Statistics of extremes and estimation of extreme rainfall: I. Theoretical investigation / Statistiques de valeurs extrêmes et estimation de précipitations extrêmes: I. Recherche théorique

Abstract

Abstract The Gumbel distribution has been the prevailing model for quantifying risk associated with extreme rainfall. Several arguments including theoretical reasoning and empirical evidence are supposed to support the appropriateness of the Gumbel distribution. These arguments are examined thoroughly in this work and are put into question. Specifically, theoretical analyses show that the Gumbel distribution is quite unlikely to apply to hydrological extremes and its application may misjudge the risk, as it underestimates seriously the largest extreme rainfall amounts. Besides, it is shown that hydrological records of typical length (some decades) may display a distorted picture of the actual distribution, suggesting that the Gumbel distribution is an appropriate model for rainfall extremes while it is not. In addition, it is shown that the extreme value distribution of type II (EV2) is a more consistent alternative. Based on the theoretical analysis, in the second part of this study an extensive empirical investigation is performed using a collection of 169 of the longest available rainfall records worldwide, each having 100–154 years of data. This verifies the inappropriateness of the Gumbel distribution and the appropriateness of EV2 distribution for rainfall extremes.     

Estimation of tail dependence coefficient in rainfall accumulation fields

Extreme rainfall events are of particular importance due to their severe impacts on the economy, the environment and the society. Characterization and quantification of extremes and their spatial dependence structure may lead to a better understanding of extreme events. An important concept in statistical modeling is the tail dependence coefficient (TDC) that describes the degree of association between concurrent rainfall extremes at different locations. Accurate knowledge of the spatial characteristics of the TDC can help improve on the existing models of the occurrence probability of extreme storms. In this study, efficient estimation of the TDC in rainfall is investigated using a dense network of rain gauges located in south Louisiana, USA. The inter-gauge distances in this network range from about 1 km to 9 km. Four different nonparametric TDC estimators are implemented on samples of the rain gauge data and their advantages and disadvantages are discussed. Three averaging time-scales are considered: 1 h, 2 h and 3 h. The results indicate that a significant tail dependency may exist that cannot be ignored for realistic modeling of multivariate rainfall fields. Presence of a strong dependence among extremes contradicts with the assumption of joint normality, commonly used in hydrologic applications.     

Statistics of extremes in hydrology

The statistics of extremes have played an important role in engineering practice for water resources design and management. How recent developments in the statistical theory of extreme values can be applied to improve the rigor of hydrologic applications and to make such analyses more physically meaningful is the central theme of this paper. Such methodological developments primarily relate to maximum likelihood estimation in the presence of covariates, in combination with either the block maxima or peaks over threshold approaches. Topics that are treated include trends in hydrologic extremes, with the anticipated intensification of the hydrologic cycle as part of global climate change. In an attempt to link downscaling (i.e., relating large-scale atmosphere–ocean circulation to smaller-scale hydrologic variables) with the statistics of extremes, statistical downscaling of hydrologic extremes is considered. Future challenges are reviewed, such as the development of more rigorous statistical methodology for regional analysis of extremes, as well as the extension of Bayesian methods to more fully quantify uncertainty in extremal estimation. Examples include precipitation and streamflow extremes, as well as economic damage associated with such extreme events, with consideration of trends and dependence on patterns in atmosphere–ocean circulation (e.g., El Niño phenomenon).     

Return period of bivariate distributed extreme hydrological events

 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. The author wishes to thank the two anonymous reviewers for their constructive comments that improving the quality of this work.     

Statistics of extremes and estimation of extreme rainfall: II. Empirical investigation of long rainfall records / Statistiques de valeurs extrêmes et estimation de précipitations extrêmes: II. Recherche empirique sur de longues séries de précipitations

Abstract

Abstract In the first part of this study, theoretical analyses showed that the Gumbel distribution is quite unlikely to apply to hydrological extremes and that the extreme value distribution of type II (EV2) is a more consistent choice. Based on these theoretical analyses, an extensive empirical investigation is performed using a collection of 169 of the longest available rainfall records worldwide, each having 100–154 years of data. This verifies the theoretical results. In addition, it shows that the shape parameter of the EV2 distribution is constant for all examined geographical zones (Europe and North America), with value κ = 0.15. This simplifies the fitting and the general mathematical handling of the distribution, which become as simple as those of the Gumbel distribution.     

CHOSEN: A synthesis of hydrometeorological data from intensively monitored catchments and comparative analysis of hydrologic extremes

Comparative hydrology has been hampered by limited availability of geographically extensive, intercompatible monitoring data on comprehensive water balance stores and fluxes. These limitations have, for example, restricted comprehensive assessment of multiple dimensions of wetting and drying related to climate change and hampered understanding of why widespread changes in precipitation extremes are uncorrelated with changes in streamflow extremes. Here, we address this knowledge gap and underlying data gap by developing a new data synthesis product and using that product to detect trends in the frequencies and magnitudes of a comprehensive set of hydroclimatic and hydrologic extremes. CHOSEN (Comprehensive Hydrologic Observatory Sensor Network) is a database of streamflow, soil moisture, and other hydroclimatic and hydrologic variables from 30 study areas across the United States. An accompanying data pipeline provides a reproducible, semi-automated approach for assimilating data from multiple sources, performing quality assurance and control, gap-filling and writing to a standard format. Based on the analysis of extreme events in the CHOSEN dataset, we detected hotspots, characterized by unusually large proportions of monitored variables exhibiting trends, in the Pacific Northwest, New England, Florida and Alaska. Extreme streamflow wetting and drying trends exhibited regional coherence. Drying trends in the Pacific Northwest and Southeast were often associated with trends in soil moisture and precipitation (Pacific Northwest) and evapotranspiration-related variables (Southeast). In contrast, wetting trends in the upper Midwest and the Rocky Mountains showed few univariate associations with other hydroclimatic extremes, but their latitudes and elevations suggested the importance of changing snowmelt characteristics. On the whole, observed trends are incompatible with a ‘drying-in-dry, wetting-in-wet’ paradigm for climate-induced hydrologic changes over land. Our analysis underscores the need for more extensive, longer-term observational data for soil moisture, snow and evapotranspiration.     

An exploratory study of spatial annual maximum of monthly precipitation in the northern region of Portugal

Adequately analyzing and modeling the extreme rainfall events is of great importance because of the effects that their magnitude and frequency can have on human life, agricultural productivity and economic aspects, among others. A single extreme event may affect several locations, and their spatial dependence has to be appropriately taken into account. Classical geostatistics is a well-developed field for dealing with location referenced data, but it is largely based on Gaussian processes and distributions, that are not appropriate for extremes. In this paper, an exploratory study of the annual maximum of monthly precipitation recorded in the northern area of Portugal from 1941 to 2006 at 32 locations is performed. The aim of this paper is to apply max-stable processes, a natural extension of multivariate extremes to the spatial set-up, to briefly describe the models considered and to estimate the required parameters to simulate prediction maps.     

A comparative study of the adaptive choice of thresholds in extreme hydrologic events

In the hydrologic analysis of extreme events such as precipitation or floods, the data can generally be divided into two types: partial duration series and annual maximum series. Partial duration series analysis is a robust method to analyze hydrologic extremes, but the adaptive choice of an optimal threshold is challenging. The main goal of this paper was to determine the best method for choosing optimal thresholds. Ten semi-parametric tail index estimators were applied to find the optimal threshold of a 24-h duration precipitation period using data from the Korean Meteorological Administration. The mean square errors of the 10 estimators were calculated to determine the optimal threshold using a semi-parametric bootstrap method. A modified generalized Jackknife estimator determined the best performance in this study among the 10 estimators evaluated with regard to estimating the mean square error of the shape estimator for the generalized Pareto distribution.     

Statistical analysis of extreme events in a non-stationary context via a Bayesian framework: case study with peak-over-threshold data

Statistical analysis of extremes currently assumes that data arise from a stationary process, although such an hypothesis is not easily assessable and should therefore be considered as an uncertainty. The aim of this paper is to describe a Bayesian framework for this purpose, considering several probabilistic models (stationary, step-change and linear trend models) and four extreme values distributions (exponential, generalized Pareto, Gumbel and GEV). Prior distributions are specified by using regional prior knowledge about quantiles. Posterior distributions are used to estimate parameters, quantify the probability of models and derive a realistic frequency analysis, which takes into account estimation, distribution and stationarity uncertainties. MCMC methods are needed for this purpose, and are described in the article. Finally, an application to a POT discharge series is presented, with an analysis of both occurrence process and peak distribution.     

Beta-�� distribution and its application to hydrologic events

The beta-κ distribution is a distinct case of the generalized beta distribution of the second kind. In previous studies, beta-p and beta-κ distributions have played important roles in representing extreme events, and thus, the present paper uses the beta-κ distribution. Further, this paper uses the method of moments and the method of L-moments to estimate the parameters from the beta-κ distribution, and to demonstrate the performance of the proposed model, the paper presents a simulation study using three estimation methods (including the maximum likelihood estimation method) and beta-κ and non beta-κ samples. In addition, this paper evaluates the performance of the beta-κ distribution by employing two widely used extreme value distributions (i.e., the GEV and Gumbel distributions) and two sets of actual data on extreme events.     

A hierarchical Bayesian approach for the analysis of climate change impact on runoff extremes

In this study the impact of climate change on runoff extremes is investigated over the Pacific Northwest (PNW). This paper aims to address the question of how the runoff extremes change in the future compared to the historical time period, investigate the different behaviors of the regional climate models (RCMs) regarding the runoff extremes and assess the seasonal variations of runoff extremes. Hydrologic modeling is performed by the variable infiltration capacity (VIC) model at a 1/8° resolution and the model is driven by climate scenarios provided by the North American Regional Climate Change Assessment Program (NARCCAP) including nine regional climate model (RCM) simulations. Analysis is performed for both the historical (1971–2000) and future (2041–2070) time periods. Downscaling of the climate variables including precipitation, maximum and minimum temperature and wind speed is done using the quantile‐mapping (QM) approach. A spatial hierarchical Bayesian model is then developed to analyse the annual maximum runoff in different seasons for both historical and future time periods. The estimated spatial changes in extreme runoffs over the future period vary depending on the RCM driving the hydrologic model. The hierarchical Bayesian model characterizes the spatial variations in the marginal distributions of the General Extreme Value (GEV) parameters and the corresponding 100‐year return level runoffs. Results show an increase in the 100‐year return level runoffs for most regions in particular over the high elevation areas during winter. The Canadian portions of the study region reflect higher increases during spring. However, reduction of extreme events in several regions is projected during summer. Copyright © 2013 John Wiley & Sons, Ltd.     

Uncertainty based modeling of rainfall-runoff: Combined differential evolution adaptive Metropolis (DREAM) and K-means clustering

Simulation of rainfall-runoff process in urban areas is of great importance considering the consequences and damages of extreme runoff events and floods. The first issue in flood hazard analysis is rainfall simulation. Large scale climate signals have been proved to be effective in rainfall simulation and prediction. In this study, an integrated scheme is developed for rainfall-runoff modeling considering different sources of uncertainty. This scheme includes three main steps of rainfall forecasting, rainfall-runoff simulation and future runoff prediction. In the first step, data driven models are developed and used to forecast rainfall using large scale climate signals as rainfall predictors. Due to high effect of different sources of uncertainty on the output of hydrologic models, in the second step uncertainty associated with input data, model parameters and model structure is incorporated in rainfall-runoff modeling and simulation. Three rainfall-runoff simulation models are developed for consideration of model conceptual (structural) uncertainty in real time runoff forecasting. To analyze the uncertainty of the model structure, streamflows generated by alternative rainfall-runoff models are combined, through developing a weighting method based on K-means clustering. Model parameters and input uncertainty are investigated using an adaptive Markov Chain Monte Carlo method. Finally, calibrated rainfall-runoff models are driven using the forecasted rainfall to predict future runoff for the watershed. The proposed scheme is employed in the case study of the Bronx River watershed, New York City. Results of uncertainty analysis of rainfall-runoff modeling reveal that simultaneous estimation of model parameters and input uncertainty significantly changes the probability distribution of the model parameters. It is also observed that by combining the outputs of the hydrological models using the proposed clustering scheme, the accuracy of runoff simulation in the watershed is remarkably improved up to 50% in comparison to the simulations by the individual models. Results indicate that the developed methodology not only provides reliable tools for rainfall and runoff modeling, but also adequate time for incorporating required mitigation measures in dealing with potentially extreme runoff events and flood hazard. Results of this study can be used in identification of the main factors affecting flood hazard analysis.     

Modelling bivariate extreme precipitation distribution for data‐scarce regions using Gumbel–Hougaard copula with maximum entropy estimation

A new method of parameter estimation in data scarce regions is valuable for bivariate hydrological extreme frequency analysis. This paper proposes a new method of parameter estimation (maximum entropy estimation, MEE) for both Gumbel and Gumbel–Hougaard copula in situations when insufficient data are available. MEE requires only the lower and upper bounds of two hydrological variables. To test our new method, two experiments to model the joint distribution of the maximum daily precipitation at two pairs of stations on the tributaries of Heihe and Jinghe River, respectively, were performed and compared with the method of moments, correlation index estimation, and maximum likelihood estimation, which require a large amount of data. Both experiments show that for the Ye Niugou and Qilian stations, the performance of MEE is nearly identical to those of the conventional methods. For the Xifeng and Huanxian stations, MEE can capture information indicating that the maximum daily precipitation at the Xifeng and Huanxian stations has an upper tail dependence, whereas the results generated by correlation index estimation and maximum likelihood estimation are unreasonable. Moreover, MEE is proved to be generally reliable and robust by many simulations under three different situations. The Gumbel–Hougaard copula with MEE can also be applied to the bivariate frequency analysis of other extreme events in data‐scarce regions.     

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.     

Climate change in Ireland from precipitation and streamflow observations

On the basis of General Circulation Model (GCM) experiments with increased CO₂, many parts of the northern latitudes including western Europe, are expected to have enhanced hydrologic cycles. Using observations of precipitation and streamflow from Ireland, we test for climatic and hydrologic change in this maritime climate of the northeast Atlantic. Five decades of hourly precipitation (at eight sites) and daily streamflow at four rivers in Ireland were investigated for patterns of climate variability. An increase in annual precipitation was found to occur after 1975. This increase in precipitation is most noticeable on the West of the island. Precipitation increases are significant in March and October and are associated with increases in the frequency of wet hours with no change in the hourly intensities. Analysis of streamflow data shows the same trends. Furthermore, analysis of extreme rainfall events show that a much greater proportion of extremes have occurred in the period since 1975. A change also occurred in the North Atlantic Oscillation (NAO) index around 1975. The increased NAO since 1975 is associated with increased westerly airflow circulation in the Northeast Atlantic and is correlated with the wetter climate in Ireland. These climatic changes have implications for water resources management particularly flood analysis and protection.     

Estimation of the generalized logistic distribution of extreme events using partial L-moments

Abstract

Statistical analysis of extreme events is often carried out to predict large return period events. In this paper, the use of partial L-moments (PL-moments) for estimating hydrological extremes from censored data is compared to that of simple L-moments. Expressions of parameter estimation are derived to fit the generalized logistic (GLO) distribution based on the PL-moments approach. Monte Carlo analysis is used to examine the sampling properties of PL-moments in fitting the GLO distribution to both GLO and non-GLO samples. Finally, both PL-moments and L-moments are used to fit the GLO distribution to 37 annual maximum rainfall series of raingauge station Kampung Lui (3118102) in Selangor, Malaysia, and it is found that analysis of censored rainfall samples of PL-moments would improve the estimation of large return period events.

Editor D. Koutsoyiannis; Associate editor K. Hamed

Citation Zakaria, Z.A., Shabri, A. and Ahmad, U.N., 2012. Estimation of the generalized logistic distribution of extreme events using partial L-moments. Hydrological Sciences Journal, 57 (3), 424–432.     

Moments of truncated continuous univariate distributions

Truncated moment expressions (TMEs), defined as moment equations for truncated or incomplete distributions, are derived for several continuous univariate distributions commonly applied to hydrologic problems, including normal, lognormal, Pearson type III, log Pearson type III, and extreme value (Weibull and Gumbel) distributions. Solutions for gamma, tanks-in-series, and exponential distributions result as special cases. For most of the distributions considered here, closed form TMEs are presented for Nth order moments for the general case of double truncation (both upper and lower bounds). For the normal and Gumbel distributions, TMEs are presented only for moments of order N={0,1,2,3} and {0,1}, respectively. The derived TMEs are used to evaluate the effect of truncation on measured moments. The relative error between the first four truncated and complete moments is calculated as a function of both upper and lower truncation point.