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.     

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.     

Extreme value Birnbaum–Saunders regression models applied to environmental data

Extreme value models are widely used in different areas. The Birnbaum–Saunders distribution is receiving considerable attention due to its physical arguments and its good properties. We propose a methodology based on extreme value Birnbaum–Saunders regression models, which includes model formulation, estimation, inference and checking. We further conduct a simulation study for evaluating its performance. A statistical analysis with real-world extreme value environmental data using the methodology is provided as illustration.     

German tanks and historical records: the estimation of the time coverage of ungauged extreme events

The use of historical data can significantly reduce the uncertainty around estimates of the magnitude of rare events obtained with extreme value statistical models. For historical data to be included in the statistical analysis a number of their properties, e.g. their number and magnitude, need to be known with a reasonable level of confidence. Another key aspect of the historical data which needs to be known is the coverage period of the historical information, i.e. the period of time over which it is assumed that all large events above a certain threshold are known. It might be the case though, that it is not possible to easily retrieve with sufficient confidence information on the coverage period, which therefore needs to be estimated. In this paper methods to perform such estimation are introduced and evaluated. The statistical definition of the problem corresponds to estimating the size of a population for which only few data points are available. This problem is generally refereed to as the German tanks problem, which arose during the second world war, when statistical estimates of the number of tanks available to the German army were obtained. Different estimators can be derived using different statistical estimation approaches, with the maximum spacing estimator being the minimum-variance unbiased estimator. The properties of three estimators are investigated by means of a simulation study, both for the simple estimation of the historical coverage and for the estimation of the extreme value statistical model. The maximum spacing estimator is confirmed to be a good approach to the estimation of the historical period coverage for practical use and its application for a case study in Britain is presented.     

Conceptual modelling to assess how the interplay of hydrological connectivity,catchment storage and tracer dynamics controls nonstationary water age estimates

Although catchment storage is an intrinsic control on the rainfall–runoff response of streams, direct measurement remains a major challenge. Coupled models that integrate long‐term hydrometric and isotope tracer data are useful tools that can provide insights into the dynamics of catchment storage and the volumes of water involved. In this study, we use a tracer‐aided hydrological model to characterize catchment storage as a dynamic control on system function related to streamflow generation, which also allows direct estimation of the nonstationarity of water ages. We show that in a wet Scottish upland catchment dominated by runoff generation from riparian peats (histosols) with high water storage, nonstationarity in water age distributions is only clearly detectable during more extreme wet and dry periods. This is explained by the frequency and longevity of hydrological connectivity and the associated relative importance of flow paths contributing younger or older waters to the stream. Generally, these saturated riparian soils represent large mixing zones that buffer the time variance of water age and integrate catchment‐scale partial mixing processes. Although storage simulations depend on model performance, which is influenced by input variability and the degree of isotopic damping in the stream, a longer‐term storage analysis of this model indicates a system that is only sensitive to more extreme hydroclimatic variability. Copyright © 2014 John Wiley & Sons, Ltd.     

基于轮廓似然估计的广义极值分布在地震中长期预测中的应用

为描述强震预测的不确定性,在地震预报极值分析模型的参数估计中,引入轮廓似然估计法。对广义极值分布中形状参数和地震重现水平的轮廓似然估计原理及数值算法进行了详细地阐述,并利用构建的广义极值分布模型对东昆仑地震带进行了地震危险性分析。关于形状参数和重现水平的点估计,以及10年以内的重现水平置信区间的估计,轮廓似然估计法与极大似然估计法效果基本相同,但在中长期地震重现水平置信区间的预测中,轮廓似然估计法得到的关于置信水平不对称的置信区间,在强震水平下对预测震级的不确定性表达更准确,预测结果更加有效。     

A regional Bayesian POT model for flood frequency analysis

Flood frequency analysis is usually based on the fitting of an extreme value distribution to the local streamflow series. However, when the local data series is short, frequency analysis results become unreliable. Regional frequency analysis is a convenient way to reduce the estimation uncertainty. In this work, we propose a regional Bayesian model for short record length sites. This model is less restrictive than the index flood model while preserving the formalism of “homogeneous regions”. The performance of the proposed model is assessed on a set of gauging stations in France. The accuracy of quantile estimates as a function of the degree of homogeneity of the pooling group is also analysed. The results indicate that the regional Bayesian model outperforms the index flood model and local estimators. Furthermore, it seems that working with relatively large and homogeneous regions may lead to more accurate results than working with smaller and highly homogeneous regions.     

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.     

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.     

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.     

A case for a reassessment of the risks of extreme hydrological hazards in the Caribbean

There is an urgent need for the development and implementation of modern statistical methodology for long-term risk assessment of extreme hydrological hazards in the Caribbean. Notwithstanding the inevitable scarcity of data relating to extreme events, recent results and approaches call into question standard methods of estimation of the risks of environmental catastrophes that are currently adopted. Estimation of extreme hazards is often based on the Gumbel model and on crude methods for estimating predictive probabilities. In both cases the result is often a remarkable underestimation of the predicted probabilities for disasters of large magnitude. Simplifications do not stop here: assumptions of data homogeneity and temporal independence are usually made regardless of potential inconsistencies with genuine process behaviour and the fact that results may be sensitive to such mis-specifications. These issues are of particular relevance for the Caribbean, given its exposure to diverse meteorological climate conditions.In this article we present an examination of predictive methodologies for the assessment of long-term risks of hydrological hazards, with particular focus on applications to rainfall and flooding, motivated by three data sets from the Caribbean region. Consideration is given to classical and Bayesian methods of inference for annual maxima and daily peaks-over-threshold models. We also examine situations where data non-homogeneity is compromised by an unknown seasonal structure, and the situation in which the process under examination has a physical upper limit. We highlight the fact that standard Gumbel analyses routinely assign near-zero probability to subsequently observed disasters, and that for San Juan, Puerto Rico, standard 100-year predicted rainfall estimates may be routinely underestimated by a factor of two.     

Seasonal effects of extreme surges

Extreme value analysis of sea levels is an essential component of risk analysis and protection strategy for many coastal regions. Since the tidal component of the sea level is deterministic, it is the stochastic variation in extreme surges that is the most important to model. Historically, this modelling has been accomplished by fitting classical extreme value models to series of annual maxima data. Recent developments in extreme value modelling have led to alternative procedures that make better use of available data, and this has led to much refined estimates of extreme surge levels. However, one aspect that has been routinely ignored is seasonality. In an earlier study we identified strong seasonal effects at one of the number of locations along the eastern coastline of the United Kingdom. In this article, we discuss the construction and inference of extreme value models for processes that include components of seasonality in greater detail. We use a point process representation of extreme value behaviour, and set our inference in a Bayesian framework, using simulation-based techniques to resolve the computational issues. Though contemporary, these techniques are now widely used for extreme value modelling. However, the issue of seasonality requires delicate consideration of model specification and parameterization, especially for efficient implementation via Markov chain Monte Carlo algorithms, and this issue seems not to have been much discussed in the literature. In the present paper we make some suggestions for model construction and apply the resultant model to study the characteristics of the surge process, especially in terms of its seasonal variation, on the eastern UK coastline. Furthermore, we illustrate how an estimated model for seasonal surge can be combined with tide records to produce return level estimates for extreme sea levels that accounts for seasonal variation in both the surge and tidal processes.

Bayesian posterior predictive return levels for environmental extremes

A key aim of most extreme value analyses is the estimation of the r-year return level; the wind speed, or sea-surge, or rainfall level (for example), we might expect to see once (on average) every r years. There are compelling arguments for working within the Bayesian setting here, not least the natural extension to prediction via the posterior predictive distribution. Indeed, for practitioners the posterior predictive return level has been cited as perhaps the most useful point summary from a Bayesian analysis of extremes, and yet little is known of the properties of this statistic. In this paper, we attempt to assess the performance of predictive return levels relative to their estimative counterparts obtained directly from the return level posterior distribution; in particular, we make comparisons with the return level posterior mean, mode and 95% credible upper bound. Differences between the predictive return level and standard summaries from the return level posterior distribution, for wind speed extremes observed in the UK, motivates this work. A large scale simulation study then reveals the superiority of the predictive return level over the other posterior summaries in many cases of practical interest.     

On the estimation of continuous 24-h precipitation maxima

Extreme value analysis of precipitation is of great importance for several types of engineering studies and policy decisions. For return level estimation of extreme 24-h precipitation, practitioners often use daily measurements (usually 08:00–08:00 local time) since high-frequency measurements are scarce. Annual maxima of daily series are smaller or equal to continuous 24-h precipitation maxima such that the resulting return levels may be systematically underestimated. In this paper we use a rule, derived earlier, on the conversion of the generalized extreme value (GEV) distribution of daily to 24-h maxima. We develop an estimator for the conversion exponent by combining daily maxima and high-frequency sampled 24-h maxima in one joint log-likelihood. Once the conversion exponent has been estimated, GEV-parameters of 24-h maxima can be obtained at sites where only daily data is available. The new methodology has been extended to spatial regression models.     

Explanatory analysis of the relationship between atmospheric circulation and occurrence of flood‐generating events in a coastal city

The aim of this study is to enhance the understanding of the occurrence of flood‐generating events in urban areas by analysing the relationship between large‐scale atmospheric circulation and extreme precipitation events, extreme sea water level events and their simultaneous occurrence, respectively. To describe the atmospheric circulation, we used the Lamb circulation type (LCT) classification and re‐grouped it into Lamb circulation classes (LCC). The daily LCCs/LCTs were connected with rare precipitation and water‐level events in Aarhus, a Danish coastal city. Westerly and cyclonic LCCs (W, C, SW and NW) showed a significantly high occurrence of extreme precipitation. Similarly, for extreme water‐level events westerly LCCs (W and SW) showed a significantly high occurrence. Significantly low occurrence of extreme precipitation and water‐level events was obtained in easterly LCCs (NE, E and SE). For concurrent events, significantly high occurrence was obtained in LCC W. We assessed the change in LCC occurrence frequency in the future based on two regional climate models (RCMs). The projections indicate that the westerly directions in LCCs are expected to increase in the future. Consequently, simultaneous occurrence of extreme water level and precipitation events is expected to increase in the future as a result of change in LCC frequencies. The RCM projections for LCC frequencies are uncertain because the representation of current LCCs is poor; a large number of days cannot be classified and the frequencies of the days that can be classified differ from the observed time series. Copyright © 2015 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.     

基于广义极值理论的潜在地震海啸源震级上限及强震重现水平的估计——以琉球海沟俯冲带为例

以琉球海沟俯冲带作为研究区,将广义极值理论用于估计潜在地震海啸源震级上限,首先分析了琉球海沟俯冲带的地震地质构造特征以及历史地震资料,界定潜在地震海啸源区,然后根据地震活动性特征按时间域进行分割,并提取各时间段发生的极限震级的地震样本,最后通过广义极值分布模型估计了该区域的震级上限值和强震重现水平,并对其进行了不确定性分析。     

An Evaluation on H/V Ratio vs. Spectral Ratio for Site-response Estimation Using the 1994 Northridge Earthquake Sequences

—The applicability of the single-station H/V method, based on the spectral ratio between the horizontal and the vertical components of strong ground motions, is examined for site-response estimation using the high quality data from the 1994 Northridge earthquake sequence. Instead of using Rayleigh-wave data from microtremors, the large amplitude-wave part of the S-wave data is used and based on the 1994 Northridge mainshock and aftershock recordings. We have found that upon averaging over a number of recordings for a given station, the station site responses, derived both from the single-station H/V ratio and from the standard spectral ratio (with respect to a reference rock-site station) are sufficiently close for practical purposes. We therefore conclude that the H/V ratio can reasonably predict the resonant frequency and the amplification level of a site response, especially for sites in the neighborhood of the epicenters. In the absence of a reference rock-site station, the H/V ratio provides a practical alternative to the standard site-response estimation.     

Parameter estimation in ensemble based snow data assimilation: A synthetic study

Estimating erroneous parameters in ensemble based snow data assimilation system has been given little attention in the literature. Little is known about the related methods’ effectiveness, performance, and sensitivity to other error sources such as model structural error. This research tackles these questions by running synthetic one-dimensional snow data assimilation with the ensemble Kalman filter (EnKF), in which both state and parameter are simultaneously updated. The first part of the paper investigates the effectiveness of this parameter estimation approach in a perfect-model-structure scenario, and the second part focuses on its dependence on model structure error. The results from first part research demonstrate the advantages of this parameter estimation approach in reducing the systematic error of snow water equivalent (SWE) estimates, and retrieving the correct parameter value. The second part results indicate that, at least in our experiment, there is an evident dependence of parameter search convergence on model structural error. In the imperfect-model-structure run, the parameter search diverges, although it can simulate the state variable well. This result suggest that, good data assimilation performance in estimating state variables is not a sufficient indicator of reliable parameter retrieval in the presence of model structural error. The generality of this conclusion needs to be tested by data assimilation experiments with more complex structural error configurations.