Camargue, France. This dyke is regularly submitted to erosion on the landward slope that needs to be repaired. Improving the resilience of the dyke calls for a reliable model of damage frequency. The developed system is a combination of copula theory, empirical wave propagation, and overtopping equations as well as a global sensitivity analysis in order to provide the return period of erosion damage on a set dyke while also providing recommendations in order for the dyke to be reinforced as well the model to be self-5 improved. The global sensitivity analysis requires to calculate a high amount of return periods over random observations of the tested parameters. This gives a distribution of the return periods, providing a more general approach to the behavior of the dyke. The results show a return period peak around the two-year mark, close to reported observation. The distribution being skewed, the mean value is however higher and is thus less reliable as a measure of dyke safety. The results of the global sensitivity analysis show that no particular category of dyke features contribute significantly more to the uncertainty of the system. The 10 highest contributing factors are the dyke height, the critical velocity as well as the coefficient of seaward slope roughness. These results underline the importance of good dyke characterization in order to improve the predictability of return periods estimations. The obtained return periods have been confirmed by current in situ observations but the uncertainty increases for the most severe events due to the lack of long-term data.
... Under the bivariate copula function, the selection of a suitable T or results in numerous combinations of flood peak and duration (Figure 2b). The points along the T or -level curve usually have different impacts from an engineering perspective (Chebana & Ouarda, 2011), so the probability of each event must be taken into account while determining the appropriate joint quantiles . In this work, the most likely realization method suggested by Salvadori et al. (2011) is employed to select the most probable flood scenario along the T or -level isoline. ...
As the planet warms, the atmosphere's water vapor holding capacity rises, leading to more intense precipitation extremes. River floods with high peak discharge or long duration can increase the likelihood of infrastructure failure and enhance ecosystem vulnerability. However, changes in the peak and duration of floods and corresponding socioeconomic exposure under climate change are still poorly understood. This study employs a bivariate framework to quantify changes in flood risks and their socioeconomic impacts in China between the past (1985–2014) and future (2071–2100) in 204 catchments. Future daily river streamflow is projected by using a cascade modeling chain based on the outputs of five bias‐corrected global climate models (GCMs) under three shared socioeconomic CMIP6 pathways (SSP1‐26, SSP3‐70, and SSP5‐85), a machine learning model and four hydrological models. We also utilize the copula function to build the joint distribution of flood peak and duration, and calculate the joint return periods of the bivariate flood hazard. Finally, the exposure of population and regional gross domestic product to floods are investigated at the national scale. Our results indicate that flood peak and duration are likely to increase in the majority of catchments by 25%–100% by the late 21st century depending on the shared socioeconomic pathway. China is projected to experience a significant increase in bivariate flood risks even under the lowest emission pathway, with 24.0 million dollars/km² and 608 people/km² exposed under a moderate emissions scenario (SSP3‐70). These findings have direct implications for hazard mitigation and climate adaptation policies in China.
... Copula functions are mathematical functions coupling time-independent variables. Research has striven to address real-world problems using Copulas: e.g., they have been used in finance and stock market [14], insurance [15], electrochemistry [16], hydrology and climatology [17], risk management [18], drought monitoring [19], flood analysis [20], extreme value analysis [21], rainfall analysis [22], etc. Although Copula functions have been used in a few numbers of wind-related studies, sometimes these studies are based on modelling of historical data [23], short-term projections without considering the effects of climate change [24] and/or using limited marginal distribution functions and few Copula functions [25]. ...
Wind energy is a key element in the ongoing push to decarbonize the energy supply. The first step in the development of a wind farm at a specific site is to conduct feasibility studies including accurate long-term wind potential assessments and estimates of the annual energy production. However, as evidence of climate change becomes more apparent recently, concerns about the planning and utilization of wind resources in the face of these new conditions have increased. Accurate projections are needed to determine the frequency distribution of wind speeds in an area and, on this basis, estimate the energy production. The purpose of this study is to analyze the wind resource, to estimate its potential and to prepare zoning maps of wind energy production to determine the most suitable sites for wind farms in Ireland. For this purpose, wind data from ten Global Circulation Models and different climate-change scenarios were used during the historical and future period of 1981–2010 and 2021–2050, respectively. These data were evaluated in the study area and then a multi-criteria decision-making method was applied to choose the best representative climate models over the area. In order to post-process the outputs of the selected models, 17 statistical distributions and 26 Copula families were applied. Results showed that the average wind speed in the region during the historical period is expected to decrease in 2021–2050 by approximately 2–7% based on the climate scenarios. Additionally, wind power density maps were produced for the study area.
... One possibility is to consider quantile regions of the risk factors distribution. In the multivariate case, a wide panel of multivariare quantiles has been reviewed in the literature [8,9,39]. The study of multivariate quantile regions has increasingly been pursued in the last decades as a tool to model multivariate risk regions, especially those based on a multivariate distribution function ( [3,18,13]), or on a depth function. ...
... The copula method is a popular approach for estimating return periods of extreme events in hydrology or finance and is com-120 monly used as a tool of risk management as univariate statistical analysis might not be enough to provide reliable probabilities with correlated variables as stated by Chebana and Ouarda (2011). One key advantage of the copula method over other methods is that it allows more flexible modeling of the dependence structure between variables. ...
The study aims to provide a complete analysis framework applied to an earthen dyke located in Camargue, France. This dyke is regularly submitted to erosion on the landward slope that needs to be repaired. Improving the resilience of the dyke calls for a reliable model of damage frequency. The developed system is a combination of copula theory, empirical wave propagation and overtopping equations as well as a global sensitivity analysis in order to provide the return period of erosional damage on a set dyke while also providing recommendations in order for the dyke to be reinforced as well the model to be self-5 improved. The results give a good correspondence, within uncertainty range, between the model prediction of return periods and the on-site observation (≈ two-year return period). The mean of the return periods is slightly higher with an average return period of six years but the peak of the distribution is located around the two years mark. The sensitivity analysis shows that the geometrical characteristics of the dyke - slope angles and dyke height - are the ones carrying the highest amount of uncertainty into the system, showing that maintaining a homogeneous dyke is of great importance. Some empirical parameters intervening inside the propagation and overtopping process are also fairly uncertain and suggest that using more robust methods at their corresponding steps could improve the reliability of the framework. The obtained return periods have been confirmed by current in situ observations but the uncertainty increases for the most severe events due to the lack of long-term data.
... The aforementioned studies, and similar ones, even though highlighting the importance of considering the heterogeneity in HFA modeling, they focused on the univariate context, considering only one single variable of a given hydrological event such as annual flood peak or minimum flow. However, it has been showed over the last years that extreme hydrologic events can be characterized by the joint behaviour of several dependent variables (Chebana & Ouarda, 2011;De Michele et al., 2013;Santhosh & Srinivas, 2013). Hence, univariate HFA does not procure a reliable assessment of the associated risk (e.g. ...
Hydrological extreme events are characterized by several correlated variables. For a better associated risk assessment, the dependence structure between these variables must be taken into account by considering copulas. On the other hand, extreme events are generated from different phenomena. In such cases, the margins and/or copula may be affected. Hence, mixture copula should be considered. Recently, there have been an increasing number of studies dealing with the parameter estimation of mixture copula. However, existing methods have several drawbacks. To overcome these drawbacks, we propose a new parameter estimation approach for the mixture copula models, based on the maximum pseudo-likelihood using a metaheuristic algorithm. A simulation study is conducted to evaluate the performance of the proposed method and to compare it with those of the widely used existing method. Results indicate that the proposed method estimates more accurately the parameters even with small sample sizes compared to the existing ones. An application to a real data set is also provided and validated with the available data.
... Since the combinations of variables leading to same joint probability are countless, the selection of design combination is usually subjective [5,18,47]. Chebana and Ouarda [51] pointed out that the different likelihood of each combination makes it possible to identify the most appropriate design result, which provides a principle for multivariate design combination optimization. Therefore, the Most-Likely Estimation (MLE) method is used to optimize the design combinations of urban stormwater and drainage systems [47,52]. ...
Waterlogging disasters cause huge loss of life and property damage every year. In this research, a Copula-based optimization method is proposed to solve the problems in bivariate design of urban stormwater and drainage systems resulting from ignorance of precipitation temporal dependence and discrepancy between different design codes. Optimized design combinations of stormwater and drainage systems conditioned on given Kendall bivariate return periods or return periods of either system can be obtained using the optimization method for the case study of Zhongshan and Zhuhai. Results show that the temporal dependencies between precipitation series with different durations should be carefully considered, which can be sufficiently described by Copula functions. Based on the optimized design combinations, it is found that the planned return periods of stormwater systems in Sponge City Plans are underestimated for both Zhongshan and Zhuhai, which restricts the full use of the drainage systems. According to the optimized results, the planned return periods of stormwater systems in Zhongshan (Zhuhai) should be adjusted to 8.04 a (6.76 a) for the downtown area and 6.52 a (5.59 a) for other areas, conditioned on the planned return periods for P24 h in Sponge City Plans. The proposed optimization method provides a useful approach for the bivariate design of stormwater and drainage systems. The results of this research can give stakeholders references in compiling engineering plans for urban waterlogging prevention and help better balance the conflicts between waterlogging safety and economic efficiency.
... However, drought is a multivariate event, and therefore, univariate approaches tend to underestimate or overestimate the drought risk (Ayantobo et al. 2018;Chebana & Ouarda 2011;K. Xu et al. 2015). ...
Meteorological drought is among the key climate-related risks affecting Ethiopia. It indicates a shortage of precipitation over a long period, usually for a season or a year. This study uses copula theory to analyze meteorological drought in the upper Tekeze river basin, Northern part of Ethiopia. Meteorological drought analysis using copula theory provides a promising opportunity to deal with such risks in advance and to improve sectoral resilience. In this study, long-year (1982–2020) rainfall and soil moisture data were used to analyze standardized precipitation index (SPI) and standardized soil moisture index (SSI), respectively. The best-fit copula family was selected to construct the joint probability distribution (JPD) of SPI and SSI. Multivariate standardized drought index (MSDI) at 3-, 6-, and 12-month timescales was analyzed using the MSDI toolbox. The nonparametric Mann–Kendall (M–K) statistical test was used for trend detection. We found that the newly developed MSDI captures all drought events during the observation period compared with SPI and SSI. MSDI particularly showed the most recent drought of 2015, with the drought duration and severity of 4 months and 6.4, respectively, and its joint return period was 5.24 years. The M–K and Sen’s Slope estimator statistical tests indicated a positive trend for all drought timescales in the basin. The spatial extent of MSDI shows most frequently extreme drought occurred in the basin. Meteorological drought analysis using multiple indices is better than a single drought index. This approach can better inform adaptation policies and interventions that are aimed at monitoring and reducing drought risk in the basin and beyond.
... Furthermore, their performance depends on prior knowledge of the marginal and joint distributions of the variables, which may not be available. Various nonparametric and semiparametric models are in use to address these concerns (e.g., Chebana & Ouarda, 2011;Karmakar & Simonovic, 2008;Lall, 1995;Reddy & Ganguli, 2012;Vittal et al., 2015). Nonparametric methods are data-driven and do not require specifying the form of the dependence (i.e., linear or nonlinear) and marginal distribution. ...
Floods can be characterized by various correlated variables such as peak flow, flood volume, duration, and time to peak. Hence, flood risk can be better assessed by performing frequency analysis in a multivariate (rather than a univariate) framework. However, multivariate approaches involve sophisticated analyses that require considerably more data than univariate approaches. In a univariate framework, flood risk assessment at an ungauged or data‐sparse location/site is generally performed through regional frequency analysis (RFA), based on information pooled from a group of sites resembling the target site. However, RFA has received little attention in the multivariate framework. The available literature recommends the use of index‐flood approach (IFA) for multivariate RFA (MRFA), even though IFA has theoretical shortcomings. Another issue is that marginals of all the flood‐related variables may not exhibit extreme behavior simultaneously. Conventionally used at‐site multivariate models are not suitable for describing the dependence structure of extremes in the regions of the support of the joint distribution where only some variables exhibit extreme behavior. This article proposes a conditional extreme values approach (CEA) to address the aforementioned issues in MRFA. Its effectiveness is demonstrated through Monte Carlo simulation experiments and a case study on watersheds from a flood‐prone region in India. Results indicate that the proposed approach can reliably predict the joint distribution of multiple flood‐related variables (and thus flood risk) at ungauged/sparsely gauged sites by effectively capturing the regional dependence structure between those variables.
... In multivariate hydrological analyses, mainly items such as showing the importance and framework of multivariate, fitting appropriate multivariate distribution and studying multivariate return periods are considered. (Chebana and Ouarda 2011). The most widely used copula cumulative joint distribution function (CDF) is Gaussian. ...
Investigating the interaction of water resources such as rainfall, river flow and groundwater level can be useful to know the
behavior of water balance in a basin. In this study, using the rainfall, river flow and groundwater level deficiency signatures
for a 60-day duration, accuracy of vine copulas was investigated by joint frequency analysis. First, while investigating correlation of pair-variables, tree sequences of C-, D- and R-vine copulas were investigated. The results were evaluated using AIC,
Log likelihood and BIC statistics. Finally, according to the physics of the problem and evaluation criteria, D-vine copula was
selected as the best copula and the relevant tree sequence was introduced. Kendall’s tau test was used to evaluate the correlation of pair-signatures. The results of the Kendall’s tau test showed that pair-signatures studied have a good correlation. Using
D-vine copula and its conditional structure, the joint return period of groundwater deficiency signature affected by rainfall
and river flow deficiency signatures was investigated. The results showed that the main changes in the groundwater level
deficiency is between 0.3 and 2 m, which due to the rainfall and the corresponding river flow deficiency, return periods will
be less than 5 years. Copula-based simulations were used to investigate the best copula accuracy in joint frequency analysis
of the studied signatures. Using copula data of the studied signatures, the groundwater deficiency signature was simulated
using D-vine copula and a selected tree sequence. The results showed acceptable accuracy of D-vine copula in simulating
the copula values of the groundwater deficiency signature. After confirming the accuracy of D-vine copula, the probability
of occurrence of groundwater deficiency signature was obtained from the joint probability of occurrence of other signatures.
This method can be used as a general drought monitoring system for better water resources management in the basin.
... There are complex relationships between ET c and P e , and single variable frequency analysis cannot fully estimate the water shortages probability. A better understanding of hydrological phenomena can be achieved by establishing joint distribution that considers all relevant variables (Chebana and Ouarda, 2011). The multivariate hydrological analysis method mainly includes multivariate probability distribution function, non-parametric, and copula methods (Yan et al., 2009). ...
The effective precipitation (Pe) and crop water requirements (ETc) can reflect the agricultural water supply and demand situations under natural precipitation conditions, and the encounter risk analysis of Pe and ETc is a prerequisite for regional water resources allocation and irrigation planning. Considering an entire growing season of rape-maize in the Hilly Area of Southwest China during 1961–2017, this study employed the popular copula functions to fit two-dimensional joint distribution of annual ETc and Pe, and analyzed the natural agriculture water shortages risk of different encounter situations. The results indicated ETc and Pe presented a negative relativity, and the Gaussian copula was found to be more suitable to estimate the joint distribution of ETc and Pe. The asynchronous encounter probability was higher two times than the synchronous encounter probability, and the pairs (rich Pe-poor ETc, poor Pe-rich ETc) had the greatest probability with value of 16.59%, indicating the natural water supply and demand usually was unmatched. The conditional probability of Pe without exceeding a certain value for different ETc states increased with increased Pe, and the conditional probability of ETc with exceeding a certain value for different Pe states decreased with increased ETc. The conditional probability of Pe without exceeding Pe 37.5%, Pe 62.5%, Pe average (ETc exceeding ETc 37.5%, ETc 62.5%, ETc average) for different ETc (Pe) states was 44.97–69.12% (the corresponding return period was 1.45–2.22 years), showing natural agriculture water shortages risk was high under general situations. However, the conditional probabilities of extremely high ETc (low Pe) with given low Pe (high ETc) were less than 3%, so extreme water shortages rarely occurred in the Sichuan Hilly Area. This study successfully applied the copula method to regional agricultural water shortages risk analysis and could provide a theoretical basis for regional water resources management and planning.
... Multivariate extreme value theory is a rapidly growing field for characterization of the joint tails of multidimensional processes in a diversity of applications. Specifically, the popular applications in hydrology include multivariate frequency analysis of floods (Chebana & Ouarda, 2011;De Michele et al., 2005;Favre et al., 2004;Karmakar & Simonovic, 2008Santhosh & Srinivas, 2013;Zhang & Singh, 2006) and droughts (De Michele et al., 2013;Ganguli & Reddy, 2014;Shiau, 2006;Shiau & Modarres, 2009;Song & Singh, 2010a, 2010bTosunoglu & Kisi, 2016), joint occurrence of flood flows at river confluences (Bender et al., 2016;Chen et al., 2012;Dutfoy et al., 2014;Favre et al., 2004;He et al., 2007;Peng et al., 2017;Prohaska et al., 2008;Raynal Abstract Estimating the dependence structure of concurrent extremes is a fundamental issue for accurate assessment of their occurrence probabilities. Identifying the extremal dependence behavior is also crucial for scientific understanding of interactions between the variables of a multidimensional environmental process. ...
Estimating the dependence structure of concurrent extremes is a fundamental issue for accurate assessment of their occurrence probabilities. Identifying the extremal dependence behavior is also crucial for scientific understanding of interactions between the variables of a multidimensional environmental process. This study investigates the suitability of parametric multivariate extreme value models to correctly represent and estimate the dependence structure of concurrent extremes. Probabilistic aspects of multivariate extreme value theory with point process representation are discussed and illustrated with application to the concurrence of rainfall deficits, soil moisture deficits, and high temperatures. Application is concerned with the investigation of extremal behavior and risk assessment in Marathwada, a drought‐prone region of Maharashtra state, India. To characterize the multivariate extremes, marginal distributions are specified first and transformed into unit Fréchet margins. Standardized distributions are represented by a Poisson point process and coordinates of data points are transformed to pseudo‐polar coordinates to make the dependence form more explicit. The extremal dependence structure is described through angular densities on the unit simplex. Strong dependence is observed between soil moisture deficits and high temperatures, whereas rainfall deficits are mildly dependent on these two variables. Overall, a weak dependence is observed between the variables considered. Estimated extremal dependence is further used to compute probabilities of a few critical extreme combinations. Results demonstrate the ability of parametric multivariate models to characterize the complex dependence structure of concurrent extremes. These models can provide a powerful new perspective for appropriate statistical analysis of dependent hydroclimatic extremes in higher dimensions.
... Standard design methods are usually based on univariate frequency analysis (Chebana and Ouarda 2011). In this study, the univariate Generalized Extreme Value (GEV) is examined for extreme sea level and precipitation. ...
Bivariate frequency analysis is an emerging method for the assessment of compound floods, especially in coastal regions. Changing climate, which usually leads to changes in characteristics of extreme hydrometeorological phenomena, makes the application of nonstationary methods more critical. In this research, a methodology is developed to apply frequency analysis on extreme sea level using physically-based hydroclimatic variables as covariates based on univariate Generalized Extreme Value (GEV) as the probability distribution function and copula methods. The results show that for extreme sea level, the location parameter of marginal distribution is directly related to the covariate variable of maximum temperature. For precipitation, the scale parameter is related to the covariate variable of minimum temperature, and the shape parameter is time-dependent. The univariate return periods of hurricanes Sandy and Irene are estimated at 85 and 12 years in nonstationary GEV distribution, respectively, while for stationary GEV distribution they are estimated at 1200 and 25 years, and in the bivariate frequency analysis of water level and precipitation, the normal copula function has more flexibility compared to other competitors. Using time-varying copula, the bivariate return periods of Hurricanes Sandy and Irene are 109 years and 136 years, respectively. The results confirm the importance of incorporating rainfall and extreme sea level in coastal flood frequency analysis. Although the proposed methodology can be applied to other hydro-climatological variables, the findings of this research suggest the necessity of considering nonstationarity in the analysis of extreme hydrologic events.
... Many studies have explored the eutrophication risk assessment using the Single-factor Index method, Principal Component Analysis method, comprehensive water quality identification index method (e.g., Fuzzy Comprehensive Evaluation method, Gray Correlation Analysis method, Artificial Neural Networks method, and copula model) (Zeng et al., 2016;Liao et al., 2019;Zhang et al., 2021). However, most previous studies mainly focused on the water quality indicators with not stressing the interdependence between water quality and water quantity indicators (Jimeno-Saez et al., 2020;Zhang et al., 2020a;Vinon-Leite et al., 2019), thus lowering their assessment accuracy (Chebana and Ouarda, 2011;Yu et al., 2016;Schumann et al., 2017). Therefore, it is necessary to consider the dependence and interaction between water quantity indicator and water quality indicator to improve the effect of eutrophication risk assessment. ...
Evaluating the eutrophication risk of a receiving reservoir is crucial for scientific water transfer schemes. However mega water transfer projects would greatly affect both water quantity and water quality processes, making the eutrophication evaluation more difficult. This study assessed the eutrophication risk of a receiving reservoir (i.e., Miyun Reservoir) after implementation of the world's largest water transfer project, namely, the South-to-North Water Transfer Project (SNWTP) in China. A new perspective of considering joint effects of water quantity indicator (i.e., water storage) and water quality indicator (i.e., Chlorophyll a concentration or Chla concentration for short) was proposed to assess the eutrophication risk. The GMM model was first introduced into the copula model to adaptively describe the marginal distribution of hydrological variables and to improve the accuracy of the marginal probability distribution for water storage. Besides, the Frank copula model was selected to establish the joint probability distribution function of water storage and Chla concentration from four candidate copula models. The eutrophication risk of the receiving reservoir was then assessed under ten water transfer scenarios concerning six amounts of water transfer in four certain periods. Results showed: 1) there was little eutrophication risk (<0.0005) in Miyun Reservoir under the selected water transfer scenarios; 2) the probability of higher Chla concentration in Miyun Reservoir would increase with larger water storage after the implementation of the SNWTP, and 3) the probability (0.00027) of water quality deterioration in the reservoir under uniform water transfer was about half (0.00045) of the centralized water transfer. These findings can contribute to the eutrophication risk assessment and adaptive management of the world's largest water transfer project.
... In univariate flood frequency analysis, the return period of a given hydrological variable is taken as the return period of structural failure. It is evident that the univariate return period of the flood peak or volume cannot provide a complete evaluation of the occurrence probability of hydrological events (Chebana & Ouarda 2011), which therefore may result in over-or underestimation of the flood risk and hydraulic design standard (Salvadori & De Michele 2004). Apparently, multivariate flood frequency analysis is preferred over univariate flood frequency analysis in hydraulic design. ...
Multivariate flood frequency analysis has been widely used in the design and risk assessment of hydraulic structures. However, analytical solutions are often obtained based on an idealized linear reservoir model in which a linear routing process is assumed, and consequently, the flood risk is likely to be over- or underestimated. The present study proposes a nonlinear reservoir model in which the relationships of reservoir water level with reservoir volume and discharge are assumed to be nonlinear in order to more accurately describe the routing process as it takes into consideration the interactions between hydrological loading and different discharge structures. The structure return period is calculated based on the copula function and compared with that based on the linear reservoir model and the bivariate return period based on the Kendall distribution function. The results show that the structure return period based on the linear model leads to an underestimation of the flood risk under the conditions of high reservoir water level. For the same reservoir, linear and nonlinear reservoir models give quite different reservoir volume-water level and discharge-water level curves; therefore, they differ substantially in the sensitivity to flood events with different combinations of flood peak and volume. We also analyze the effects of the parameters involved in the reservoir volume-water level and discharge-water level relationships on the maximum water level at different return periods in order to better understand the applicability and effectiveness of the proposed method for different hydraulic projects. HIGHLIGHTS
The structure return period is calculated based on the nonlinear reservoir model.;
Nonlinear relationships of water level with volume and discharge are assumed.;
Interactions between hydrological loading and different structures are analyzed.;
Effects of different storage and discharge capacity parameters are discussed.;
... To estimate the desired design combination of wave height and surge accurately under extreme conditions can be rather challenging. Many studies have outlined that a univariate frequency analysis may not be capable of assessing the occurrence probability of extremes if the events are characterized by interrelated random variables (Chebana and Ouarda, 2011;Masina et al., 2015). According to Marcos et al. (2019), the return periods of extreme sea levels are underestimated in 30 % of the coasts around the world if dependence is neglected. ...
In designing coastal and nearshore structures, the joint probability of the wave heights and storm surges is essential in determining the possible highest total water level. The key elements to accurately estimate the joint probability are the appropriate sampling of the extreme values and selection of probability functions for the analysis. This study is to provide a full assessment of the performance of the different methods employed in the joint probability analysis. The bivariate extreme wave height and surge samples are analysed using 2 different probability distributions and the performance of 4 copulas, namely: Gumbel–Hougaard copula, Clayton copula, Frank copula and Galambos copula, is assessed. The possible highest total water levels for 100-year return period along the coastline of the mainland China are estimated by the joint probability method with the Gumbel–Hougaard copula. The results show that the wave heights and surges are highly correlated in the areas of dense typhoon paths. The distributions of the possible highest total water levels show a higher value in the southeast coast and lower value in the north. The results also indicate that at the locations where the sea states are energetic, the joint probability approach can improve the accuracy of design.
... The case in which one of the two return periods is much higher than the other may be statistically valid and a limit case of the copulas, but it is not hydrologically plausible since the subbasins are expected to react similarly, at least for extreme events where mostly global instead of local drivers cause the flood events. To overcome this problem, critical ranges can be estimated (Chebana & Ouarda, 2011;Volpi & Fiori, 2012). These critical ranges include only those combinations that are likely to occur, based on past observations. ...
In contrast to the basic assumption of a homogeneous population underlying common approaches to flood frequency analysis, flood events often arise from different runoff‐generating processes. In many large river basins, the diversity of these processes within tributary basins and the superposition of their flood waves increase the complexity of statistical flood modeling. Under these circumstances, the allocation of the most effective flood protection measures requires a spatially explicit analysis of flood‐generating processes and the determination of the probability of downstream flood scenarios. For large basins, flood scenarios are often derived using individual historical floods along with model‐based simulations. We, instead, performed hydrograph‐based flood‐type classification and volume‐based runoff analyses for the Upper Danube River to estimate the contributions of subbasins to floods at downstream locations. Using this information, we generated long synthetic samples of peak‐volume‐pairs to apply a multivariate statistical flood‐frequency model that yields a conditional probability of a flood peak given the peaks in tributary stations. The results show that only certain combinations of flood types may result in extreme peaks downstream of confluences. They also highlight the need to distinguish runoff‐generation mechanisms for the larger floods from ones that drive smaller, more frequent events. Through an example with the Rhine River, we demonstrate how the statistical model can be generalized for complex river networks featuring several tributary confluences. Finally, design floods for different scenarios of flood‐type combinations and assigned probabilities are derived, an approach that can be used to possible climate impacts to flood frequency.
... Once the total flood volume and the number of sub-regions have been determined, there are infinite regional flood allocation schemes; however, not all the schemes are consistent with the constitutional law of hydrological events (Chebana and Ouarda, 2011;Salvadori et al., 2011). Moreover, although some schemes are rational, their probability of occurrence is relatively small (such as the flood volume in the main runoff-producing sub-region only accounting for a small proportion of the total flood, and the flood volume of the secondary runoff-producing sub-region representing a more significant proportion of the total flood). ...
Lakes are typical plain reservoirs, with many similar functions to valley reservoirs; however, their flood processes differ significantly due to their specific topography. Flooding of the dam site, the entire reservoir inflow, is considered during flood control analysis of valley reservoirs. Whereas, the flood inundation risk of flood detention areas in the lake basin should be considered for lakes. Therefore, more attention should be paid to the flood processes of each sub-region. Here, based on the flood processes characteristics of large water-carrying lakes, a complete framework for establishing design floods for sizeable rivers-connected lakes is introduced. First, multiple Copula-based joint distribution functions are constructed based on the water system structure of the lake. Then, several confidence intervals are obtained using a methodology that identifies the boundary for multivariate combinations. Finally, through these confidence intervals are used in conjunction with each other, the design flood processes for water-carrying lake system can be determined. The proposed methodology was applied to the Hongze Lake, China. The results indicate that this method not only effectively avoids the randomness of the traditional method but also balances the characteristics of the flood process in each sub-region with the features of the entire flood process. The proposed method has a strong statistical theoretical foundation and expands the applicability of multi-variable flood frequency analysis techniques to water conservancy projects.
... As more than one explanatory variable are often used in a PFHA and in the case these variables are dependent, the dependency structure must be modeled and a consistent theoretical framework must be introduced for the calculation of the return periods and design quantiles with multivariate analysis based on copulas (e.g., Salvadori et al., 2011). Indeed, numerous studies have shown that, in the case of multivariate hazards, a univariate frequency analysis does not allow the estimation in a complete way of the probability of occurrence of an extreme event (Chebana and Ouarda, 2011;Hamdi et al., 2016). According to Salvadori and De Michele (2004), modeling the dependency allows a better understanding of the hazard and avoids under-/overestimating the risk. ...
Coastal facilities such as nuclear power plants (NPPs) have to be
designed to withstand extreme weather conditions and must, in particular, be
protected against coastal floods because it is the most important source of
coastal lowland inundations. Indeed, considering the combination of tide
and extreme storm surges (SSs) is a key issue in the evaluation of the risk
associated with coastal flooding hazard. Most existing studies are generally
based on the assumption that high tides and extreme SSs are
independent. While there are several approaches to analyze and characterize
coastal flooding hazard with either extreme SSs or sea levels, only few
studies propose and compare several approaches combining the tide density
with the SS variable. Thus this study aims to develop a method for modeling
dependence and coincidence of SSs and high tide. In this work, we have used
existing methods for tide and SS combination and tried to improve the
results by proposing a new alternative approach while showing the
limitations and advantages of each method. Indeed, in order to estimate
extreme sea levels, the classic joint probability method (JPM) is used by
making use of a convolution between tide and the skew storm surge (SSS).
Another statistical indirect analysis using the maximum instantaneous storm
surge (MSS) is proposed in this paper as an alternative to the first method
with the SSS variable. A direct frequency analysis using the extreme total
sea level is also used as a reference method. The question we are trying to
answer in this paper is then the coincidence and dependency essential for a
combined tide and SS hazard analysis. The results brought to light a bias in
the MSS-based procedure compared to the direct statistics on sea levels, and this
bias is more important for high return periods. It was also concluded that
an appropriate coincidence probability concept, considering the dependence
structure between SSs, is needed for a better assessment of the risk using
the MSS. The city of Le Havre in France was used as a case study. Overall,
the example has shown that the return level (RL) estimates using the MSS
variable are quite different from those obtained with the method using
the SSSs, with acceptable uncertainty. Furthermore, the shape parameter is
negative from all the methods with a much heavier tail when the SSS and the
extreme sea levels (ESLs) are used as variables of interest.
... e naïve part is composed of two segments starting at the end of both extremities of the proper part. Detailed description and proprieties of the bivariate quantile function can be found in Chebana and Ouarda [77]. For the grid 1682 example in winter, for risk value of p � 0.9, the univariate quantile values of precipitation and temperature are 10 mm and 5°C, respectively. ...
Precipitation and temperature are among major climatic variables that are used to characterize extreme weather events, which can have profound impacts on ecosystems and society. Accurate simulation of these variables at the local scale is essential to adapt urban systems and policies to future climatic changes. However, accurate simulation of these climatic variables is difficult due to possible interdependence and feedbacks among them. In this paper, the concept of copulas was used to model seasonal interdependence between precipitation and temperature. Five copula functions were fitted to grid (approximately 10 km × 10 km) climate data from 1960 to 2013 in southern Ontario, Canada. Theoretical and empirical copulas were then compared with each other to select the most appropriate copula family for this region. Results showed that, of the tested copulas, none of them consistently performed the best over the entire region during all seasons. However, Gumbel copula was the best performer during the winter season, and Clayton performed best in the summer. More variability in terms of best copula was found in spring and fall seasons. By examining the likelihoods of concurrent extreme temperature and precipitation periods including wet/cool in the winter and dry/hot in the summer, we found that ignoring the joint distribution and confounding impacts of precipitation and temperature lead to the underestimation of occurrence of probabilities for these two concurrent extreme modes. This underestimation can also lead to incorrect conclusions and flawed decisions in terms of the severity of these extreme events.
... flood peak), may not be sufficient to describe those hydrological extremes containing multivariate characteristics. Thus the univariate frequency/risk analysis methods may be unable to obtain reliable risk inferences for the failure probability or recurrence intervals of interdependent extreme events (Chebana and Ouarda, 2011;Requena et al., 2013;Salvadori et al., 2016;Sadegh et al., 2017). ...
Extensive uncertainties exist in hydrologic risk analysis. Particularly for interdependent hydrometeorological extremes, the random features in individual variables and their dependence structures may lead to bias and uncertainty in future risk inferences. In this study, an iterative factorial copula (IFC) approach is proposed to quantify parameter uncertainties and further reveal their contributions to predictive uncertainties in risk inferences. Specifically, an iterative factorial analysis (IFA) approach is developed to diminish the effect of the sample size and provide reliable characterization for parameters' contributions to the resulting risk inferences. The proposed approach is applied to multivariate flood risk inference for the Wei River basin to demonstrate the applicability of IFC for tracking the major contributors to resulting uncertainty in a multivariate risk analysis framework. In detail, the multivariate risk model associated with flood peak and volume will be established and further introduced into the proposed iterative factorial analysis framework to reveal the individual and interactive effects of parameter uncertainties on the predictive uncertainties in the resulting risk inferences. The results suggest that uncertainties in risk inferences would mainly be attributed to some parameters of the marginal distributions, while the parameter of the dependence structure (i.e. copula function) would not produce noticeable effects. Moreover, compared with traditional factorial analysis (FA), the proposed IFA approach would produce a more reliable visualization for parameters' impacts on risk inferences, while the traditional FA would remarkably overestimate the contribution of parameters' interaction to the failure probability in AND (i.e. all variables would exceed the corresponding thresholds) and at the same time underestimate the contribution of parameters' interaction to the failure probabilities in OR (i.e. one variable would exceed its corresponding threshold) and Kendall (i.e. the correlated variables would exceed a critical multivariate threshold).
... The copula model has been commonly used to derive the multivariate probability distribution H(X,Y) (Beersma and Buishand 2004;Kao and Govindaraju 2008;Chebana and Ouarda 2011;Liu et al. 2016), which is advantageous in that the modeling of marginal and joint distributions is separated. For two continuous random variables X and Y with the marginal probability U and V, the copula-based joint distribution function can be expressed as (Nelsen 2006): ...
The past decades have witnessed a surge in the study of compound droughts and hot extremes, which are among the most stressful weather and climate extremes with disastrous impacts. A suitable tool for tracking properties of compound droughts and hot extremes taking into account their dependence is desirable. In this study, we proposed a compound drought-hot extreme index (CDHI) for characterizing compound extremes based on the joint distribution using copula models. Results show that lower CDHI values indicate more severe droughts and hot extremes, which enables the comparison of the joint status of droughts and hot events. The monitoring of historical compound droughts and hot extremes is evaluated based on the CDHI values and associated categories for the period of 1980, 2001, and 2011 in Texas, which demonstrates the usefulness of the proposed index for tracking the joint status of droughts and hot extremes at temporal and spatial scales. The proposed index would provide a useful tool for monitoring compound droughts and hot extremes and providing early warning information.
Changing climatic conditions have escalated the risk of compound disaster, and there remains a scarcity of quantitative research at river basin scale. An integrated research framework is proposed in this study to quantitatively analyze and assess the risk of future compound flood in Xijiang River Basin based on external driving factor and internal variables. Under this framework, a multi-model ensemble of 10 preferred CMIP6 GCMs is carried out based on statistical downscaling and Bayesian weighted average method, and the multi-scale variation characteristics of precipitation and runoff during 2020 ~ 2099 are analyzed based on the ensemble data. Combined with univariate and multivariate trend analysis considering intra-correlation, the multi-class copula functions are utilized to estimate the joint probability and return period of compound flood. The results show that: 1) The precipitation and runoff increase by 8.25%, 14.5%, and 34.05%, 55.18% respectively compare to the baseline period under SSP2-4.5 and SSP5-8.5, with both displaying an increasing trend at rates of 1.03%/10a, 2.66%/10a, and 2.74%/10a, 4.62%/10a on the interdecadal scale under the two scenarios, respectively. 2) The internal variables of the compound flood represented by the annual maximum peak flow (AMPF) and the annual consecutive maximum 7-day flood volume (AM7dFV) present a significant increasing trend under the two scenarios, but the annual maximum precipitation (AMPre) of the external driving factor does not show a significant trend while the annual total precipitation (ATPre) of the external driving factor increases significantly under both scenarios. It is noteworthy that both the internal variables and the external driving factors of compound flood show significant increases in the multivariate analysis. 3) The joint variable of compound flood demonstrates a substantially increasing trend under both scenarios, along with an increase in the magnitude of the once-in-a-century flood. Discounting the intra-correlation between multivariate, the degree of disaster would be underestimated.
Increasing frequency and intensification of flooding pose significant threats to critical structures, such as dams and levees. Failure of these structures can lead to substantial economic losses and significant adverse environmental and social consequences. Improving the resilience of these structures against climate-related impacts is important to avoid future risks of failure due to the potential intensification of flooding. National-level guidance on integrating resilience-based frameworks and addressing climate risks and uncertainties in existing design flood estimation methodologies for dams and levees are lacking. To address these gaps, this study first reviews projected climate change patterns for Canada and then discusses regional vulnerabilities of dams by considering significant historical floods and their consequences. Subsequently, a review of existing design flood estimation procedures, with a focus on frequency- and probable maximum flood-based approaches, is conducted to identify areas where climate change-related aspects can be integrated. By examining the challenges associated with various stages of design flood estimation procedures, the review discusses a framework for enhancing climate resiliency of dams and levees considering four pillars of resilience. Furthermore, Canadian design flood estimation practices are compared with international practices to identify areas that require attention. The study highlights the importance of a resilience-based framework in providing design and operation guidance to ensure that dams and levees are resilient to climate impacts. Policymakers and engineers can prioritize consideration of climate-resilience in the design and operation of these structures in order to safeguard communities and infrastructure from the growing risks of future floods associated with climate change.
This study proposed a framework to evaluate multivariate return periods of hurricanes using event‐based frequency analysis techniques. The applicability of the proposed framework was demonstrated using point‐based and spatial analyses on a recent catastrophic event, Hurricane Ian. Univariate, bivariate, and trivariate frequency analyses were performed by applying generalized extreme value distribution and copula on annual maximum series of flood volume, peak discharge, total rainfall depth, maximum wind speed, wave height and storm surge. As a result of point‐based analyses, return periods of Hurricane Ian was investigated by using our framework; univariate return periods were estimated from 39.2 to 60.2 years, bivariate from 824.1 to 1,592.6 years, and trivariate from 332.1 to 1,722.9 years for the Daytona‐St. Augustine Basin. In the Florida Bay‐Florida Keys Basin, univariate return periods were calculated from 7.5 to 32.9 years, bivariate from 36.5 to 114.9 years, and trivariate from 25.0 to 214.8 years. Using the spatial analyses, we were able to generate the return period map of Hurricane Ian across Florida. Based on bivariate frequency analyses, 18% of hydrologic unit code 8 (HUC8) basins had an average return period of more than 30 years. Sources of uncertainty, due to the scarcity of analysis data, stationarity assumption and impact of other weather systems such as strong frontal passages, were also discussed. Despite these limitations, our framework and results will be valuable in disaster response and recovery.
It was demonstrated in the previous chapters of this part how important the consideration of spatial dependency and of the different flood-generating mechanisms in the catchments is for the generation of consistent flood events in large river basins. This idea is further extended in this chapter, where a methodology to obtain type-specific flood scenarios based on the multisite spatial model in Chap. 11 is proposed. Such scenarios are most relevant for many practical applications for water management. The assignment of probabilities to each scenario allows for an evaluation of the most critical scenarios. The proposed model is the natural extension of the type-specific hydrograph scenarios given in Chap. 9 but in this case the different combinations of flood types of the upstream catchments are considered to allow for a spatially consistent scenario evaluation of the flood events.
The joint probability analysis of river water temperature (RWT) and low flow (LF) characteristics is essential as their combined effect can negatively affect aquatic species, e.g., ectotherm fish. Traditional multivariate models may not be as effective as copula-based methodologies. This study introduces a new multivariate approach, the nonparametric copula density framework, free from any distribution assumption in their univariate margins and copula joint density. The proposed framework utilized RWT and LF datasets collected at five different river stations in Switzerland. The study evaluates a nonparametric Gaussian kernel with six bandwidth selectors to model marginal densities. It employs nonparametric-based Beta kernel density, Bernstein estimator, and Transformation kernel estimator to approximate copula density with nonparametric and parametric margins. The performance of some parametric copula densities was also compared. The most justifiable models were employed to estimate bivariate joint exceedance probabilities and return periods (RPs). The Beta kernel copula with Gaussian kernel margins outperformed other models for most stations; Bernstein and Transformation copula with Gaussian kernel margins were better for only one station each. The univariate RPs (RWT or LF) are lower than the AND-joint but higher than OR joint case. As the percentile value of LF events (serve as a conditioning variable) increases, the bivariate joint RPs of RWT also increase. Higher values in RWT events result in higher RPs than lower values at the fixed percentile value of LF. All such estimated risk statistics are beneficial to analyze their mutual risk in aquatic habitats and freshwater ecosystems.
Accurate statistical description of extreme environmental conditions is needed for risk assessment and management of marine structures and is a crucial input to design of any structure that need to withstand loads from environmental forces. Such descriptions are essentially multivariate extreme value problems, where the environmental loads are due to concurrent extreme combinations of several environmental variables. Typically, in coastal and ocean engineering applications, the simultaneous joint behaviour of significant wave height and wave period is of particular interest and is needed to describe the wave loads on marine structures. Environmental contours are often used to explore the extreme wave loads, and essentially consider extreme combinations of simultaneous significant wave height and wave period, and are based on a joint statistical distribution fitted to relevant metocean data. However, typical applications of environmental contours do not account for temporal dependencies in the environmental variables, and this may lead to an overestimation of extreme conditions. In this paper, an approach for partially accounting for serial dependence in the construction of environmental contours is proposed, based on simulating time-series of a primary variable which preserves both its marginal distribution and auto-correlation structure. It is shown that this gives lower estimates of extreme environmental conditions compared to conventional application of environmental contours that do not account for serial dependence. Hence, more accurate description of the extreme environment can be available for design and construction of structures exposed to environmental loads.
Extreme rainfall is a critical “agent” driving flash floods in urban areas. In rainfall frequency analysis (RFA), however, storms are usually assumed to be uniform in space and fixed in time. Spatially and temporally uniform design storms and area reduction factors are oftentimes used in conjunction with RFA results in engineering practice for infrastructure design and planning. The consequences of such assumptions are poorly understood. This study examines how spatiotemporal rainfall heterogeneity impacts RFA, using a newly-introduced bivariate framework consisting of copula theory and stochastic storm transposition (SST). A large number of regionally-extreme storms with specific features—rainfall depth, duration, intensity, and level of intra-storm spatial organization—were collected. Rainfall intensity-duration-frequency (IDF) estimates exhibiting these bivariate features were then generated by synthesizing long records of rainfall via SST. The results show that dependencies exist among spatiotemporal storm characteristics. Bivariate frequency results exhibit smaller uncertainties but more complex physical meanings that the results from conventional methods. In particular, the highly spatially-organized storms play a leading role in frequency estimates.
In recent decades, substantial efforts have been devoted in flood monitoring, prediction, and risk analysis for aiding flood event preparedness plans and mitigation measures. Introducing an initial framework of spatially probabilistic analysis of flood research, this study highlights an integrated statistical copula and satellite data-based approach to modelling the complex dependence structures between flood event characteristics, i.e., duration (D), volume (V) and peak (Q). The study uses Global daily satellite-based Climate Hazards Group InfraRed Precipitation with Station data (CHIRPS) (spatial resolution of ∼5km) during 1981–2019 to derive a Standardized Antecedence Precipitation Index (SAPI) and its characteristics through a time-dependent reduction function for Myanmar. An advanced vine copula model was applied to model joint distributions between flood characteristics for each grid cell. The southwest (Rakhine, Bago, Yangon, and Ayeyarwady) and south (Kayin, Mon, and Tanintharyi) regions are found to be at high risk, with a probability of up to 40% of flood occurrence in August and September in the south (Kayin, Mon, and Tanintharyi) and southwest regions (Rakhine, Bago, Yangon, and Ayeyarwady). The results indicate a strong correlation among flood characteristics; however, their mean and standard deviation are spatially different. The findings reveal significant differences in the spatial patterns of the joint exceedance probability of flood event characteristics in different combined scenarios. The probability that duration, volume, and peak concurrently exceed 50th-quantile (median) values are about 60–70% in the regions along the administrative borders of Chin, Sagaing, Mandalay, Shan, Nay Pyi Taw, and Keyan. In the worst case and highest risk areas, the probability that duration, volume, and peak exceed the extreme values, i.e., the 90th-quantile, about 10–15% in the southwest of Sagaing, southeast of Chin, Nay Pyi Taw, Mon and areas around these states and up to 30% in the southeast of Dekkhinathiri township (Nay Pyi Taw). The proposed approach could improve the evaluation of exceedance probabilities used for flood early warning and risk assessment and management. The proposed framework is also applicable at larger scales (e.g., regions, continents and globally) and in different hydrological design events and for risk assessments (e.g., insurance).
Hurricane-induced compound events (HICEs), such as coastal surges and winds, usually exhibit a high degree of nonlinear dependence, and thus, a single disaster modeling method cannot effectively evaluate and design the corresponding engineering applications. Therefore, this research aims at developing a statistical model suitable for HICEs to analyze and design multivariate hazard scenarios. Simultaneously, a risk-driven weighting function is constructed, considering the likelihood of event occurrence and response of the targets, to identify the riskiest design event in the critical event set. We apply the proposed model to an industrial area on the Galveston coast; use numerically synthesized HICEs to explore the dependence of the flood height, wind speed, and current velocity; and discuss the effects of different weighting rules on the design events. The modeling results show that three marginal variables are significantly correlated with one another, and the correlation between the flood height and wind speed in extreme events is enhanced. Additionally, on the same set of critical events, the riskiest event is typically not the most likely event, and the difference between them decreases as the return period increases. Moreover, the risk-driven weighting function provides a reliable scheme for disaster prevention design events of special petrochemical facilities.
In this study, a framework is proposed to estimate the nonstationary bivariate design flood, which includes three key steps. First, a nonstationary copula model was constructed to analyze the temporal variation in the bivariate joint distribution. Second, the equivalent reliability method was used to calculate the design value of the dominant variable for a return period and the design lifespan length under non-stationarity. Third, the design value of the secondary variable that was conditioned on the design value of the dominant variable, was calculated using the conditional most likely combination and conditional expectation combination strategies. Through the above three steps, the typical design value combination of the nonstationary bivariate was obtained for a specific design standard. A case study, based on the nonstationary annual maximum 1-day (AM1) and 15-day (AM15) flood volume at the Yichang station, was conducted to illustrate the framework. The results indicate that the joint distribution of the AM1 and AM15 flood volumes varied over time. For a specific design standard, the design values of the AM15 and AM1 flood volumes decreased with an increase in the design lifespan length. Moreover, the conditional most likely estimation of the design value for the AM1 flood volume that was conditioned on the design value for the AM15 flood volume was greater than its conditional expectation estimation.
In this study, a factorial multimodel Bayesian copula (FMBC) method is proposed to investigate various uncertainties in the copula-based multivariate risk models and further track the major contributors to the imprecise predictions for different risk indices. In FMBC, the copula models with different marginals and dependence structures will be firstly established with the parameter uncertainties being quantified by the adaptive Metropolis algorithm. A multilevel factorial analysis approach is then adopted to characterize the individual and interactive effects of marginals, copula functions and the associated parameter uncertainties on different risk indices. Moreover, a copula-based dependent sampling algorithm is proposed in the factorial analysis process to generate parameter samples under consideration of their correlation. The applicability of the FMBC approach is demonstrated through the multivariate flood risk analysis at two gauging stations in Wei River basin. The results indicate that extensive fluctuation exists in the multivariate return periods resulting from different marginal and dependence structures as well as the associated parameter uncertainties. For risk indices of the failure probabilities in And and Kendall, their predictive variability can be mainly attributed to the uncertainties in model parameters and copula structure, with their total contribution more than 75%. In comparison, the failure probability in OR would be mainly influenced by the parameter uncertainty and also marginal structures, with their total contribution more than 80%. The copula structure would not have a visible effect on the failure probability in OR, with its contribution less than 5% for most scenarios. The obtained results can provide scientific support for reliable hydrological risk inferences within a multivariate context.
Climate change will not only affect a single hydrological series, but also may affect the dependence structure. However, traditional hydrological approaches often fail to analyse the multiple characteristics of runoff within a bivariate framework under climate change. This study explores the influence of future climate change (2031−2080) on the runoff volume series in spring, summer and autumn as well as their dependence structures in the Luanhe River Basin, China. The results indicate that the time-varying dependence structure between spring and summer runoff series, as well as that between the summer and autumn runoff series, are mainly related to the summer precipitation. The differences between copula models with constant dependence parameters and those with time-varying dependence parameters are presented under different cases. This study incorporates the impacts of climate change on the bivariate frequency analysis of seasonal runoff series, which may provide important guidance for water resource management under climate change.
The complex hydrological events such as storm, flood and drought are often characterized by a number of correlated random variables. Copulas can model the dependence structure independently of the marginal distribution functions and provide multivariate distributions with different margins and the dependence structure. In this study, the conditional behavior of two signatures was investigated by analyzing the joint signatures of groundwater level deficiency and rainfall deficiency in Naqadeh sub-basin in Lake Urmia Basin using copula functions. The study results of joint changes in the two signatures showed that a 90–135 mm reduction in rainfall in the area increased groundwater level between 1.2 and 1.7 m. The study results of the conditional density of bivariate copulas in the estimation of groundwater level deficiency values by reducing rainfall showed that changes in values of rainfall deficiency signature in the sub-basin led to the generation of probability curves of groundwater level deficiency signature. Regarding the maximum groundwater level deficiency produced, the relationship between changes in rainfall deficiency and groundwater level deficiency was calculated in order to estimate the groundwater level deficiency signature values. The conditional density function presented will be an alternative method to the conditional return period. HIGHLIGHTS
The purpose of the present study is to investigate the conditional density analysis of bivariate copulas to estimate the groundwater level deficiency using the rainfall deficiency.;
In this regard, the diagonal section of the copulas was used to reduce the complexity of the conditional density of the pairwise variables.;
By using conditional density and combining it with copula simulation, the dependent values can be simulated and predicted.;
Due to the lack of application of the conditional return period, the best alternative method is to use the proposed method based on conditional density.;
The presented equation can be used as an alarm and monitoring system.;
This paper presents a bivariate regional frequency analysis applied to vectors of extreme significant wave heights and concurrent zero up-crossing wave period over an area in the North Atlantic ocean. The analysis is based on a bivariate index-wave/period approach and the assumption of a common regional growth curve within homogeneous regions. The main benefits of performing a regional frequency analysis as opposed to at-site analysis based on data from one location only are twofold: It may give more accurate predictions of extreme conditions due to the increased amount of data made available for the statistical analysis, and results for ungauged locations can be obtained easily by interpolation of the index wave/period. Previous regional frequency analyses for ocean waves have been restricted to the univariate case. However, design conditions for marine and coastal structures will typically be characterised by the joint distribution of several metocean variables. Hence, it is useful to perform multivariate analyses of the extreme ocean environment. As a minimum, information about the joint distribution of significant wave height and period is often needed. This paper outlines the various steps involved in a bivariate regional frequency analysis and presents the results of such an analysis applied to data covering the North Atlantic ocean over a period of 30 years.
The three-parameter generalized extreme-value (GEV) distribution has found wide application for describing annual floods, rainfall, wind speeds, wave heights, snow depths, and other maxima. Previous studies show that small-sample maximum-likelihood estimators (MLE) of parameters are unstable and recommend L moment estimators. More recent research shows that method of moments quantile estimators have for -0.25 < kappa < 0.30 smaller root-mean-square error than L moments and MLEs. Examination of the behavior of MLEs in small samples demonstrates that absurd values of the GEV-shape parameter kappa can be generated. Use of a Bayesian prior distribution to restrict kappa values to a statistically/physically reasonable range in a generalized maximum likelihood (GML) analysis eliminates this problem. In our examples the GML estimator did substantially better than moment and L moment quantile estimators for -0.4 less than or equal to kappa less than or equal to 0.
Several types of hydrological events are described with multivariate characteristics (droughts, floods, rain storms, etc.). When carrying out a multivariate regional frequency analysis for these events it is important to jointly consider all these characteristics. The aim of this paper is to extend the statistical homogeneity test of Hosking and Wallis (1993) to the multivariate case. As a tool, multivariate L-moments are used to define the statistics and general copula models to describe the statistical behavior of dependent variables. The usefulness of the methodology is illustrated on flood events. Monte-Carlo simulations are also performed for a bivariate Gumbel logistic model with Gumbel marginal distributions. Results illustrate the power of the proposed multivariate L-moment homogeneity test to detect heterogeneity on the whole structure of the model and on the marginal distributions. In a bivariate flood setting, a comparison is carried out with the classical homogeneity test of Hosking and Wallis based on several types of regions.
An analytical derivation of the storm volume distribution is given. The single storm is modeled as a rectangular pulse having non-independent random duration and intensity. The structure of the dependence between storm duration and intensity is given via a suitable 2-Copula, and the marginal distributions are endowed with Generalized Pareto laws, as recently pointed out in the hydrological Literature. The statistical properties of the rainfall volume are investigated, both analytically and using simulated samples; an application to rainfall data is also shown.
Regional frequency analysis (RFA) consists generally of two steps: (1) delineation of hydrological homogeneous regions and (2) regional estimation. Existing regionalization methods which adopt this two-step approach suffer from two principal drawbacks. First, the restriction of the regional estimation to a particular region by excluding some sites can correspond to a loss of some information. Second, the definition of a region generates a border effect problem. To overcome these problems, a new method is proposed in the present paper. The proposed method is based on three elements: (1) a weight function to treat the border effect problem, (b) a function to evaluate how ``similar'' each site is to the target one, and (c) an iterative procedure to improve estimation results. Element (b) is treated using the statistical notion of depth functions which is introduced to provide a ranking of stations in a multivariate context. Furthermore, the properties of depth functions meet the characteristics sought in RFA. It is shown that the proposed method is flexible and general and that traditional RFA methods represent special cases of the depth-based approach corresponding to particular weight functions. A comparison is carried out with the canonical correlation analysis (CCA) approach. Results indicate that the depth-based approach performs better than does CCA both in terms of relative bias and relative root mean squares error.
The Halphen family of distributions is a flexible and complete system to fit sets of observations independent and identically
distributed. Recently, it is shown that this family of distributions represents a potential alternative to the generalized
extreme value distributions to model extreme hydrological events. The existence of jointly sufficient statistics for parameter
estimation leads to optimality of the method of maximum likelihood (ML). Nevertheless, the ML method requires numerical approximations
leading to less accurate values. However, estimators by the method of moments (MM) are explicit and their computation is fast.
Even though MM method leads to good results, it is not optimal. In order to combine the advantages of the ML (optimality)
and MM (efficiency and fast computations), two new mixed methods were proposed in this paper. One of the two methods is direct
and the other is iterative, denoted respectively direct mixed method (MMD) and iterative mixed method (MMI). An overall comparison
of the four estimation methods (MM, ML, MMD and MMI) was performed using Monte Carlo simulations regarding the three Halphen
distributions. Generally, the MMI method can be considered for the three Halphen distributions since it is recommended for
a majority of cases encountered in hydrology. The principal idea of the mixed methods MMD and MMI could be generalized for
other distributions with complicated density functions.
: The knowledge of the volume and duration of low-flow events in river channels is essential for water management and the design of hydraulics structures. In this study, both preceding characteristics, X
1 and X
2, are considered simultaneously via two types of bivariate distributions whose marginals are exponential. One of these bivariate distributions has been presented by Nagao and Kadoya (1971) and the other has been used by Singh and Singh (1991) to the study of rainfall intensity and rainfall depth. The results are applied to the low-flow series (“peaks-below-threshold”) of Lepreau River (station 01AQ001) in New Brunswick, Canada. These results show that the model that was successfully employed by Singh and Singh (1991) to study rainfall, presents certain difficulties when a very strong correlation, ρ, between the two random variables X
1 and X
2, exists. The model by Nagao and Kadoya (1971) seems to be more satisfactory for such situations, although this model seems also to be quite sensitive to variations in ρ.
Many proposals have been made recently for goodness-of-fit testing of copula models. After reviewing them briefly, the authors concentrate on "blanket tests", i.e., those whose implementation requires neither an arbitrary categorization of the data nor any strategic choice of smoothing parameter, weight function, kernel, window, etc. The authors present a critical review of these procedures and suggest new ones. They describe and interpret the results of a large Monte Carlo experiment designed to assess the effect of the sample size and the strength of dependence on the level and power of the blanket tests for various combinations of copula models under the null hypothesis and the alternative. To circumvent problems in the determination of the limiting distribution of the test statistics under composite null hypotheses, they recommend the use of a double parametric bootstrap procedure, whose implementation is detailed. They conclude with a number of practical recommendations.
After five decades, the field of Statistical Hydrology continues to evolve and remains a very active area of investigation. Researchers continue to examine various distributions, methods of estimation of parameters, and problems related to regionalization. However, much of this research appears in journals and reports and usually in a form not easily accessible to practitioners and students-producing a gap between research and practice.
Flood Frequency Analysis fills this gap by presenting many of these distributions and estimation procedures in a unified format within a single, self-contained book. Focusing on distribution families popular within the hydrologic community, the authors discuss three parameter estimation methods for each distribution: the method of moments, the maximum likelihood method, and the method of probability weighted moments. They present the details behind the procedures to provide the basis for the computations, and they illustrate each procedure with real data. Most of the computations discussed have been programmed for use with personal computers, and executable versions of these programs are available on CD-ROM from the senior author.
Only increased use of new methods and distributions can produce a consensus on their validity. With other books on the subject either limited in scope or seriously outdated, Flood Frequency Analysis provides the ideal vehicle for practicing hydrologists and engineers to explore and apply the latest methods and research results, and in doing so, contribute to the advancement of the field.
The study of the statistics of extreme events is an essential first step in the mitigation of natural catastrophies, that often cause severe economic losses worldwide. This book is about the theoretical and practical aspects of the statistics of Extreme Events in Nature.
Most importantly, this is the first text in which Copulas are introduced and used in Geophysics.
Several topics are fully original, and show how standard models and calculations can be improved by exploiting the opportunities offered by Copulas. In addition, new quantities useful for design and risk assessment are introduced. Practicioners in all research areas of Geosciences and extreme events (including Finance and Insurance, closely related to natural disasters) will definitely benefit from the new Copula-approach outlined in the book.
Audience
This volume will be of interest to researchers and practitioners in the fields of civil and environmental engineering, geophysics, geosciences, geography and environmental science. Also scientists and undergraduate up to post graduate level students in water resources and hydrology will find valuable information in this book.
The study of the statistics of extreme events is an essential first step in the mitigation of natural catastrophies, that often cause severe economic losses worldwide. This book is about the theoretical and practical aspects of the statistics of Extreme Events in Nature.
For a number of cases, weighted and generalized least squares regression procedures were shown by Stedinger and Tasker (1985, 1986) to provide better estimators of the parameters of models of hydrologic statistics as a function of drainage basin characteristics. These procedures require estimation of the true model's standard error of prediction. Evaluated here are three model error estimators for weighted least squares (WLS) regression and two for generalized least squares (GLS) regression. The generalized mean square estimator employed by Stedinger and Tasker (1985) was nearly unbiased, and comparable in accuracy to and easier to compute than the maximum likelihood estimator. GLS and WLS regression estimators of the log-streamflows' standard deviation are shown to be substantially more accurate than ordinary least squares estimators. Finally, the consequences and impact of nonlognormal streamflows were evaluated. In the cases considered, GLS procedures continued to perform well even when the normality assumption was violated.
This article considers two examples of bivariate extremal distributions when the margins follow the first asymptotic distribution of the largest values. The variables are assumed to be reduced and each distribution contains only one parameter indicating the association between the extremes. The probability and the density functions of these distributions at the characteristic value, median, and mode have been analyzed. A procedure has been developed to estimate the single parameter of the distributions. A criterion for distinguishing between the two bivariate extremal distributions has been worked out. Finally, the various theories developed are applied to a numerical example.
The relation between flood frequency estimates and the economic decision-making process is briefly discussed and selected developments in frequency analysis are traced in tabular form. The main types of frequency analysis procedures and models are surveyed including at-site, at-site/regional and regional only cases. The at-site/regional cases include Index Flood, Bayesian and TCEV methods. Criteria for selecting a frequency analysis procedure are discussed under two headings, descriptive ability and predictive ability. The former relates to a model’s ability to preserve statistics of observed flood series while the latter relates to quantile estimating ability in a robust manner. The relative merits of different types of flood quantile estimators are discussed, beginning with at-site estimators and the search for a robust at-site estimator. This is followed by the principal results available about at-site/regional quantile estimators. Further topics considered are regional homogeneity of flood statistical behaviour, the effect of spatial and temporal interdependence of flood magnitudes, and the detection and use of historical floods.
An extension of the concept of quantiles in multidimensions that uses the geometry of multivariate data clouds has been considered. The approach is based on blending as well as generalization of the key ideas used in the construction of spatial median and regression quantiles, both of which have been extensively studied in the literature. These geometric quantiles are potentially useful in constructing trimmed multivariate means as well as many other L estimates of multivariate location, and they lead to a directional notion of central and extreme points in a multidimensional setup. Such quantiles can be defined as meaningful and natural objects even in infinite-dimensional Hilbert and Banach spaces, and they yield an effective generalization of quantile regression in multiresponse linear model problems. Desirable equivariance properties are shown to hold for these multivariate quantiles, and issues related to their computation for data in finite-dimensional spaces are discussed. n consistency and asymptotic normality of sample geometric quantiles estimating the corresponding population quantiles are established after deriving a Bahadur-type linear expansion. The sampling variation of geometric quantiles is carefully investigated, and estimates for dispersion matrices, which may be used in developing confidence ellipsoids, are constructed. In course of this development of sampling distributions and related statistical properties, we observe several interesting facts, some of which are quite counterintuitive. In particular, many of the intriguing properties of spatial medians documented in the literature appear to be inherited by geometric quantiles.
Inclusion of historical information in flood-frequency analysis increases the accuracy of flood estimates; however, some of the major factors affecting this accuracy are the a-priori specification of a particular probability distribution function and the method of estimating its parameters. In this study, a new nonparametric procedure is proposed that altogether eliminates the specification of a distribution and greatly simplifies parameter-estimation problems. The nonparametric method, however, is not particularly efficient in extrapolating distribution function beyond an available record length. Thus, to overcome such a problem, a new kernal is introduced in the form of an extreme-value distribution. Also, the smoothing parameter is estimated by a cross-validation procedure, and a new mixture- distribution model is proposed for inclusion of historical data into analysis. A simulation study employing a two-parameter log-normal distribution shows that the accuracy of flood estimates does not greatly increase with the addition of data length beyond 10 years. The present paper shows that inclusion of historical information into nonparametric analysis improves extrapolation.
Using the copula method, bivariate distributions of flood peak and volume, and flood volume and duration were derived. A major advantage of this method is that marginal distributions of individual variables (i.e., flood peak, volume, and duration) can be of any form and the variables can be correlated. The copula method was applied to obtain the conditional return periods that are needed for hydrologic design. The derived distributions were tested using flood data from Amite River at Denham Springs, La., and the Ashuapmushuan River at Saguenay, Quebec, Canada. The derived distributions were also compared with the Gumbel mixed and the bivariate Box-Cox transformed normal distributions. The copula-based distributions were found to be in better agreement with plotting position-based frequency estimates than were other distributions.
Droughts cause severe damage in terms of both natural environments and human lives, and hydrologists and water resources managers are concerned with estimating the relative frequencies of these events. Univariate parametric methods for frequency analysis may not reveal significant relationships among drought characteristics. Alternatively, nonparametric methods provide local estimates of the univariate and multivariate density function by using weighted moving averages of the data in a small neighborhood around the point of estimation and opposed to parametric methods. A methodology for estimating the return period of droughts using a nonparametric kernel estimator is presented in order to examine the univariate as well as the bivariate behavior of droughts. After evaluating and validating a nonparametric kernel estimator, a drought frequency analysis is conducted to estimate the return periods of droughts for the Conchos River Basin in Mexico. The results show that, for the univariate analysis, the return periods of the severe drought occurring in the 1990s are 100 years or higher. For the bivariate analysis, the return periods are approximately 50 years for joint distributions and more than 120 years for the conditional distributions of severity and duration.
A bivariate distribution function H(x, y) with marginals F(x) and G(y) is said to be generated by an Archimedean copula if it can be expressed in the form H(x, y) = ϕ[ϕ{F(x)} + ϕ{G(y)}] for some convex, decreasing function ϕ defined on [0, 1] in such a way that ϕ(1) = 0. Many well-known systems of bivariate distributions belong to this class, including those of Gumbel, Ali-Mikhail-Haq-Thélot, Clayton, Frank, and Hougaard. Frailty models also fall under that general prescription. This article examines the problem of selecting an Archimedean copula providing a suitable representation of the dependence structure between two variates X and Y in the light of a random sample (X1, Y1), …, (Xn, Yn). The key to the estimation procedure is a one-dimensional empirical distribution function that can be constructed whether the uniform representation of X and Y is Archimedean or not, and independently of their marginals. This semiparametric estimator, based on a decomposition of Kendall's tau statistic, is seen to be √n-consistent, and an explicit formula for its asymptotic variance is provided. This leads to a strategy for selecting the parametric family of Archimedean copulas that provides the best possible fit to a given set of data. To illustrate these procedures, a uranium exploration data set is reanalyzed. Although the presentation is restricted to problems involving a random sample from a bivariate distribution, extensions to situations involving multivariate or censored data could be envisaged.
The problem of selecting the appropriate design flood is a constant concern to dam engineering and, in general, in the hydrological practice. Overtopping represents more than 40% of dam failures in the world. The determination of the design flood is based in some cases on the T-year quantile of flood peak, and in other cases considering also the T-year quantile of flood volume. However, flood peak and flood volume have a positive (strong or weak) dependence. To model properly this aspect a bivariate probability distribution is considered using the concept of 2-Copulas, and a bivariate extreme value distribution with generalized extreme value marginals is proposed. The peak-volume pair can then be transformed into the correspondent flood hydrograph, representing the river basin response, through a simple linear model. The hydrological safety of dams is considered checking adequacy of dam spillway. The reservoir behavior is tested using a long synthetic series of flood hydrographs. An application to an existing dam is given.
The two-parameter generalized Pareto distribution (GPD) has been recommended for the frequency analysis of environmental extreme events. In the present paper, we concentrate on one form of the GPD (which we will call GPDB) which can be useful in the frequency analysis of two types of hydrological variables: (1) when the shape parameter α is positive, the distribution (denoted by GPDB-2) can be used to study phenomena such as flood flows, which are bounded from below but have a long right tail; (2) when α is negative, the resulting distribution (GPDB-3) could be used to study variables such as low flows which are bounded from both sides but with a left tail. Six versions of the generalized method of moments (GMM) for fitting GPDB are investigated. The flexibility of the GMM provides the user with the possibility of choosing a version of the method (i.e. the moment pair that is used in fitting the distribution) which assigns larger weight to the larger elements of the sample, or another version which gives more weight to the smaller elements, depending on the problem at hand. A general formula for the asymptotic variance of the T-year event XT obtained by combining any two moments of GPDB is presented and applied. It is shown that the adequate choice of the order of these two moments to fit the distribution can lead, in some cases, to a considerable reduction in the variance of the estimator of XT, in comparison with estimation by the traditional method of moments (which uses moments of order one and two). The performance indices that are used to compare the different versions of the GMM are based on root mean square error, bias, and variance—both asymptotic and observed (based on simulation)—of GPDB quantiles and parameters. It is shown that moments of order (0),−1) and (2,−1) lead to the best results when the shape parameter a is positive (GPDB − 2), and the traditional method of moments can be considered as most efficient for negative values of α (GPDB −3). The GMM with moments of order 0 and one is shown to be rather consistent and moderately satisfactory for both GPDB-2 and GPDB-3.
This book, the fourth of a four volume set, contains five sections encompassing major aspects of regional flood frequency analysis. Each section starts usually with an invited state-of-the-art paper followed by contributed papers. The first section provides an assessment of regional flood frequency analysis. Methods for performing regional frequency analysis for ungaged watersheds are presented in Section 2. More discussion on regional frequency analysis is provided in Section 3. Selection and comparison of regional frequency methods are dealt with in Section 4; these are of great interest to the user. Increasing attention is being focused these days on paleohydrologic flood analysis. This topic is covered in Section 5.
Hydrology and more generally sciences involved in water resources
management, researches and technological or operational development face
a fundamental difficulty: the extreme variability of hydrological
fields. It clearly appears today that this variability is a function of
the observation scale and yield natural hazards such as floods or
droughts. The estimation of return periods for extreme precipitation and
flooding events requires a model of the natural (unperturbed)
statistical behaviour of the probability tails and the possible
clustering (including possible long-range dependencies) of the extremes.
Appropriate approaches for handling such non classical variability over
wide ranges of time and space scale do exist. They are based on a
fundamental property of the non-linear equations: scale invariance. Its
specific framework is that of multifractals. In this framework
hydrological variability builds up scale by scale leading to
non-classical statistics; this provides the key element needed to better
understand and predict floods. Scaling is a verifiable physical
principle which can be exploited to model hydrological processes and
estimate their statistics over wide ranges of space-time scales. We
first present the Multifractal Flood Frequency Analysis (MFFA) tool and
illustrate some results of its application to a large database (for
more than 16000 selected stations over USA and Canada). We then discuss
its efficiency by showing how the mean flow information - coupled with
universal multifractal parametrizations with power law tails - can be
used to estimate return times for extreme flood events.
Extreme environmental events, such as floods, droughts, rainstorms, and
high winds, have severe consequences for human society. Regional
frequency analysis helps to solve the problem of estimating the
frequency of these rare events at one site by using data from several
sites. This book is the first complete account of the L-moment approach
to regional frequency analysis. Regional Frequency Analysis
comprehensively describes the theoretical background to the subject, is
rich in practical advice for users, and contains detailed examples that
illustrate the approach. This book will be of great value to
hydrologists, atmospheric scientists and civil engineers, concerned with
environmental extremes.
Let (X, Y) be a bivariate random vector whose distribution function H(x, y) belongs to the class of bivariate extreme-value distributions. If F1 and F2 are the marginals of X and Y, then H(x, y) = C{F1(x),F2(y)}, where C is a bivariate extreme-value dependence function. This paper gives the joint distribution of the random variables Z = {log F1(X)}/{log F1(X)F2(Y)} and W = C{F1{(X),F2(Y)}. Using this distribution, an algorithm to generate random variables having bivariate extreme-value distribution is présentés. Furthermore, it is shown that for any bivariate extreme-value dependence function C, the distribution of the random variable W = C{F1(X),F2(Y)} belongs to a monoparametric family of distributions. This property is used to derive goodness-of-fit statistics to determine whether a copula belongs to an extreme-value family.Soit (X, Y) un couple de variables aléatoires dont la fonction de repartition H(x, y) est une loi de valeurs extrěmes bidimensionnelles. Si F1 et F2 sont les lois marginales de X et Y, on a H(x, y) = C{F1(x), F2(y)}, où C est une copule de valeurs extrěmes bidimensionnelles. Dans un premier temps, on détermine la distribution conjointe de Z = {log F1(X)}/{log F1(X)F2(Y)} et de W = C{ F1(X),F2(Y)}. Ce résultat a plusieurs applications intéressantes. II permet d'abord de construire un algorithme relativement simple pour simuler des lois de valeurs extrěmes bidimensionnelles. De plus, quelque soit la copule des valeurs extrěmes C, ce résultat montre également que la loi marginale de W = C{F1(X),F2(Y)} appartient à une famille de distributions indicée par un paramètre. Cette observation permet de construire un test d'ajustement pour déterminer si une copule C appartient à la famille des copules de valeurs extrěmes.
Extreme hydrological events are inevitable and stochastic in nature. Characterized by multiple properties, the multivariate
distribution is a better approach to represent this complex phenomenon than the univariate frequency analysis. However, it
requires considerably more data and more sophisticated mathematical analysis. Therefore, a bivariate distribution is the most
common method for modeling these extreme events. The return periods for a bivariate distribution can be defined using either
separate single random variables or two joint random variables. In the latter case, the return periods can be defined using
one random variable equaling or exceeding a certain magnitude and/or another random variable equaling or exceeding another
magnitude or the conditional return periods of one random variable given another random variable equaling or exceeding a certain
magnitude. In this study, the bivariate extreme value distribution with the Gumbel marginal distributions is used to model
extreme flood events characterized by flood volume and flood peak. The proposed methodology is applied to the recorded daily
streamflow from Ichu of the Pachang River located in Southern Taiwan. The results show a good agreement between the theoretical
models and observed flood data.
Within the context of a general bivariate distribution an intuitive method is presented in order to study the dependence structure of the two distributions. A set of points—level curve—which accumulate the same probability for a fixed quadrant is considered. This procedure provides four level curves which can be considered as the boundary of a generalization of the real interquantile interval. It is shown that the accumulated probability among the level curves depends on the dependence structure of the distribution function where the dependence structure is given by the notion of copula. Furthermore, the case when the marginal distributions are independent is investigated. This result is used to find out positive or negative dependence properties for the variables. Finally, a nonparametric test for independence with a local dependence meaning is performed and applied to different data sets.
Many hydrological engineering planning, design, and management problems require a detailed knowledge of flood event characteristics, such as flood peak, volume and duration. Flood frequency analysis often focuses on flood peak values, and hence, provides a limited assessment of flood events. This paper proposes the use of the Gumbel mixed model, the bivariate extreme value distribution model with Gumbel marginals, to analyze the joint probability distribution of correlated flood peaks and volumes, and the joint probability distribution of correlated flood volumes and durations. Based on the marginal distributions of these random variables, the joint distributions, the conditional probability functions, and the associated return periods are derived. The model is tested and validated using observed flood data from the Ashuapmushuan river basin in the province of Quebec, Canada. Results indicate that the model is suitable for representing the joint distributions of flood peaks and volumes, as well as flood volumes and durations.
The existing attitude that the criterion for choice of plotting position is arbitrary is rebuked and it is shown that a worthwhile criterion can be based on desired statistical properties of the plot, rather than on comparison of plotting positions with estimates of probability for individual sample values. These properties are that any quantile estimate made from the plot should be unbiased and should have smallest mean square error among all such estimates. This leads to specification of plotting position initially in terms of reduced variate rather than probability value. The unbiased plotting position is E(y(i)), the mean of the ith order statistic in samples from the reduced variate population, which differs from one distribution to another. A good approximation for each distribution is available in the probability domain. These take the general form with in the normal case and α = 0.44 in the extreme-value type-1 (EV1) and exponential cases. The Weibull formula, α = 0, is correct for the uniform distribution alone and is shown to be biased for other distributions. Hazen's formula shows up much better in terms of bias than many would expect. If a single simple distribution free formula were required then would be the best compromise. The plotting position postulates which have supported the Weibull formula for many years are examined and some are seen to be unreasonable in view of statistical facts.
Despite its potential advantages, canonical correlation analysis (CCA) has been little used in the fields of hydrology and water resources. In a regional flood frequency analysis, canonical correlations can be used to investigate the correlation structure between the two sets of variables represented by watershed characteristics and flood peaks. This paper presents a clear theoretical framework for the use of canonical correlations in regional flood frequency analysis. Some additional results are also presented for the case of gauged target-basins. The approach described in this paper allows one to carry out the determination of homogeneous hydrologic neighborhoods and identifies the variables to use during the step of regional estimation. A data set of 106 stations from the province of Ontario (Canada) is used to demonstrate the advantages of this method and investigate various aspects in relation with its robustness. Results indicate that the method is robust to such factors as the number of stations and the type of parametric distribution being used. Step-by-step algorithms for the delineation of hydrologic neighborhoods in the cases of gauged and ungauged basins are also presented.
For random vectors taking values in $\mathbb{R}^d$ we introduce a notion of multivariate quantiles defined in terms of a class of sets and study an associated process which we call the generalized quantile process. This process specializes to the well known univariate quantile process. We obtain functional central limit theorems for our generalized quantile process and show that both Gaussian and non-Gaussian limiting processes can arise. A number of interesting example are included.
This paper defines two distribution free goodness-of-fit test statistics for copulas. It states their asymptotic distributions under some composite parametric assumptions in an independent identically distributed framework. A short simulation study is provided to assess their power performances.
Let [alpha] [set membership, variant] (0, 1); the [alpha]-quantile of an -valued (k [greater-or-equal, slanted] 1) random variable X is a point minimizing the expectation E(||X - [theta]||p,[alpha] - ||X||p,[alpha]), where ||·||p,[alpha] is defined in terms of the lp-norm, 1 [less-than-or-equals, slant] p [less-than-or-equals, slant] [infinity], and [alpha] [set membership, variant] (0, 1). The properties of such an [alpha]-quantile extend those obtained previously for [alpha] = 0.5, i.e. for the median (see Kemperman, 1987). Computational aspects are also discussed.
Despite the absence of a natural ordering of Euclidean space for dimensions greater than one, the effort to define vector-valued quantile functions for multivariate distributions has generated several approaches. To support greater discrimination in comparing, selecting and using such functions, we introduce relevant criteria, including a notion of “medianoriented quantile function”. On this basis we compare recent quantile approaches and several multivariate versions of trimmed mean and interquartile range. We also discuss a univariate “generalized quantile” approach that enables particular features of multivariate distributions, for example scale and kurtosis, to be studied by two-dimensional plots. Methods based on statistical depth functions are found to be especially attractive for quantile-based multivariate inference.