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Various regionalization approaches are in use to delineate watersheds in an area into homogeneous groups/regions for regional frequency analysis of hydrologic extremes (floods, droughts). Each approach differs in the underlying assumptions and strategy for regionalization and yields regions that differ in composition. There is ambiguity in the choice of approaches, as none is established to be universally superior in yielding true regions that are unknown. This article proposes a novel fuzzy ensemble clustering (FEC) approach to deal with uncertainty in the composition of regions obtained using different regionalization approaches. It forms fuzzy meta-regions by integrating information on similarities in watershed groupings found in an ensemble of regions derived using several regionalization approaches. The FEC approach's potential to form effective regions (than those found in the ensemble) and their utility in regional flood frequency analysis to predict flood quantiles at ungauged sites is demonstrated through Monte Carlo simulation experiments and a case study on peninsular India. An ensemble of regions for use with FEC is formed using the region of influence approach, clustering approach based on Gaussian mixture model (GMM), and a hybrid approach which combines canonical correlation analysis with GMM based clustering. The FEC is shown to be effective in grouping similar watersheds even when the ensemble comprises regions having some improperly/wrongly grouped watersheds. © 2021. American Geophysical Union. All Rights Reserved.

Using a multi-century reconstruction of drought, we investigate how rare the 2012-2015 California drought is. A Bayesian approach to a nonstationary, bivariate probabilistic model, including the estimation of Copula parameters is used to assess the time varying return period of the current drought. Both the duration and severity of drought exhibit similar multi-century trends. The period from 800-1200 AD was perhaps more similar to the recent period than the period from 1200 to 1800 AD. The median return period of the recent drought accounting for both duration and severity, varies from approximately 667 to 2652 years, if the model parameters from the different time periods are considered. However, we find that the recent California drought is of unprecedented severity, especially given the relatively modest duration of the drought. The return period of the severity of the recent drought given its 4-year duration is estimated to be nearly 21,000 years. This article is protected by copyright. All rights reserved.

Regional frequency analysis (RFA) is a well-established methodology to provide an estimate of the flood frequency curve at ungauged (or scarcely gauged) sites. Different RFA approaches exist, depending on the way the information is transferred to the site of interest, but it is not clear in the literature if a specific method systematically outperforms the others. The aim of this study is to provide a framework wherein carrying out the intercomparison by building up a virtual environment based on synthetically generated data. The considered regional approaches include: (i) a unique regional curve for the whole region; (ii) a multiple-region model where homogeneous subregions are determined through cluster analysis; (iii) a Region-of-Influence model which defines a homogeneous subregion for each site; (iv) a spatially-smooth estimation procedure where the parameters of the regional model vary continuously along the space. Virtual environments are generated considering different patterns of heterogeneity, including step change and smooth variations. If the region is heterogeneous, with the parent distribution changing continuously within the region, the spatially-smooth regional approach outperforms the others, with overall errors 10%-50% lower than the other methods. In the case of a step-change, the spatially-smooth and clustering procedures perform similarly if the heterogeneity is moderate, while clustering procedures work better when the step-change is severe. To extend our findings, an extensive sensitivity analysis has been performed to investigate the effect of sample length, number of virtual stations, return period of the predicted quantile, variability of the scale parameter of the parent distribution, number of predictor variables and different parent distribution. Overall, the spatially-smooth approach appears as the most robust approach as its performances are more stable across different patterns of heterogeneity, especially when short records are considered. This article is protected by copyright. All rights reserved.

Precipitation quantile estimates are used in engineering, agriculture, and a variety of other disciplines. Index flood regional frequency methods pool normalized gauge data in the case of homogeneity among the constituent gauges of the region. Unitless regional quantile estimates are outputted and rescaled at each gauge. Because violation of the homogeneity hypothesis is a major component of quantile estimation error in regional frequency analysis, heterogeneity estimators should be “reasonable proxies” of the error of quantile estimation. In this study, three Monte Carlo heterogeneity statistics tested in Hosking and Wallis (1997) are plotted against Monte Carlo estimates of quantile error for all five-or-more-gauge regionalizations in a 12 gauge network in the Twin Cities region of Minnesota. Upper-tail quantiles with nonexceedance probabilities of 0.75 and above are examined at time steps ranging from daily to monthly. A linear relationship between heterogeneity and error estimates is found and quantified using Pearson's r score. Two of Hosking and Wallis's (1997) heterogeneity measures, incorporating the coefficient of variation in one case and additionally the skewness in the other, are found to be reasonable proxies for quantile error at the L-moment ratio values characterizing these data. This result, in addition to confirming the utility of a commonly used coefficient of variation-based heterogeneity statistic, provides evidence for the utility of a heterogeneity measure that incorporates skewness information.

[1] Droughts, like floods, are extreme expressions of the river flow dynamics. Here, droughts are intended as episodes during which the streamflow is below a given threshold, and are described as multivariate events characterized by two variables: average intensity and duration. In this work, we introduce the new concept of Dynamic Return Period, formulated using the theory of Copulas, and calculated via a Survival Kendall's approach. We show how it can be used (i) to monitor the temporal evolution of a drought event, and (ii) to perform real time assessment. In addition, a randomization strategy is introduced, in order to get rid of repeated measurements, which may adversely affect the statistical analysis of the available data, as well as the calculation of the return periods of interest: a practical example is shown, involving the fit of the drought duration distribution. The case study of the Po river basin (Northern Italy) is used as an illustration.

Regional frequency analyses based on index flood procedures have been used within the hydrologic community since 1960. It appears that when the index flood method was first suggested, the index flood was taken to be the at-site population mean, which, in turn, in the last two or three decades, has been estimated by the at-site sample mean. The objectives of this paper are to investigate the consequences of replacing a population characteristic with its sample counterpart and to propose an analytically correct regional model dubbed as the population index flood (PIF) method. In this method the homogeneity of the region is embedded in the structure of the parameter space of the underlying distribution model. Simulation experiments are conducted to test the proposed PIF method based on the generalized extreme value distribution with parameters estimated using the method of maximum likelihood (MLE) and the method of probability- weighted moments (PWM). Furthermore, in the simulation experiments the PIF method is compared with the Hosking and Wallis [1997] regional estimation scheme (HW scheme). Comparing among all index flood methods investigated herein, the PIF method with parameters estimated using MLE provides the best overall results for the 0.95 and the 0.99 quantites in terms of both bias and root-mean-square error for moderate to sufficiently large sample sizes, but for the 0.995 quantile the HW scheme seems to perform best for the investigated sample sizes.

We examine some performance indices (PIs) that are used to compare regional and at-site flood quantile estimation methods. These include the relative bias, the regional average root-mean-square-error (RMSE), the regional average relative root-mean-square-error (RRMSE), and the average RMSE and RRMSE ratios of quantile estimators. We study the dependence of these PIs on the relative variability (coefficient of variation) of the data. This is done by examining the effect of a location shift in the data on these PIs. The aim is to bring awareness to the fact that when comparing hydrological quantile estimators, some PIs are more greatly affected than others by data shifts in location. Among the PIs considered, we identify those that are invariant to a location shift in the data and those that are not. This is done under both assumptions of homogeneous and heterogeneous hydrological region. The generalized extreme value distribution is used to demonstrate some of the results, but the conclusions are applicable to other distributions with a location parameter. It is argued that because of the lack of invariance to location shift of certain quantile estimation methods and PIs, additional precautions need to be taken when comparing these methods. Although we focus discussion around flood frequency analysis, the points raised should be viewed within the broader context of hydrological frequency analysis.

A common assumption in frequency analysis is that hydrologic extremes (floods or heavy precipitation) are generated by a random process. This implies that natural climatic variability does not change the distribution of extreme events. A regional frequency analysis approach is proposed to test the hypothesis of randomness over secular timescales. Observed regional occurrences of extreme events are compared to those from a random process. Significant departures may indicate nonrandomness due to climatic variability. Application of the approach to a region in the Southern Plains indicates nonrandomness in annual maximum precipitation.

Regional frequency analysis uses data from a number of measuring sites. A “region” is a group of sites each of which is assumed to have data drawn from the same frequency distribution. The analysis involves the assignment of sites to regions, testing whether the proposed regions are indeed homogeneous, and choice of suitable distributions to fit to each region's data. This paper describes three statistics useful in regional frequency analysis: a discordancy measure, for identifying unusual sites in a region; a heterogeneity measure, for assessing whether a proposed region is homogeneous; and a goodness-of-fit measure, for assessing whether a candidate distribution provides an adequate fit to the data. Tests based on the statistics provide objective backing for the decisions involved in regional frequency analysis. The statistics are based on the L moments [Hosking, 1990] of the at-site data.

[1] Conventional multivariate hydrological frequency analysis utilizes only the concurrent parts of data sets, leaving a lot of nonconcurrent data unutilized. Simultaneous inclusion of such nonconcurrent data can significantly reduce uncertainty in hydrologic design estimates. The methodology proposed in this paper allows varied length multivariate data to be combined and analyzed in an integrated framework through a “Composite Likelihood Approach.” The method employs copula-based multivariate distributions in order to provide necessary flexibility of admitting arbitrary marginals. The paper presents the theoretical basis of the approach and highlights its advantages through two applications. A significant reduction in uncertainty in design flood quantiles of a relatively shorter flood series is achieved by utilizing an associated downstream flood data. The advantage of the methodology is further demonstrated by establishing significant information gain for six different combinations of Gaussian and non-Gaussian marginals. The proposed approach marks a paradigm shift in hydrologic design procedures, particularly for partially gauged basins, wherein a higher precision in hydrologic designs is achieved by leveraging associated information that has hitherto remained unutilized. It is opined that the approach will enable offsetting the impact of dwindling hydrological observation networks around the world by enhancing information that is derivable from existing networks.

[1] Flood risk analysis is subject to often severe uncertainties, which can potentially undermine flood management decisions. This paper explores the use of information gap theory to analyze the sensitivity of flood management decisions to uncertainties in flood inundation models and flood frequency analysis. Information gap is a quantified nonprobabilistic theory of robustness. To analyze uncertainties in flood modeling, an energy-bounded information gap model is established and applied first to a simplified uniform channel and then to a more realistic 2-D flood model. Information gap theory is then applied to the estimation of flood discharges using regional frequency analysis. The use of an information gap model is motivated by the notion that hydrologically similar sites are clustered in the space of their L moments. The information gap model is constructed around a parametric statistical flood frequency analysis, resulting in a hybrid model of uncertainty in which natural variability is handled statistically while epistemic uncertainties are represented in the information gap model. The analysis is demonstrated for sites in the Trent catchment, United Kingdom. The analysis is extended to address ungauged catchments, which, because of the attendant uncertainties in flood frequency analysis, are particularly appropriate for information gap analysis. Finally, the information gap model of flood frequency is combined with the treatment of hydraulic model uncertainties in an example of how both sources of uncertainty can be accounted for using information gap theory in a flood risk management decision.

Because of their multivariate nature, several hydrological phenomena can be described by more than one correlated characteristic. These characteristics are generally not independent and should be jointly considered. Consequently, univariate regional frequency analysis (FA) cannot provide complete assessment of true probabilities of occurrence. The objective of the present paper is to propose a procedure for regional flood FA in a multivariate framework. In the present paper, the focus is on the estimation step of regional FA. The proposed procedure represents a multivariate version of the index flood model and is based on copulas and a multivariate quantile version with a focus on the bivariate case. The model offers increased flexibility to designers by leading to several scenarios associated with the same risk. The univariate quantiles represent special cases corresponding to the extreme scenarios. A simulation study is carried out to evaluate the performance of the model in a bivariate framework. Simulation results show that bivariate FA provides the univariate quantiles with equivalent accuracy. Similarity is observed between results of the bivariate model and those of the univariate one in terms of the behavior of the corresponding performance criteria. The procedure performs better when the regional homogeneity is high. Furthermore, the impacts of small variations in the record length at gauged sites and the region size on the performance of the proposed procedure are not significant.

[1] 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.

This paper considers some practical aspects concerning the application of the Weibull distribution to low-flow frequency analysis. Two- and three-parameter forms of the distribution are fitted to a total of 987 distributions derived from the daily flow data of 134 catchments located in southeastern Australia. The relative performance of three estimation methods (moments, maximum likelihood, and probability weighted moments) is investigated, and it is found that the different estimation methods provide distinct sets of quantile estimates. The method of probability weighted moments is more likely to give unsatisfactory estimates of the smallest drought and in general tends to yield less severe estimates of drought volumes relative to the other two methods. The method of maximum likelihood, however, occasionally provides estimates of drought volumes that are many times greater than that yielded by the methods moments or probability weighted moments. In addition, the differences between low-flow frequency estimates based on calendar and hydrologic years is investigated, and recommendations are given for the selection of an appropriate subset of the data to be fitted.

[1] Metaelliptical copulas are introduced as a flexible tool for modeling multivariate data in hydrology. The properties of this broad class of dependence functions are reviewed, along with associated rank-based procedures for copula parameter estimation and goodness-of-fit testing. A new graphical diagnostic tool is also proposed for selecting an appropriate metaelliptical copula. Peak, volume, and duration of the annual spring flood for the Romaine River (Quebec, Canada) are used for illustration purposes.

[1] The discordancy measure in terms of the sample L-moment ratios (L-CV, L-skewness, L-kurtosis) of the at-site data is widely recommended in the screening process of atypical sites in the regional frequency analysis (RFA). The sample mean and the covariance matrix of the L-moments ratios, on which the discordancy measure is based, are not robust against outliers in the data, and consequently, this measure can be strongly affected by the discordant sites present in the region. We propose to replace the classical mean and covariance matrix estimates by their robust alternatives on the basis of the minimum covariance determinant estimator. The performance of the classical and robust measures for discordant sites identification is assessed in a series of Monte Carlo simulation experiments within the framework of the RFA. The simulation study shows that the robust discordant measure outperforms the classical one and is consistent with the heterogeneity measure H. Thus we recommend its use as a tool for discordant sites detection and formation of homogeneous regions in RFA.

[1] This article presents the modeling of multivariate extreme values using copulas. Our approach allows us to model the dependence structure independently of the marginal distributions, which is not possible with standard classical methods. The methodology has been applied on two different problems in hydrology. The first application is concerned with the combined risk in the framework of frequency analysis. Four copulas have been tested on peak flows from the watershed of Peribonka in Quebec, Canada. The second application relates to the joint modeling of peak flows and volumes. Three copulas have been applied to the watershed of the Rimouski River in Quebec, Canada. This approach using copulas is promising since it allows us to take into account a wide range of correlation which can happen in hydrology.

[1] The expected moments algorithm (EMA) [Cohn et al., 1997] and the Bulletin 17B [Interagency Committee on Water Data, 1982] historical weighting procedure (B17H) for the log Pearson type III distribution are compared by Monte Carlo computer simulation for cases in which historical and/or paleoflood data are available. The relative performance of the estimators was explored for three cases: fixed-threshold exceedances, a fixed number of large floods, and floods generated from a different parent distribution. EMA can effectively incorporate four types of historical and paleoflood data: floods where the discharge is explicitly known, unknown discharges below a single threshold, floods with unknown discharge that exceed some level, and floods with discharges described in a range. The B17H estimator can utilize only the first two types of historical information. Including historical/paleoflood data in the simulation experiments significantly improved the quantile estimates in terms of mean square error and bias relative to using gage data alone. EMA performed significantly better than B17H in nearly all cases considered. B17H performed as well as EMA for estimating X100 in some limited fixed-threshold exceedance cases. EMA performed comparatively much better in other fixed-threshold situations, for the single large flood case, and in cases when estimating extreme floods equal to or greater than X500. B17H did not fully utilize historical information when the historical period exceeded 200 years. Robustness studies using GEV-simulated data confirmed that EMA performed better than B17H. Overall, EMA is preferred to B17H when historical and paleoflood data are available for flood frequency analysis.

[1] The population index flood (PIF) method is an analytical model that has been recently suggested for regional frequency analysis. In this paper, explicit equations based on Fisher's information are derived for estimating the standard error of at-site quantile estimators for two regional PIF methods utilizing the generalized extreme value distribution with maximum likelihood estimation. These explicit equations are used to calculate the asymptotic gain in using regional frequency analysis as opposed to single site frequency analysis. Simulation experiments for different sized regions and different values of the shape parameter show that the suggested methods for estimating the standard error of at-site quantile estimators give values close to the actual or true values. In addition, similar simulation experiments are also used to test the accuracy of a newly suggested procedure for estimating the standard errors of at-site quantile estimators for the Hosking and Walls regional index flood method. The results of the simulations indicate that these estimated standard errors can in some cases give unreliable results. In general, this study shows that the PIF models are a useful addition to existing regional frequency analysis models. Their analytic structure, which is not present in other regional models, has important theoretical and practical implications.

[1] A spatial-temporal model of rainfall, based on a Neyman-Scott stochastic point process, is fitted to hourly data taken from nine sites in the Arno Basin, Italy. The stochastic model is an extension of the temporal Neyman-Scott rectangular pulses model into two-dimensional space and introduces a further parameter into the model. In the model, storms arrive in a Poisson process, where each storm consists of discs representing rain cells, with centers distributed over an area according to a spatial Poisson process. The cells have a random radius, lifetime, and intensity, with the intensity remaining constant over the area of the disc and cell lifetime. A fitting procedure is proposed which couples the results obtained in two preceding papers: the second-order properties of the spatial-temporal model and the third moment function of the single site model [Cowpertwait, 1995, 1998]. The model is validated by comparing extreme historical hourly data and equivalent data simulated using the fitted spatial-temporal model. These comparisons are made using a regional frequency analysis, based on L moments, and log-log plots of the upper distribution tail. The results indicate that the model is able to preserve regional extremes and support the use of the model in hydrological applications.

Frequency analysis is commonly used in hydrology to define flood plains, design hydraulic structures, and aid in watershed planning and management. Although design event magnitudes may be determined analytically by using probability distributions, the analytical determination of confidence limits for the design event is not easy. Using the alternate method of moments to determine confidence limits involves an assumption of normality of the distribution of design events. It is shown by data generation that this assumption is valid for quite small samples. Tables are given to compute confidence limits in this manner for several common probability distributions.

Historically, meander planform data have been analyzed by various statistical and geometrical techniques, but uncertainty persists about the wavelength, frequency, and reference geometry. Therefore meander wavelength was analyzed in the Fourier frequency domain by power spectral analysis in terms of hierarchical centerlines as a reference geometry, eliminating some of the subjective bias associated with wavelength measurements. Equally spaced sampling points were located along the talweg of meandering streams. Sample values were measured as the perpendicular distance to a reference line connecting meander belt midpoints. This line, itself meandering, constitutes a river's second-order meander frequency, which is analyzed with respect to a still higher-order reference line. Sampling along the talweg attenuates the spectral estimates of natural frequencies of meanders. Corrections are obtained by multiplying frequency peaks by the sinuosity of the study reach. Power spectra obtained from hierarchical reference lines of first- and second-order meanders of the White River in southern Indiana indicate discrete frequencies in each order. First- and second-order harmonics are observed at high and low frequencies. Complex meandering along the Dolores River in western Colorado is characterized by very low-frequency energy in third-order meanders and successively higher frequency energy in second- and first-order meanders. A comparison of power spectra for hierarchical and arbitrary reference lines shows that arbitrary references condense spectral estimates of meander energy within artificially averaged frequency bands. Another comparison, using the theory of minimum variance for meander formation, confirms that hierarchical reference lines improve meander characterization.

A practical approach is developed for frequency analysis of multiyear drought durations of annual streamflow series. The proposed approach utilizes a technique which smooths the frequency-curve irregularity of drought durations and, therefore, reduces the statistical uncertainties associated with sample-size limitations. Several hazard-function models are examined with regard to their ability to represent the duration-dependent termination rate of a drought data set. The logistic model with an exponential correction for short durations is considered to have sufficient flexibility for describing most portions of the termination-rate function. The parameters of the logistic model are identified using a piecewise linear regression for the linearized termination probability curve. Once the form of the hazard model is specified arid its parameters are estimated, the probability distribution and exceedance probability of drought durations can be determined from their corresponding relationships with the hazard model. The historic droughts of Feather River, California, are used to demonstrate the methodology. Results indicate that the observed nine-year drought of Feather River has a return period of 360 years.

A distribution for frequency analysis of the logarithms of annual floods has been derived from data from 78 catchments in eastern Australia. The statistics of data from these catchments demonstrate the need for three-parameter distributions instead of two-parameter distributions for analysis of annual floods. A nonlinear relationship between frequency factor and In In [T/[T − 1)] function of recurrence interval is demonstrated, and this relationship is used as the basis for the distribution. Methods of fitting the distribution to data sets, including sets which contain zero values, are described. A comparison is made of results from the new distribution, the log Pearson type 3, the log normal, the log Gumbel, and the Gumbel distributions.

Expressions in terms of sample size and parameters of the probability distribution are obtained for the standard errors of sample estimates of flood magnitudes corresponding to given return periods for normal and double exponential universes. These errors are shown to converge toward fixed proportions of the estimates for very high return periods. (Key words: Floods; frequency analysis; statistics.)

A method is considered for selecting mathematical models to fit frequency distributions of hydrologic data among ten commonly used methods of frequency analysis. The methods are those of Gumbel, Lieblein, Chow, and the seven types of Gringorten. The best fitting model is chosen on the basis of comparison for goodness of fit indicated by the coefficients of determination. Friedman's nonparametric statistical test was employed to detect significant differences between the models.