Extreme value theory based on the r largest annual events: a robust approach

In Smith (1986, J. Hydrol. 86, 27–43), a family of statistical distributions and estimators for extreme values based on a fixed number r > = 1 of the largest annual events are presented. The method of estimation was numerical maximum likelihood. In this paper, we consider the robust estimation of parameters in such families of distributions. The estimation technique, which is based on optimal B-robust estimates, will assign weights to each observation and give estimates of the parameters based on the data which are well modeled by the distribution. Thus, observations which are not consistent with the proposed distribution can be identified and the validity of the model can be assessed. The method is illustrated on Venice sea level data.     

Weighted estimate of extreme quantile: an application to the estimation of high flood return periods

Parametric models are commonly used in frequency analysis of extreme hydrological events. To estimate extreme quantiles associated to high return periods, these models are not always appropriate. Therefore, estimators based on extreme value theory (EVT) are proposed in the literature. The Weissman estimator is one of the popular EVT-based semi-parametric estimators of extreme quantiles. In the present paper we propose a new family of EVT-based semi-parametric estimators of extreme quantiles. To built this new family of estimators, the basic idea consists in assigning the weights to the k observations being used. Numerical experiments on simulated data are performed and a case study is presented. Results show that the proposed estimators are smooth, stable, less sensitive, and less biased than Weissman estimator.     

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.     

Parameter and quantile estimation of the 2-parameter kappa distribution by maximum likelihood

Asymptotic properties of maximum likelihood parameter and quantile estimators of the 2-parameter kappa distribution are studied. Eight methods for obtaining large sample confidence intervals for the shape parameter and for quantiles of this distribution are proposed and compared by using Monte Carlo simulation. The best method is highlighted on the basis of the coverage probability of the confidence intervals that it produces for sample sizes commonly found in practice. For such sample sizes, confidence intervals for quantiles and for the shape parameter are shown to be more accurate if the quantile estimators are assumed to be log normally distributed rather than normally distributed (same for the shape parameter estimator). Also, confidence intervals based on the observed Fisher information matrix perform slightly better than those based on the expected value of this matrix. A hydrological example is provided in which the obtained theoretical results are applied.     

Expected probability weighted moment estimator for censored flood data

Two well-known methods for estimating statistical distributions in hydrology are the Method of Moments (MOMs) and the method of probability weighted moments (PWM). This paper is concerned with the case where a part of the sample is censored. One situation where this might occur is when systematic data (e.g. from gauges) are combined with historical data, since the latter are often only reported if they exceed a high threshold. For this problem, three previously derived estimators are the “B17B” estimator, which is a direct modification of MOM to allow for partial censoring; the “partial PWM estimator”, which similarly modifies PWM; and the “expected moments algorithm” estimator, which improves on B17B by replacing a sample adjustment of the censored-data moments with a population adjustment. The present paper proposes a similar modification to the PWM estimator, resulting in the “expected probability weighted moments (EPWM)” estimator. Simulation comparisons of these four estimators and also the maximum likelihood estimator show that the EPWM method is at least competitive with the other four and in many cases the best of the five estimators.     

Using regional regression within index flood procedures and an empirical Bayesian estimator

Studies have illustrated the performance of at-site and regional flood quantile estimators. For realistic generalized extreme value (GEV) distributions and short records, a simple index-flood quantile estimator performs better than two-parameter (2P) GEV quantile estimators with probability weighted moment (PWM) estimation using a regional shape parameter and at-site mean and L-coefficient of variation (L-CV), and full three-parameter at-site GEV/PWM quantile estimators. However, as regional heterogeneity or record lengths increase, the 2P-estimator quickly dominates. This paper generalizes the index flood procedure by employing regression with physiographic information to refine a normalized T-year flood estimator. A linear empirical Bayes estimator uses the normalized quantile regression estimator to define a prior distribution which is employed with the normalized 2P-quantile estimator. Monte Carlo simulations indicate that this empirical Bayes estimator does essentially as well as or better than the simpler normalized quantile regression estimator at sites with short records, and performs as well as or better than the 2P-estimator at sites with longer records or smaller L-CV.     

Bias compensation in flood frequency analysis

Abstract

Flood frequency analysis (FFA) is essential for water resources management. Long flow records improve the precision of estimated quantiles; however, in some cases, sample size in one location is not sufficient to achieve a reliable estimate of the statistical parameters and thus, regional FFA is commonly used to decrease the uncertainty in the prediction. In this paper, the bias of several commonly used parameter estimators, including L-moment, probability weighted moment and maximum likelihood estimation, applied to the general extreme value (GEV) distribution is evaluated using a Monte Carlo simulation. Two bias compensation approaches: compensation based on the shape parameter, and compensation using three GEV parameters, are proposed based on the analysis and the models are then applied to streamflow records in southern Alberta. Compensation efficiency varies among estimators and between compensation approaches. The results overall suggest that compensation of the bias due to the estimator and short sample size would significantly improve the accuracy of the quantile estimation. In addition, at-site FFA is able to provide reliable estimation based on short data, when accounting for the bias in the estimator appropriately.

Editor D. Koutsoyiannis; Associate editor Sheng Yue     

Regional Frequency Analysis of Extreme Groundwater Levels

Flood risk is generally perceived as being a consequence of surface water inundation. However, large damage is also caused by high groundwater levels. In surface hydrology, statistical frequency analysis is a standard tool to estimate discharge with a given return period or exceedance probability. First, a suitable probability distribution is fit to a series of annual maximum peaks. Second, this distribution is used to determine the discharge corresponding to the desired return period. Where only short series of recorded data are available, the estimates can often be improved by regional frequency analysis (RFA). Unfortunately, there is little information in the literature on analogous approaches for the estimation of extreme groundwater levels. In this contribution, the applicability of l ‐moments‐based RFA for the estimation of extreme groundwater levels is investigated. The main issues specific to groundwater levels are (1) appropriate transformation of the data, (2) criteria for identification of statistically homogeneous regions, (3) consideration of correlation between sites, and (4) choice of distribution function. This study is based on data from more than 1100 observation sites in four shallow Austrian Aquifers with a record length of 10 to 50 years. Results show that homogeneous regions for l ‐moments‐based RFA can be identified covering about one half of the total area of the aquifers. The confidence intervals for the 30‐ and 100‐year return levels can be significantly reduced by RFA. Out of the four investigated distribution functions, none is to be preferred generally.     

A distribution function based bandwidth selection method for kernel quantile estimation

The key problem in nonparametric frequency analysis of flood and droughts is the estimation of the bandwidth parameter which defines the degree of smoothing. Most of the proposed bandwidth estimators have been based on the density function rather than the cumulative distribution function or the quantile that are the primary interest in frequency analysis. We propose a new bandwidth estimator derived from properties of quantile estimators. The estimator builds on work by Altman and Léger (1995). The estimator is compared to the well-known method of least squares cross-validation (LSCV) using synthetic data generated from various parametric distributions used in hydrologic frequency analysis. Simulations suggest that our estimator performs at least as well as, and in many cases better than, the method of LSCV. In particular, the use of the proposed plug-in estimator reduces bias in the estimation as compared to LSCV. When applied to data sets containing observations with identical values, typically the result of rounding or truncation, the LSCV and most other techniques generally underestimates the bandwidth. The proposed technique performs very well in such situations.     

Comparison and evaluation of uncertainties in extreme flood estimations of the upper Yangtze River by the Delta and profile likelihood function methods

Frequency calculation for extreme flood and methods used for its uncertainty estimation are popular subjects in hydrology research. In this study, uncertainties in extreme flood estimations of the upper Yangtze River were investigated using the Delta and profile likelihood function (PLF) methods, which were used to calculate confidence intervals of key parameters of the generalized extreme value distribution and quantiles of extreme floods. Datasets of annual maximum daily flood discharge (AMDFD) from six hydrological stations located in the main stream and tributaries of the upper Yangtze River were selected in this study. The results showed that AMDFD data from the six stations followed the Weibull distribution, which has a short tail and is bounded above with an upper bound. Of the six stations, the narrowest confidence interval can be detected in the Yichang station, and the widest interval was found in the Cuntan station. Results also show that the record length and the return period are two key factors affecting the confidence interval. The width of confidence intervals decreased with the increase of record length because more information was available, while the width increased with the increase of return period. In addition, the confidence intervals of design floods were similar for both methods in a short return period. However, there was a comparatively large difference between the two methods in a long return period, because the asymmetry of the PLF curve increases with an increase in the return period. This asymmetry of the PLF method is more proficient in reflecting the uncertainty of design flood, suggesting that PLF method is more suitable for uncertainty analysis in extreme flood estimations of the upper Yangtze River Basin.     

Multigaussian kriging for point-support estimation: incorporating constraints on the sum of the kriging weights

In the geostatistical analysis of regionalized data, the practitioner may not be interested in mapping the unsampled values of the variable that has been monitored, but in assessing the risk that these values exceed or fall short of a regulatory threshold. This kind of concern is part of the more general problem of estimating a transfer function of the variable under study. In this paper, we focus on the multigaussian model, for which the regionalized variable can be represented (up to a nonlinear transformation) by a Gaussian random field. Two cases are analyzed, depending on whether the mean of this Gaussian field is considered known or not, which lead to the simple and ordinary multigaussian kriging estimators respectively. Although both of these estimators are theoretically unbiased, the latter may be preferred to the former for practical applications since it is robust to a misspecification of the mean value over the domain of interest and also to local fluctuations around this mean value. An advantage of multigaussian kriging over other nonlinear geostatistical methods such as indicator and disjunctive kriging is that it makes use of the multivariate distribution of the available data and does not produce order relation violations. The use of expansions into Hermite polynomials provides three additional results: first, an expression of the multigaussian kriging estimators in terms of series that can be calculated without numerical integration; second, an expression of the associated estimation variances; third, the derivation of a disjunctive-type estimator that minimizes the variance of the error when the mean is unknown.     

A Robust Method for Relative Gravity Data Estimation with High Efficiency

When gravimetric data observations have outliers, using standard least squares (LS) estimation will likely give poor accuracies and unreliable parameter estimates. One of the typical approaches to overcome this problem consists of using the robust estimation techniques. In this paper, we modified the robust estimator of Gervini and Yohai (2002) called REWLSE (Robust and Efficient Weighted Least Squares Estimator), which combines simultaneously high statistical efficiency and high breakdown point by replacing the weight function by a new weight function. This method allows reducing the outlier impacts and makes more use of the information provided by the data. In order to adapt this technique to the relative gravity data, weights are computed using the empirical distribution of the residuals obtained initially by the LTS (Least Trimmed Squares) estimator and by minimizing the mean distances relatively to the LS-estimator without outliers. The robustness of the initial estimator is maintained by adapted cut-off values as suggested by the REWLSE method which allows also a reasonable statistical efficiency. Hereafter we give the advantage and the pertinence of REWLSE procedure on real and semi-simulated gravity data by comparing it with conventional LS and other robust approaches like M- and MM-estimators.     

Probability density estimation in stochastic environmental models using reverse representations

The estimation of probability densities of variables described by stochastic differential equations has long been done using forward time estimators, which rely on the generation of forward in time realizations of the model. Recently, an estimator based on the combination of forward and reverse time estimators has been developed. This estimator has a higher order of convergence than the classical one. In this article, we explore the new estimator and compare the forward and forward–reverse estimators by applying them to a biochemical oxygen demand model. Finally, we show that the computational efficiency of the forward–reverse estimator is superior to the classical one, and discuss the algorithmic aspects of the estimator.     

Local hydraulic gradient estimator analysis of long-term monitoring networks

Three measurements of head at unique locations form a three-point estimator of the local magnitude and orientation of the hydraulic gradient. The relative head measurement error (RHME) is defined here as the measurement error normalized by the head drop across the three-point estimator. Monte Carlo simulation results show that estimators with base to height ratios between 0.5 and 5.0 and that are large enough to keep the RHME below 0.05 create the most accurate gradient estimates and provide criteria for identifying good estimators. These criteria are applied to an example ground water monitoring network design problem in the Culebra dolomite near the Waste Isolation Pilot Plant repository to both analyze temporal changes and modify and expand the current monitoring network. Limiting the three-point estimators to those that meet the shape and RHME criteria reduces the number of possible estimators by >50% and leads to approximately 1 order of magnitude decrease in the average estimated magnitude of the gradient relative to using all estimators. Application of these criteria also reduces the variability in estimated gradient magnitude and orientation between the two time periods of measurements. Redundant wells in the network are identified by removing each existing well in turn and determining which removals yield the smallest decrease in the number of acceptable estimators. Optimal new well locations are identified by mapping the increase in total number of acceptable estimators for a single new well placed in the study domain.     

Maximum-likelihood estimation of unsaturated hydraulic parameters

Maximum-likelihood estimators properly represent measurement error, thus provide a statistically sound basis for evaluating the adequacy of a model fit and for finding the multivariate parameter confidence region. We demonstrate the advantages of using maximum-likelihood estimators rather than simple least-squares estimators for the problem of finding unsaturated hydraulic parameters. Inversion of outflow data given independent retention data can be treated by an extension to a Bayesian estimator. As an example, we apply the methodology to retention and transient unsaturated outflow observations, both obtained on the same medium sand sample. We found the van Genuchten expression to be adequate for the retention data, as the best fit was within measurement error. The Cramer–Rao confidence bound described the true parameter uncertainty approximately. The Mualem–van Genuchten expression was, however, inadequate for our outflow observations, suggesting that the parameters (, n) may not always be equivalent in describing both retention and unsaturated conductivity.     

Sequential kriging and cokriging: Two powerful geostatistical approaches

A sequential linear estimator is developed in this study to progressively incorporate new or different spatial data sets into the estimation. It begins with a classical linear estimator (i.e., kriging or cokriging) to estimate means conditioned to a given observed data set. When an additional data set becomes available, the sequential estimator improves the previous estimate by using linearly weighted sums of differences between the new data set and previous estimates at sample locations. Like the classical linear estimator, the weights used in the sequential linear estimator are derived from a system of equations that contains covariances and cross-covariances between sample locations and the location where the estimate is to be made. However, the covariances and cross-covariances are conditioned upon the previous data sets. The sequential estimator is shown to produce the best, unbiased linear estimate, and to provide the same estimates and variances as classic simple kriging or cokriging with the simultaneous use of the entire data set. However, by using data sets sequentially, this new algorithm alleviates numerical difficulties associated with the classical kriging or cokriging techniques when a large amount of data are used. It also provides a new way to incorporate additional information into a previous estimation.     

Application of the Bayesian approach in regional analysis of extreme rainfalls

Based on the Partial Duration Series model a regional Bayesian approach is introduced in the modelling of extreme rainfalls from a country-wide system of recording raingauges in Denmark. The application of the Bayesian principles is derived in case of both exponential and generalized Pareto-distributed exceedances. The method is applied to, respectively, the total precipitation depth and the maximum 10 minutes rain intensity of individual storms from 41 stations. By means of the regional analysis prior distributions of the parameters in the Partial Duration Series model are estimated. It is shown that the regional approach significantly reduces the uncertainty of the T-year event estimator compared to estimation based solely on at-site data. In addition, the regional approach provides quantile estimates at non-monitored sites.     

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.     

The effect of nested grid sampling on the parameter estimation of a spatial Gompertz diffusion

This paper evaluates the effects of using data observed on regular nested grids on the parameter estimates of a two-parameter Gompertz diffusion model. This new spatial diffusion process represents a technically more complex stage of Gompertz modeling. Firstly, the diffusion model is introduced through an appropriate transformation of a two-parameter Gaussian diffusion process. Probabilistic characteristics of this model, such as the transition densities and the trend functions, are obtained. Secondly, statistical estimation is considered using data obtained on a regular or irregular grid; the explicit expression of the likelihood equations and the parameter estimators are given for regular grids. Finally, a simulation experiment illustrates the results of this paper.