Bivariate frequency analysis: discussion of some useful concepts in hydrological application

Basic concepts such as conditional probability distributions, conditional return periods, and joint return periods are important to understand and to interpret multivariate hydrological events such as floods and storms. However, these concepts are not well documented in the open literature. This paper assembles and clarifies these concepts, and illustrates their practical utility. Relationships between joint return periods and univariate return periods are also derived. These concepts and relationships are demonstrated by applying a bivariate extreme value distribution to represent the joint distribution of flood peak and volume from an actual basin. Copyright © 2002 John Wiley & Sons, Ltd.     

The bivariate lognormal distribution to model a multivariate flood episode

Complex hydrological events such as floods always appear to be multivariate events that are characterized by a few correlated variables. A complete understanding of these events needs to investigate joint probabilistic behaviours of these correlated variables. The lognormal distribution is one of frequently selected candidates for flood‐frequency analysis. The multivariate lognormal distribution will serve as an important tool for analysing a multivariate flood episode. This article presents a procedure for using the bivariate lognormal distribution to describe the joint distributions of correlated flood peaks and volumes, and correlated flood volumes and durations. Joint distributions, conditional distributions, and the associated return periods of these random variables can be readily derived from their marginal distributions. The approach is verified using observed streamflow data from the Nord river basin, located in the Province of Quebec, Canada. The theoretical distributions show a good fit to observed ones. Copyright © 2000 John Wiley & Sons, Ltd.     

A bivariate gamma distribution for use in multivariate flood frequency analysis

A gamma distribution is one of the most frequently selected distribution types for hydrological frequency analysis. The bivariate gamma distribution with gamma marginals may be useful for analysing multivariate hydrological events. This study investigates the applicability of a bivariate gamma model with five parameters for describing the joint probability behavior of multivariate flood events. The parameters are proposed to be estimated from the marginal distributions by the method of moments. The joint distribution, the conditional distribution, and the associated return periods are derived from marginals. The usefulness of the model is demonstrated by representing the joint probabilistic behaviour between correlated flood peak and flood volume and between correlated flood volume and flood duration in the Madawask River basin in the province of Quebec, Canada. Copyright © 2001 John Wiley & Sons, Ltd.     

Uncertainty analysis in statistical modeling of extreme hydrological events

With the increase of both magnitude and frequency of hydrological extreme events such as drought and flooding, the significance of adequately modeling hydrological extreme events is fully recognized. Estimation of extreme rainfall/flood for various return periods is of prime importance for hydrological design or risk assessment. However, due to knowledge and data limitation, uncertainty involved in extrapolating beyond available data is huge. In this paper, different sources of uncertainty in statistical modeling of extreme hydrological events are studied in a systematic way. This is done by focusing on several key uncertainty sources using three different case studies. The chosen case studies highlight a number of projects where there have been questions regarding the uncertainty in extreme rainfall/flood estimation. The results show that the uncertainty originated from the methodology is the largest and could be >40% for a return period of 200 years, while the uncertainty caused by ignoring the dependence among multiple hydrological variables seems the smallest. In the end, it is highly recommended that uncertainty in modeling extreme hydrological events be fully recognized and incorporated into a formal hydrological extreme analysis.     

Trivariate generalized extreme value distribution in flood frequency analysis

Abstract

The multivariate extension of the logistic model with generalized extreme value (GEV) marginals is applied to provide a regional at-site flood estimate. The maximum likelihood estimators of the parameters were obtained numerically by using a multivariable constrained optimization algorithm. The asymptotic results were checked by distribution sampling techniques in order to establish whether or not those results can be utilized for small samples. A region in northern Mexico with 21 gauging stations was selected to apply the model. Results were compared with those obtained by the most popular univariate distributions, the bivariate approach of the logistic model and three regional methods: station-year, index flood and L-moments. These show that there is a reduction in the standard error of fit when estimating the parameters of the marginal distribution with the trivariate distribution instead of its univariate and bivariate counterpart, and differences between at-site and regional at-site design events can be significant as return period increases.     

A bivariate pareto model for drought

Univariate Pareto distributions have been so widely used in hydrology. It seems however that bivariate or multivariate Pareto distributions have not yet found applications in hydrology, especially with respect to drought. In this note, a drought application is described by assuming a bivariate Pareto model for the joint distribution drought durations and drought severity in the State of Nebraska. Based on this model, exact distributions are derived for the inter arrival time, magnitude and the proportion of droughts. Estimates of 2, 5, 10, 20, 50 and 100 year return periods are derived for the three variables, drought duration, drought severity and the pairwise combinations: (drought duration, drought severity), (inter arrival time of drought, proportion of drought) and (drought duration, drought magnitude). These return period estimates could have an important role in hydrology, for example, with respect to measures of vegetation water stress for plants in water-controlled ecosystems.     

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.     

On the distribution of flood volume in partial duration series analysis of flood phenomena

A methodology based on the theory of stochastic processes is applied to the analysis of floods. The approach will be based on some results of the theory of extreme values over a threshold. In this paper, we focus on the estimation of the distribution of the flood volume in partial duration series analysis of flood phenomena, by using a bivariate exponential distribution of discharge exceedances and durations over a base level.     

On the distribution of flood volume in partial duration series analysis of flood phenomena

A methodology based on the theory of stochastic processes is applied to the analysis of floods. The approach will be based on some results of the theory of extreme values over a threshold. In this paper, we focus on the estimation of the distribution of the flood volume in partial duration series analysis of flood phenomena, by using a bivariate exponential distribution of discharge exceedances and durations over a base level.     

FLOOD EXPOSURE RECOMMENDED AS A PARAMETER FOR DESCRIBING THE FATIGUE LOADING ON FLOOD CONTROL STRUCTURES

Abstract

Highwater stages of frequently very long duration which commonly develop on plain-land watercourses having flat slopes, constitute a fatigue loading on the flood levees built along such streams. In order to describe by a single parameter the magnitude of this fatigue load, the concept of flood exposure has been introduced and defined as the area under the flood hydrograph exceeding a certain stage (the toe of the levee). Consequently both stage and duration of highwater are taken into consideration.

Positive relationships have been found to exist between the magnitude of flood exposure and adverse phenomena associated with highwaters (groundwater emergence, underseepage, leakage, boil formation, saturation slumping and wave action). The magnitude of the labour force, fleet of mechanical equipment and materials used in flood fighting, consequently the costs thereof depend to a significant extent on the magnitude of flood exposure, which can thus be used conveniently for economic analysis as well.

Mathematical statistical analysis has shown the logarithmic normal distribution to fit best to the empirical distribution of flood exposure.

The full set of data was found to be homogeneous in the gaging sections examined on the Danube and Tisza Rivers. Sets of data have been grouped according to the dates of major interference into the life of the watercourse. It was found further that in the sections considered the duration of highwaters could also be regarded as a homogeneous random variable.     

A distribution-free ordinal classification of floods based on moments

ABSTRACT

Classification of floods is often based on return periods of their peaks estimated from probability distributions and hence depends on assumptions. The choice of an appropriate distribution function and parameter estimation are often connected with high uncertainties. In addition, limited length of data series and the stochastic characteristic of the occurrence of extreme events add further uncertainty. Here, a distribution-free classification approach is proposed based on statistical moments. By using robust estimators the sampling effects are reduced and time series of different lengths can be analysed together. With a developed optimization procedure, locally and regionally consistent flood categories can be defined. In application, it is shown that the resulting flood categories can be used to assess the spatial extent of extreme floods and their coincidences. Moreover, groups of gauges, where simultaneous events belong to the same classes, are indicators for homogeneous groups of gauges in regionalization.     

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.     

A Bayesian analysis of Generalized Pareto Distribution of runoff minima

Global climate change models have predicted the intensification of extreme events, and these predictions are already occurring. For disaster management and adaptation of extreme events, it is essential to improve the accuracy of extreme value statistical models. In this study, Bayes' Theorem is introduced to estimate parameters in Generalized Pareto Distribution (GPD), and then the GPD is applied to simulate the distribution of minimum monthly runoff during dry periods in mountain areas of the Ürümqi River, Northwest China. Bayes' Theorem treats parameters as random variables and provides a robust way to convert the prior distribution of parameters into a posterior distribution. Statistical inferences based on posterior distribution can provide a more comprehensive representation of the parameters. An improved Markov Chain Monte Carlo (MCMC) method, which can solve high‐dimensional integral computation in the Bayes equation, is used to generate parameter simulations from the posterior distribution. Model diagnosis plots are made to guarantee the fitted GPD is appropriate. Then based on the GPD with Bayesian parameter estimates, monthly runoff minima corresponding to different return periods can be calculated. The results show that the improved MCMC method is able to make Markov chains converge faster. The monthly runoff minima corresponding to 10a, 25a, 50a and 100a return periods are 0.60 m³/s, 0.44 m³/s, 0.32 m³/s and 0.20 m³/s respectively. The lower boundary of 95% confidence interval of 100a return level is below zero, which implies that the Ürümqi River is likely to cease to flow when 100a return level appears in dry periods. Copyright © 2015 John Wiley & Sons, Ltd.     

Bivariate frequency analysis of flood and extreme precipitation under changing environment: case study in catchments of the Loess Plateau,China

Floods have changed in a complex manner, triggered by the changing environment (i.e., intensified human activities and global warming). Hence, for better flood control and mitigation in the future, bivariate frequency analysis of flood and extreme precipitation events is of great necessity to be performed within the context of changing environment. Given this, in this paper, the Pettitt test and wavelet coherence transform analysis are used in combination to identify the period with transformed flood-generating mechanism. Subsequently, the primary and secondary return periods of annual maximum flood (AMF) discharge and extreme precipitation (Pr) during the identified period are derived based on the copula. Meanwhile, the conditional probability of occurring different flood discharge magnitudes under various extreme precipitation scenarios are estimated using the joint dependence structure between AMF and Pr. Moreover, Monte Carlo-based algorithm is performed to evaluate the uncertainties of the above copula-based analyses robustly. Two catchments located on the Loess plateau are selected as study regions, which are Weihe River Basin (WRB) and Jinghe River Basin (JRB). Results indicate that: (1) the 1994–2014 and 1981–2014 are identified as periods with transformed flood-generating mechanism in the WRB and JRB, respectively; (2) the primary and secondary return periods for AMF and Pr are examined. Furthermore, chance of occurring different AMF under varying Pr scenarios also be elucidated according to the joint distribution of AMF and Pr. Despite these, one thing to notice is that the associate uncertainties are considerable, thus greatly challenges measures of future flood mitigation. Results of this study offer technical reference for copula-based frequency analysis under changing environment at regional and global scales.     

Wind storm risk management: sensitivity of return period calculations and spread on the territory

Modeling and forecasting damage from wind storms is a major issue for insurance companies. In this article, we focus on the sensitivity of estimations of return periods for extreme events with respect to modeling assumptions and the type of input data. Numerous variables play a role: the quality of data concerning the location of insured buildings and weather report homogeneity, missing updates for correcting non-stationarities concerning the insurance portfolio history, ground roughness or climate change, the evolution of the model after an unprecedented event such as the Lothar storm observed in 1999 in Europe, temporal aggregation of daily events over several days, where events could span over several days up to one week, and storm trajectories, which could change due to global warming or sweep larger areas. Our work explores three important aspects. First, we highlight the geographic heterogeneity of the spatial distribution of wind speeds and the resulting damages. Therefore, we propose to partition the French territory into 6 relatively homogeneous storm zones, based on the dependence among observed wind speeds and geographic distance. Second, we extend a storm index—defined in Mornet et al. (Risk Anal 35:2029–2056, 2015)—to take into account geographic heterogeneity, and we analyze its tail behavior to show the difficulties met to obtain reliable results on extreme events. Third, we explore the calculation of Solvency Capital Requirements based on a model that we propose for the annual claim amount distribution. The purpose of our analysis is to quantify and to point out the high level of uncertainty in the computation of return periods and of other quantities strongly influenced by extreme events.     

New estimates of future changes in extreme rainfall across the UK using regional climate model integrations. 2. Future estimates and use in impact studies

Under enhanced greenhouse conditions, climate models suggest an increase in rainfall intensities in the northern Hemisphere. Major flood events in the UK during autumn 2000 and central Europe in August 2002, have focussed attention on the dramatic impacts these changes may have on many sectors of society. In the companion paper [Fowler et al., J. Hydrol. (2004) this issue], we suggested that the HadRM3H model may be used with some confidence to estimate extreme rainfall distributions, showing good predictive skill in estimating statistical properties of extreme rainfall during the baseline period, 1961–1990. In this study, we use results from the future integration of HadRM3H (following the IPCC SRES scenario A2 for 2070–2100) to assess possible changes in extreme rainfall across the UK using two methods: regional frequency analysis and individual grid box analysis. Results indicate that for short duration events (1–2 days), event magnitude at a given return period will increase by 10% across the UK. For longer duration events (5–10 days), event magnitudes at given return periods show large increases in Scotland (up to +30%), with greater relative change at higher return periods (25–50 years). In the rest of the UK, there are small increases in the magnitude of more frequent events (up to +10%) but reductions at higher return periods (up to −20%). These results provide information to alter design storm depths to examine climate change impacts on various structures. The uncertainty bounds of the estimated changes and a ‘scaling’ methodology are additionally detailed. This allows the estimation of changes for the 2020s, 2050s and 2080s, and gives some confidence in the use of these estimates in impact studies.     

Flood coincidence analysis of Poyang Lake and Yangtze River: risk and influencing factors

We build copula function-based joint distribution models for the annual maximum flood peaks of the Yangtze River and Poyang Lake, to analyze the coincidence probabilities, using scenarios that combine with the impoundment of three Gorges, define influencing indexes and relative contribution rates on flood coincidence at varying frequencies. The study shows the probabilities for coincidence of floods with 1000, 100, and 10-year return periods in both Yangtze main stem and Poyang Lake are respectively 0.02, 0.19 and 2.87%, with higher coincidence probabilities for shorter return periods; when 1000-year flood occurs in the Yangtze, the probabilities for Poyang Lake to encounter flood of the 1000, 100, or 10-year magnitude are higher than 16.08, 42.48 or 74.77% respectively; Poyang–Yangtze flood coincidence is affected by operation of the hydraulic engineering. The lowering of flood peaks caused by the Three Gorges impoundment and regulation of the lake have respectively reduced the probabilities of Poyang–Yangtze flood coincidence by about 7.0 and 1.97%, with average relative contribution rates ? 33.82 and ? 17.1%; influenced by hydrological projects in Poyang basin, variations in Poyang’s inflow flood have displayed an average contribution rate of 20.4% for the negative effect on extreme (P < 5% or P > 90%) flood coincidence, while having a positive contribution rate of 38.2% on floods of other return periods. The results can help increase our understanding of flood coincidence, and support flood control efforts in Poyang Lake; its analytical approach may also be useful to other applications of copula functions.     

The generalized Pareto sum

The generalized Pareto distribution has received much popularity as models for extreme events in hydrological sciences. In this note, the important problem of the sum of two independent generalized Pareto random variables is considered. Exact analytical expressions for the probability distribution of the sum are derived and a detailed application to drought data from Nebraska is provided. Copyright © 2007 John Wiley & Sons, Ltd.     

Theoretical discussion and Monte-Carlo simulations for a Negative Binomial process paradox

The idea of an over-threshold sampling is to retain all the events of a time-series exceeding a given threshold. The probabilistic analysis implies estimating two statistical models, one describing the occurrence of the events (date of the events), the other describing their magnitude (value of the local maximum). These two models are then combined to obtain the distribution of the annual maxima. A well-known result of a Poisson process is that waiting time, defined as the duration between two successive events exceeding the threshold, is exponentially distributed. The assertion that the waiting time of a Negative Binomial process is also exponentially distributed seems to be in obvious contradiction with the Poisson process properties. A theoretical discussion and Monte-Carlo simulations are presented to solve this apparent paradox.