Dam risk assessment based on univariate versus bivariate statistical approaches: a case study for Argentina

Abstract

Considering floods as multivariate events allows a better statistical representation of their complexity. In this work the relevance of multivariate analysis of floods for designing or assessing the safety of hydraulic structures is discussed. A flood event is characterized by its peak flow and volume. The dependence between the variables is modelled with a copula. One thousand random pairs of variables are transformed to hydrographs, applying the Beta distribution function. Synthetic floods are routed through a reservoir to assess the risk of overtopping a dam. The resulting maximum water levels are compared to estimations considering the peak flow and volume separately. The analysis is performed using daily flows observed in the River Agrio in Neuquén Province, Argentina, a catchment area of 7300 km². The bivariate approach results in higher maximum water level values. Therefore the multivariate approach should be preferred for the estimation of design variables.

Editor D. Koutsoyiannis; Associate editor S. Grimaldi     

Design event selection in bivariate hydrological frequency analysis

Abstract

In the bivariate analysis of hydrological events, such as rainfall storms or flood hydrographs, the choice of an appropriate return period for structure design leads to infinite combinations of values of the related random variables (e.g. peak and volume in the analysis of floods). These combinations are generally not equivalent, from a practical point of view. In this paper, a methodology is proposed to identify a subset of the critical combinations set that includes a fixed and arbitrarily chosen percentage in probability of the events, on the basis of their probability of occurrence. Therefore, several combinations can be selected within the subset, taking into account the specific characteristic of the design problem, in order to evaluate the effects of different hydrological loads on a structure. The proposed method is applicable to any type of bivariate distribution, thus providing a simple but effective rule to narrow down the infinite possible choices for the hydrological design variables. In order to illustrate how the proposed methodology can be easily used in practice, it is applied to a study case in the context of bivariate flood frequency analysis.

Editor Z.W. Kundzewicz; Associate editor Sheng Yue

Citation Volpi, E. and Fiori, A., 2012. Design event selection in bivariate hydrological frequency analysis. Hydrological Sciences Journal, 57 (8), 1506–1515.     

Probabilistic modelling of flood events using the entropy copula

The estimation of flood frequency is vital for the flood control strategies and hydraulic structure design. Generating synthetic flood events according to statistical properties of observations is one of plausible methods to analyze the flood frequency. Due to the statistical dependence among the flood event variables (i.e. the flood peak, volume and duration), a multidimensional joint probability estimation is required. Recently, the copula method is widely used for multivariable dependent structure construction, however, the copula family should be chosen before application and the choice process is sometimes rather subjective. The entropy copula, a new copula family, employed in this research proposed a way to avoid the relatively subjective process by combining the theories of copula and entropy. The analysis shows the effectiveness of the entropy copula for probabilistic modelling the flood events of two hydrological gauges, and a comparison of accuracy with the popular copulas was made. The Gibbs sampling technique was applied for trivariate flood events simulation in order to mitigate the calculation difficulties of extending to three dimension directly. The simulation results indicate that the entropy copula is a simple and effective copula family for trivariate flood simulation.     

Stochastic modelling of rainfall occurrences in continuous time

Abstract

The paper is concerned with the modelling of rainfall occurrence in continuous time. The Alternating Renewal Process is employed for the evaluation of probability distribution functions for total wet and dry periods over a homogeneous time interval (0, t). The derived general solution is simplified by assuming that the individual wet and dry intervals are random variables following an Erlang distribution, in particular an exponential distribution. Data on a continuous time scale from the Mikra Station in Greece are used to illustrate the proposed methodology.     

A new seasonal design flood method based on bivariate joint distribution of flood magnitude and date of occurrence

Abstract

Seasonal design floods which consider information on seasonal variation are very important for reservoir operation and management. The seasonal design flood method currently used in China is based on seasonal maximum (SM) samples and assumes that the seasonal design frequency is equal to the annual design frequency. Since the return period associated with annual maximum floods is taken as the standard in China, the current seasonal design flood cannot satisfy flood prevention standards. A new seasonal design flood method, which considers dates of flood occurrence and magnitudes of the peaks (runoff), was proposed and established based on copula function. The mixed von Mises distribution was selected as marginal distribution of flood occurrence dates. The Pearson Type III and exponential distributions were selected as the marginal distribution of flood magnitude for annual maximum flood series and peak-over-threshold samples, respectively. The proposed method was applied at the Geheyan Reservoir, China, and then compared with the currently used seasonal design flood methods. The case study results show that the proposed method can satisfy the flood prevention standard, and provide more information about the flood occurrence probabilities in each sub-season. The results of economic analysis show that the proposed design flood method can enhance the floodwater utilization rate and give economic benefits without lowering the annual flood protection standard.

Citation Chen, L., Guo, S. L., Yan, B. W., Liu, P. & Fang, B. (2010) A new seasonal design flood method based on bivariate joint distribution of flood magnitude and date of occurrence. Hydrol. Sci. J. 55(8), 1264–1280.     

Multivariate modeling of droughts using copulas and meta-heuristic methods

This study investigated the utility of two meta-heuristic algorithms to estimate parameters of copula models and for derivation of drought severity–duration–frequency (S–D–F) curves. Drought is a natural event, which has huge impact on both the society and the natural environment. Drought events are mainly characterized by their severity, duration and intensity. The study adopts standardized precipitation index for drought characterization, and copula method for multivariate risk analysis of droughts. For accurate estimation of copula model parameters, two meta-heuristic methods namely genetic algorithm and particle swarm optimization are applied. The proposed methodology is applied to a case study in Trans Pecos, an arid region in Texas, USA. First, drought severity and duration are separately modeled by various probability distribution functions and then the best fitted models are selected for copula modeling. For modeling the joint dependence of drought variables, different classes of copulas, namely, extreme value copulas, Plackett and Student’s t copulas are employed and their performance is evaluated using standard performance measures. It is found that for the study region, the Gumbel–Hougaard copula is the best fitted copula model as compared to the others and is used for the development of drought S–D–F curves. Results of the study suggest that the meta-heuristic methods have greater utility in copula-based multivariate risk assessment of droughts.     

Asymmetric copula in multivariate flood frequency analysis

The univariate flood frequency analysis is widely used in hydrological studies. Often only flood peak or flood volume is statistically analyzed. For a more complete analysis the three main characteristics of a flood event i.e. peak, volume and duration are required. To fully understand these variables and their relationships, a multivariate statistical approach is necessary. The main aim of this paper is to define the trivariate probability density and cumulative distribution functions. When the joint distribution is known, it is possible to define the bivariate distribution of volume and duration conditioned on the peak discharge. Consequently volume–duration pairs, statistically linked to peak values, become available. The authors build trivariate joint distribution of flood event variables using the fully nested or asymmetric Archimedean copula functions. They describe properties of this copula class and perform extensive simulations to highlight differences with the well-known symmetric Archimedean copulas. They apply asymmetric distributions to observed flood data and compare the results those obtained using distributions built with symmetric copula and the standard Gumbel Logistic model.     

Comparison of direct statistical and indirect statistical-deterministic frameworks in downscaling river low-flow indices

ABSTRACT

This work explores the ability of two methodologies in downscaling hydrological indices characterizing the low flow regime of three salmon rivers in Eastern Canada: Moisie, Romaine and Ouelle. The selected indices describe four aspects of the low flow regime of these rivers: amplitude, frequency, variability and timing. The first methodology (direct downscaling) ascertains a direct link between large-scale atmospheric variables (the predictors) and low flow indices (the predictands). The second (indirect downscaling) involves downscaling precipitation and air temperature (local climate variables) that are introduced into a hydrological model to simulate flows. Synthetic flow time series are subsequently used to calculate the low flow indices. The statistical models used for downscaling low flow hydrological indices and local climate variables are: Sparse Bayesian Learning and Multiple Linear Regression. The results showed that direct downscaling using Sparse Bayesian Learning surpassed the other approaches with respect to goodness of fit and generalization ability.

Editor D. Koutsoyiannis; Associate editor K. Hamed     

The impact of seasonal flood peak dependence on annual maxima design quantiles

If the maximum annual peak flow series are a mixture of summer and winter flows, a seasonal approach to flood frequency analysis is necessary. While considering seasonal maxima as mutually independent events, the annual maxima distribution is defined as the product of seasonal distributions. However, if the independency assumption does not hold, a bivariate approach with dependent margins should be applied, i.e. the copula approach. The impact of dependency on design quantiles is investigated here in the context of the Fréchet-Hoeffding inequality defining copula bounds and the definition of dependency. The results of the two approaches are compared using six catchments in the San River basin, where in four cases the dependency of seasonal maxima has been identified as positive significant and no strong dominance of any one season is observed. The product model leads to higher estimates of design quantiles than do models where the dependency is taken into account and, therefore, is safe.

EDITOR R. Woods ASSOCIATE EDITOR A. Fiori     

Exploration of optimal time steps for daily precipitation bias correction: a case study using a single grid of RCM on the River Exe in southwest England

ABSTRACT

Bias correction is a necessary post-processing procedure in order to use regional climate model (RCM)-simulated local climate variables as the input data for hydrological models due to systematic errors of RCMs. Most of the present bias-correction methods adjust statistical properties between observed and simulated data based on a predefined duration (e.g. a month or a season). However, there is a lack of analysis of the optimal period for bias correction. This study attempted to address the question whether there is an optimal number for bias-correction groups (i.e. optimal bias-correction period). To explore this we used a catchment in southwest England with the regional climate model HadRM3 precipitation data. The proposed methodology used only one grid of RCM in the Exe catchment, one emissions scenario (A1B) and one member (Q0) among 11 members of HadRM3. We tried 13 different bias-correction periods from 3-day to 360-day (i.e. the whole of one year) correction using the quantile mapping method. After the bias correction a low pass filter was used to remove the high frequencies (i.e. noise) followed by estimating Akaike’s information criterion. For the case study catchment with the regional climate model HadRM3 precipitation, the results showed that a bias-correction period of about 8 days is the best. We hope this preliminary study on the optimum number bias-correction period for daily RCM precipitation will stimulate more research to improve the methodology with different climatic conditions. Future efforts on several unsolved problems have been suggested, such as how strong the filter should be and the impact of the number of bias correction groups on river flow simulations.

Editor M.C. Acreman Associate editor S. Kanae     

Data‐based analysis of bivariate copula tail dependence for drought duration and severity

In recent decades, copula functions have been applied in bivariate drought duration and severity frequency analysis. Among several potential copulas, Clayton has been mostly used in drought analysis. In this research, we studied the influence of the tail shape of various copula functions (i.e. Gumbel, Frank, Clayton and Gaussian) on drought bivariate frequency analysis. The appropriateness of Clayton copula for the characterization of drought characteristics is also investigated. Drought data are extracted from standardized precipitation index time series for four stations in Canada (La Tuque and Grande Prairie) and Iran (Anzali and Zahedan). Both duration and severity data sets are positively skewed. Different marginal distributions were first fitted to drought duration and severity data. The gamma and exponential distributions were selected for drought duration and severity, respectively, according to the positive skewness and Kolmogorov–Smirnov test. The results of copula modelling show that the Clayton copula function is not an appropriate choice for the used data sets in the current study and does not give more drought risk information than an independent model for which the duration and severity dependence is not significant. The reason is that the dependence of two variables in the upper tail of Clayton copula is very weak and similar to the independent case, whereas the observed data in the transformed domain of cumulative density function show high association in the upper tail. Instead, the Frank and Gumbel copula functions show better performance than Clayton function for drought bivariate frequency analysis. Copyright © 2012 John Wiley & Sons, Ltd.     

Bivariate drought frequency curves and confidence intervals: a case study using monthly rainfall generation

Although water resources management practices recently use bivariate distribution functions to assess drought severity and its frequency, the lack of systematic measurements is the major hindrance in achieving quantitative results. This study aims to suggest a statistical scheme for the bivariate drought frequency analysis to provide comprehensive and consistent drought severities using observed rainfalls and their uncertainty using synthesized rainfalls. First, this study developed a multi-variate regression model to generate synthetic monthly rainfalls using climate variables as causative variables. The causative variables were generated to preserve their correlations using copula functions. This study then focused on constructing bivariate drought frequency curves using bivariate kernel functions and estimating their confidence intervals from 1,000 likely replica sets of drought frequency curves. The confidence intervals achieved in this study may be useful for making a comprehensive drought management plan through providing feasible ranges of drought severity.     

Copula joint function and its application in probability seismic hazard analysis

A mature mathematical technique called copula joint function is introduced in this paper, which is commonly used in the financial risk analysis to estimate uncertainty. The joint function is generalized to the n-dimensional Frank’s copula. In addition, we adopt two attenuation models proposed by YU and Boore et al, respectively, and construct a two-dimensional copula joint probabilistic function as an example to illustrate the uncertainty treatment at low probability. The results show that copula joint func...     

Multivariate probabilistic estimates of heat stress for rice across China

Heat stress, a major threat to rice (Oryza sativa) production across China, would tend to increase in frequency and intensity under warming climate. Unlike probabilistic analysis via a univariate character, heat stress events, characterized by three variables (i.e., duration, peak and accumulated detrimental intensity), were identified in the past years. Nine distribution functions (i.e., Beta, Cauchy, Logistic, Normal, Exponential, Gamma, Lognormal, Weibull and Generalized Extreme Value) were firstly introduced and compared to select the best-fit marginal distribution of univariable by using Kolmogorov–Smirnov test, and seven copula functions (i.e., Normal and t, Gumbel–Hougaard, Clayton, Frank, Joe, Ali-Mikhail-Haq) were applied in the distributions of multivariables by Akaike Information Criterion statistics. It was obvious that higher magnitude was in the eastern parts in the context of heat stress frequency and characteristic variables. Critical values of heat stress variables corresponding to the certain return periods (i.e. 5, 10, 20 and 50 years) successively expanded in intensity and spatial scope. Inter-correlations of heat stress variables were significant, enlightening the importance of copula in connecting heat stress variables. The combined and co-occurrence bivariate and trivariate return period at certain univariate value corresponding to the given return periods, were consistent at the spatial scale. Accordingly, it was highlighted that eastern parts, especially Zhejiang, central-northern Fujian and eastern Jiangxi, were prone to heat stress, as a consequence of not only univariate but also multivariate probabilistic analysis. These results can be helpful in quantitatively assessing the vulnerability of rice to heat stress and provide us desired information of prevention strategies for heat stress.     

Mapping groundwater quality variables using PCA and geostatistics: a case study of Bajo Andarax,southeastern Spain

Abstract

A case study is presented for the application of statistical and geostatistical methods to the problem of estimating groundwater quality variables. This methodology has been applied to the investigation of the detrital aquifer of the Bajo Andarax (Almería, Spain). The use of principal components analysis is proposed, as a first step, for identifying relevant types of groundwater and the processes that bring about a change in their quality. As a result of this application, three factors were obtained, which were used as three new variables (VI: sulphate influence; V2: thermal influence; and V3: marine influence). Analysis of their spatial distribution was performed through the calculation of experimental and theoretical variograms, which served as input for geostatistical modelling using ordinary block kriging. This analysis has allowed a probabilistic representation of the data to be obtained by mapping the three variables throughout the aquifer for each sampling point. In this way, one can evaluate the spatial and temporal variation of the principal physico-chemical processes associated with the three variables VI, V2 and V3 implicated in the groundwater quality of the detrital aquifer.     

Bivariate flood frequency analysis using the copula function: a case study of the Litija station on the Sava River

As an alternative to the commonly used univariate flood frequency analysis, copula frequency analysis can be used. In this study, 58 flood events at the Litija gauging station on the Sava River in Slovenia were analysed, selected based on annual maximum discharge values. Corresponding hydrograph volumes and durations were considered. Different bivariate copulas from three families were applied and compared using different statistical, graphical and upper tail dependence tests. The parameters of the copulas were estimated using the method of moments with the inversion of Kendall's tau. The Gumbel–Hougaard copula was selected as the most appropriate for the pair of peak discharge and hydrograph volume (Q‐V). The same copula was also selected for the pair hydrograph volume and duration (V‐D), and the Student‐t copula was selected for the pair of peak discharge and hydrograph duration (Q‐D). The differences among most of the applied copulas were not significant. Different primary, secondary and conditional return periods were calculated and compared, and some relationships among them were obtained. Copyright © 2014 John Wiley & Sons, Ltd.     

DISTRIBUTION OF DYNAMIC WATER LEVEL IN A SHALLOW LAKE

ABSTRACT

The shore of a large and shallow reservoir or lake may incur damages caused by high or low static water level, as well as from dynamic water level rises induced by wind; thus, the random variables representing, respectively, static water level and wind-induced rise must be added. The case study of Lake Balaton, Hungary, illustrates a proposed methodology to estimate, on the one hand, the distribution function of monthly static water level and on the other hand, that of monthly maximum rise caused by wind (seiche plus waves). We consider one section of lake shore which is homogeneous from the viewpoints of types of structure, dominant winds and corresponding values of fetch, so that a well-defined damage function can be used later for that section. A convolution of the two distribution functions is performed to yield the distribution function of monthly maximum water level. On the basis of existing data, normal distributions are suggested for either static or dynamic water levels. Extensions and transferability of the methodology are discussed.     

Multivariate drought characteristics using trivariate Gaussian and Student t copulas

This study aims to investigate the changing properties of drought events in Weihe River basin, China, by modeling the multivariate joint distribution of drought duration, severity and peak using trivariate Gaussian and Student t copulas. Monthly precipitations of Xi'an gauge are used to illustrate the meta‐elliptical copula‐based methodology for a single‐station application. Gaussian and Student t copulas are found to produce a better fit comparing with other six symmetrical and asymmetrical Archimedean copulas, and, checked by the goodness‐of‐fit tests based on a modified bootstrap version of Rosenblatt's transformation, both of them are acceptable to model the multivariate joint distribution of drought variables. Gaussian copula, the best fitting, is employed to construct the dependence structures of positively associated drought variables so as to obtain the multivariate joint and conditional probabilities of droughts. A Kendall's return period (KRP) introduced by Salvadori and De Michele (2010) is then adopted to assess the multivariate recurrent properties of drought events, and its spatial distributions indicate that prolonged droughts are likely to break out with rather short recurrence intervals in the whole region, while drought status in the southeast seems to be severer than the northwest. The study is of some merits in terms of multivariate drought modeling using a preferable copula‐based method, the results of which could serve as a reference for regional drought defense and water resources management. Copyright © 2011 John Wiley & Sons, Ltd.     

Utility of high-resolution climate model simulations for water resources prediction over the Korean Peninsula: a sensitivity study / Utilité de simulations de modèle climatique à haute résolution pour la prévision des ressources en eau dans la Péninsule Coréenne: étude de sensibilité

Abstract

Abstract The utility of simulations of Global Climate Models (GCMs) for regional water resources prediction and management on the Korean Peninsula was assessed by a probabilistic measure. Global Climate Model simulations of an indicator variable (e.g. surface precipitation or temperature) were used for discriminating high vs low regional observations of a target variable (e.g. watershed precipitation or reservoir inflow). The formulation uses the significance probability of the Kolmogorov-Smirnov test for detecting differences between two distributions. High resolution Atmospheric Model Intercomparison Project-II (AMIP-II) type GCM simulations performed by the European Centre for Medium-Range Weather Forecasts (ECMWF) and AMIP-I type GCM simulations performed by the Korean Meteorological Research Institute (METRI) were used to obtain information for the indicator variables. Observed mean areal precipitation and temperature, and watershed-outlet discharge values for seven major river basins in Korea were used as the target variables. The results suggest that the use of the climate model nodal output from both climate models in the vicinity of the target basin with monthly resolution will be beneficial for water resources planning and management analysis that depends on watershed mean areal precipitation and temperature, and outlet discharge.     

Joint probability analysis of water and sediment and predicting sediment load based on copula function

The Jinsha River comprises the upper reaches of the Yangtze River, which is the river section with the highest sediment content. Monitoring of sediment transport in the Jinsha River is done to the guarantee for the normal operation of the Three Gorges Reservoir. In the current study, a copula function was used to do a joint probability analysis of the water and sediment in the Jinsha River Basin (JRB), further a sediment load prediction model based on the copula function also was constructed. The results show that the average annual flow from 2001 to 2018 at the outlet of the Jinsha River (Yibin station) is about 60.43 billion m³, and the average annual sediment load is about 58.82 million t. The linear correlation coefficient between annual flow and annual sediment load is 0.28. The best marginal distribution for annual flow and sediment load is Pearson Type Three (PE3) and Generalized Normal (GNO), respectively, and the best fit for the combined distribution of the two variables is the Frank copula function. The synchronous probability of water and sediment occurrence is 0.459, and the asynchronous probability is 0.541. Based on the copula prediction model, the sediment load can be effectively simulated, and the correlation coefficient between the simulated sequence and the measured sequence reached 0.93. The current study provides important significance for the analysis of water and sediment in the JRB, which is beneficial to the management of Three Gorges Reservoir sediment discharge in the upstream and downstream.