A note of caution when interpreting parameters of the distribution of excesses

In climatology and hydrology, univariate Extreme Value Theory has become a powerful tool to model the distribution of extreme events. The Generalized Pareto Distribution (GPD) is routinely applied to model excesses in space or time by letting the two GPD parameters depend on appropriate covariates. Two possible pitfalls of this strategy are the modeling and the interpretation of the scale and shape GPD parameters estimates which are often and incorrectly viewed as independent variables. In this note we first recall a statistical technique that makes the GPD estimates less correlated within a Maximum Likelihood (ML) estimation approach. In a second step we propose novel reparametrizations for two method-of-moments particularly popular in hydrology: the Probability Weighted Moment (PWM) method and its generalized version (GPWM). Finally these three inference methods (ML, PWM and GPWM) are compared and discussed with respect to the issue of correlations.     

Frequency analysis of climate extreme events in Zanjan, Iran

In this study, generalized extreme value distribution (GEV) and generalized Pareto distribution (GPD) were fitted to the maximum and minimum temperature, maximum wind speed, and maximum precipitation series of Zanjan. Maximum (minimum) daily and absolute annual observations of Zanjan station from 1961 to 2011 were used. The parameters of the distributions were estimated using the maximum likelihood estimation method. Quantiles corresponding to 2, 5, 10, 25, 50, and 100 years return periods were calculated. It was found that both candidate distributions fitted to extreme events series, were statistically reasonable. Most of the observations from 1961 to 2011 were found to fall within 1–10 years return period. Low extremal index (θ) values were found for excess maximum and minimum temperatures over a high threshold, indicating the occurrence of consecutively high peaks. For the purpose of filtering the dependent observations to obtain a set of approximately independent threshold excesses, a declustering method was performed, which separated the excesses into clusters, then the de-clustered peaks were fitted to the GPD. In both models, values of the shape parameters of extreme precipitation and extreme wind speed were close to zero. The shape parameter was less negative in the GPD than the GEV. This leads to significantly lower return period estimates for high extremes with the GPD model.     

Probability estimation of rare extreme events in the case of small samples: Technique and examples of analysis of earthquake catalogs

The most general approach to studying the recurrence law in the area of the rare largest events is associated with the use of limit law theorems of the theory of extreme values. In this paper, we use the Generalized Pareto Distribution (GPD). The unknown GPD parameters are typically determined by the method of maximal likelihood (ML). However, the ML estimation is only optimal for the case of fairly large samples (>200–300), whereas in many practical important cases, there are only dozens of large events. It is shown that in the case of a small number of events, the highest accuracy in the case of using the GPD is provided by the method of quantiles (MQs). In order to illustrate the obtained methodical results, we have formed the compiled data sets characterizing the tails of the distributions for typical subduction zones, regions of intracontinental seismicity, and for the zones of midoceanic (MO) ridges. This approach paves the way for designing a new method for seismic risk assessment. Here, instead of the unstable characteristics—the uppermost possible magnitude M_max—it is recommended to use the quantiles of the distribution of random maxima for a future time interval. The results of calculating such quantiles are presented.     

A Bayesian analysis of Generalized Pareto Distribution of runoff minima

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

青藏高原东北缘地震活动性广义帕累托模型的全域敏感性分析

基于广义帕累托分布构建地震活动性模型,因其输入参数取值难以避免不确定性,导致依据该模型所得的地震危险性估计结果具有不确定性。鉴于此,本文选取青藏高原东北缘为研究区,提出了基于全域敏感性分析的地震危险性估计的不确定性分析流程和方法。首先,利用地震活动性广义帕累托模型,进行研究区地震危险性估计;然后,选取地震记录的起始时间和震级阈值作为地震活动性模型的输入参数,采用具有全域敏感性分析功能的E-FAST方法,对上述两个参数的不确定性以及两参数之间的相互作用对地震危险性估计不确定性的影响进行定量分析。结果表明：地震危险性估计结果（不同重现期的震级重现水平、震级上限及相应的置信区间）对两个输入参数中的震级阈值更为敏感;不同重现期的地震危险性估计结果对震级阈值的敏感程度不同;对不同的重现期而言,在影响地震危险性估计结果的不确定性上,两个输入参数之间存在非线性效应,且非线性效应程度不同。本文提出的不确定性分析流程和方法,可以推广应用于基于其它类型地震活动性模型的地震危险性估计不确定性分析。     

Types of probability distributions in the evaluation of extreme floods

The development of an optimal scheme for evaluation of maximal water discharges is discussed, including adequate probability distribution laws, an effective procedure for their approximation based on observational data, and reliable goodness-of-fit tests for analytical and empirical distributions. One-dimensional probability distribution laws are systematized. Promising distributions were identified, including generalized distribution of extreme values, lognormal distribution, Pearson type V power distribution, and GPD, for evaluating maximal discharges. The available methods for approximating analytical curves, including the up-to-date method of L-moments are considered. Parameter estimation algorithm based on L-moment method for Pearson type III distribution is considered. Pearson type III distribution, lognormal distribution, GEV, and GPD are compared in the approximation of maximal water discharges in rivers of Austria, Siberia, Far East, and the Hawaiian Islands.     

Characterization of the Frequency of Extreme Earthquake Events by the Generalized Pareto Distribution

Recent results in extreme value theory suggest a new technique for statistical estimation of distribution tails (Embrechts et al., 1997), based on a limit theorem known as the Gnedenko-Pickands-Balkema-de Haan theorem. This theorem gives a natural limit law for peak-over-threshold values in the form of the Generalized Pareto Distribution (GPD), which is a family of distributions with two parameters. The GPD has been successfully applied in a number of statistical problems related to finance, insurance, hydrology, and other domains. Here, we apply the GPD approach to the well-known seismological problem of earthquake energy distribution described by the Gutenberg-Richter seismic moment-frequency law. We analyze shallow earthquakes (depth h<70 km) in the Harvard catalog over the period 1977–2000 in 12 seismic zones. The GPD is found to approximate the tails of the seismic moment distributions quite well over the lower threshold approximately M 10²⁴ dyne-cm, or somewhat above (i.e., moment-magnitudes larger than m _W =5.3). We confirm that the b-value is very different (b=2.06 ± 0.30) in mid-ocean ridges compared to other zones (b=1.00 ± 0.04) with a very high statistical confidence and propose a physical mechanism contrasting crack-type rupture with dislocation-type behavior. The GPD can as well be applied in many problems of seismic hazard assessment on a regional scale. However, in certain cases, deviations from the GPD at the very end of the tail may occur, in particular for large samples signaling a novel regime.     

Frequency analysis of precipitation extremes in Heihe River basin based on generalized Pareto distribution

Precipitation extremes could cause a series of social, environmental and ecological problems. This paper, taking Heihe River basin, the second largest inland river basin in China, as the study area, focused on the frequency analysis of precipitation extremes based on the historical daily precipitation records (1960–2010) at nine stations. Generalized Pareto distribution (GPD) was employed for fitting the peaks over threshold (POT) series, in which Hill plot, percentile method and the average annual occurrence number were used to select the threshold in GPD. Maximum likelihood estimate and L-moment were used to estimate the parameters. The inherent assumptions for POT series were investigated by auto-correlation coefficient, Mann–Kendall test, Spearman’s ρ test, cumulative deviation test and Worsley likelihood ratio test. 10, 20, 50 and 100 year precipitation extremes for Heihe River basin were calculated and analyzed as well. It was found the POT series derived from several methods involved were approximately independent and stationary, and GPD could give a satisfactory fit to the POT series for each station. For the upper and lower reaches, the frequency of precipitation extremes at long return periods (20, 50 year or longer) presented increasing in recent years, and the intensity of the highest precipitation were getting stronger as well. The intensity of the highest precipitation extremes for the lower reach (21 and 35 %) increased higher than those for the upper reach (10 and 11 %). For the middle reach, the frequency of precipitation extremes (over 20 year return level) was not found to be increased. The uneven spatial and temporal distribution of precipitation extremes for the basin especially for the upper and lower reaches were getting more and more serious, which would bring great challenges for the local water allocation and management.     

Comment on Pisarenko et al., “Characterization of the Tail of the Distribution of Earthquake Magnitudes by Combining the GEV and GPD Descriptions of Extreme Value Theory”

In this short note, I comment on the research of Pisarenko et al. (Pure Appl. Geophys 171:1599–1624, 2014) regarding the extreme value theory and statistics in the case of earthquake magnitudes. The link between the generalized extreme value distribution (GEVD) as an asymptotic model for the block maxima of a random variable and the generalized Pareto distribution (GPD) as a model for the peaks over threshold (POT) of the same random variable is presented more clearly. Inappropriately, Pisarenkoet al. (Pure Appl. Geophys 171:1599–1624, 2014) have neglected to note that the approximations by GEVD and GPD work only asymptotically in most cases. This is particularly the case with truncated exponential distribution (TED), a popular distribution model for earthquake magnitudes. I explain why the classical models and methods of the extreme value theory and statistics do not work well for truncated exponential distributions. Consequently, these classical methods should be used for the estimation of the upper bound magnitude and corresponding parameters. Furthermore, I comment on various issues of statistical inference in Pisarenkoet al. and propose alternatives. I argue why GPD and GEVD would work for various types of stochastic earthquake processes in time, and not only for the homogeneous (stationary) Poisson process as assumed by Pisarenko et al. (Pure Appl. Geophys 171:1599–1624, 2014). The crucial point of earthquake magnitudes is the poor convergence of their tail distribution to the GPD, and not the earthquake process over time.     

Risk assessment of extreme air pollution based on partial duration series: IDF approach

The occurrences of extreme pollution events have serious effects on human health, environmental ecosystems, and the national economy. To gain a better understanding of this issue, risk assessments on the behavior of these events must be effectively designed to anticipate the likelihood of their occurrence. In this study, we propose using the intensity–duration–frequency (IDF) technique to describe the relationship of pollution intensity (i) to its duration (d) and return period (T). As a case study, we used data from the city of Klang, Malaysia. The construction of IDF curves involves a process of determining a partial duration series of an extreme pollution event. Based on PDS data, a generalized Pareto distribution (GPD) is used to represent its probabilistic behaviors. The estimated return period and IDF curves for pollution intensities corresponding to various return periods are determined based on the fitted GPD model. The results reveal that pollution intensities in Klang tend to increase with increases in the length of time between return periods. Although the IDF curves show different magnitudes for different return periods, all the curves show similar increasing trends. In fact, longer return periods are associated with higher estimates of pollution intensity. Based on the study results, we can conclude that the IDF approach provides a good basis for decision-makers to evaluate the expected risk of future extreme pollution events.     

The instability of the <Emphasis Type="Italic">M</Emphasis><Subscript>max</Subscript> parameter and an alternative to its using

The work presents statistical methods for estimating the distribution parameters of rare, strong earthquakes. Using the two main theorems of extreme value theory (EVT), the distribution of T-maximum (the maximum magnitude over the time period T). Two methods for estimating the parameters of this distribution are proposed using the Generalized Pareto Distribution (GPD) and the General Extreme Value Distribution (GEV). In addition, the that allow the determination of the distribution of the T-maximum for an arbitrary value of T are proposed. The approach being used clarifies the nature of the instability of the widely accepted M max parameter. In the work, instead of unstable values of the M max parameter, the robust parameter Q _T (q), the q level quantile for the distribution of the T-maximum, is proposed to be used. The described method has been applied to the Harvard Catalogue of Seismic Moments of 1977–2006 and to the Magnitude Catalogue for Fennoscandia in 1900–2005. Moreover, the estimates of parameters of the corresponding GPD and GEV distributions, in particular, the most interesting shape parameter and the values of the M _max and Q _T (q) parameters are given.     

Non-stationary frequency analysis of extreme precipitation in South Korea using peaks-over-threshold and annual maxima

The conventional approach to the frequency analysis of extreme precipitation is complicated by non-stationarity resulting from climate variability and change. This study utilized a non-stationary frequency analysis to better understand the time-varying behavior of short-duration (1-, 6-, 12-, and 24-h) precipitation extremes at 65 weather stations scattered across South Korea. Trends in precipitation extremes were diagnosed with respect to both annual maximum precipitation (AMP) and peaks-over-threshold (POT) extremes. Non-stationary generalized extreme value (GEV) and generalized Pareto distribution (GPD) models with model parameters made a linear function of time were applied to AMP and POT respectively. Trends detected using the Mann–Kendall test revealed that the stations showing an increasing trend in AMP extremes were concentrated in the mountainous areas (the northeast and southwest regions) of South Korea. Trend tests on POT extremes provided fairly different results, with a significantly reduced number of stations showing an increasing trend and with some stations showing a decreasing trend. For most of stations showing a statistically significant trend, non-stationary GEV and GPD models significantly outperformed their stationary counterparts, particularly for precipitation extremes with shorter durations. Due to a significant-increasing trend in the POT frequency found at a considerable number of stations (about 10 stations for each rainfall duration), the performance of modeling POT extremes was further improved with a non-homogeneous Poisson model. The large differences in design storm estimates between stationary and non-stationary models (design storm estimates from stationary models were significantly lower than the estimates of non-stationary models) demonstrated the challenges in relying on the stationary assumption when planning the design and management of water facilities. This study also highlighted the need of caution when quantifying design storms from POT and AMP extremes by showing a large discrepancy between the estimates from those two approaches.     

Emulation of simulated earthquake catalogues

In earthquake occurrence studies, the so-called q value can be considered both as one of the parameters describing the distribution of interevent times and as an index of non-extensivity. Using simulated datasets, we compare four kinds of estimators, based on principle of maximum entropy (POME), method of moments (MOM), maximum likelihood (MLE), and probability weighted moments (PWM) of the parameters (q and τ ₀) of the distribution of inter-events times, assumed to be a generalized Pareto distribution (GPD), as defined by Tsallis (1988) in the frame of non-extensive statistical physics. We then propose to use the unbiased version of PWM estimators to compute the q value for the distribution of inter-event times in a realistic earthquake catalogue simulated according to the epidemic type aftershock sequence (ETAS) model. Finally, we use these findings to build a statistical emulator of the q values of ETAS model. We employ treed Gaussian processes to obtain partitions of the parameter space so that the resulting model respects sharp changes in physical behaviour. The emulator is used to understand the joint effects of input parameters on the q value, exploring the relationship between ETAS model formulation and distribution of inter-event times.     

Characterization of the Tail of the Distribution of Earthquake Magnitudes by Combining the GEV and GPD Descriptions of Extreme Value Theory

The present work is a continuation and improvement of the method suggested in Pisarenko et al. (Pure Appl Geophys 165:1–42, 2008) for the statistical estimation of the tail of the distribution of earthquake sizes. The chief innovation is to combine the two main limit theorems of Extreme Value Theory (EVT) that allow us to derive the distribution of T-maxima (maximum magnitude occurring in sequential time intervals of duration T) for arbitrary T. This distribution enables one to derive any desired statistical characteristic of the future T-maximum. We propose a method for the estimation of the unknown parameters involved in the two limit theorems corresponding to the Generalized Extreme Value distribution (GEV) and to the Generalized Pareto Distribution (GPD). We establish the direct relations between the parameters of these distributions, which permit to evaluate the distribution of the T-maxima for arbitrary T. The duality between the GEV and GPD provides a new way to check the consistency of the estimation of the tail characteristics of the distribution of earthquake magnitudes for earthquake occurring over an arbitrary time interval. We develop several procedures and check points to decrease the scatter of the estimates and to verify their consistency. We test our full procedure on the global Harvard catalog (1977–2006) and on the Fennoscandia catalog (1900–2005). For the global catalog, we obtain the following estimates: \( \hat{M}_{{\rm max} } \)  = 9.53 ± 0.52 and \( \hat{Q}_{10} (0.97) \)  = 9.21 ± 0.20. For Fennoscandia, we obtain \( \hat{M}_{{\rm max} } \)  = 5.76 ± 0.165 and \( \hat{Q}_{10} (0.97) \)  = 5.44 ± 0.073. The estimates of all related parameters for the GEV and GPD, including the most important form parameter, are also provided. We demonstrate again the absence of robustness of the generally accepted parameter characterizing the tail of the magnitude-frequency law, the maximum possible magnitude M _max, and study the more stable parameter Q _T (q), defined as the q-quantile of the distribution of T-maxima on a future interval of duration T.     

Robust numerical solution of the reservoir routing equation

The robustness of numerical methods for the solution of the reservoir routing equation is evaluated. The methods considered in this study are: (1) the Laurenson–Pilgrim method, (2) the fourth-order Runge–Kutta method, and (3) the fixed order Cash–Karp method. Method (1) is unable to handle nonmonotonic outflow rating curves. Method (2) is found to fail under critical conditions occurring, especially at the end of inflow recession limbs, when large time steps (greater than 12 min in this application) are used. Method (3) is computationally intensive and it does not solve the limitations of method (2). The limitations of method (2) can be efficiently overcome by reducing the time step in the critical phases of the simulation so as to ensure that water level remains inside the domains of the storage function and the outflow rating curve. The incorporation of a simple backstepping procedure implementing this control into the method (2) yields a robust and accurate reservoir routing method that can be safely used in distributed time-continuous catchment models.     

用于弹性波方程数值模拟的有限差分系数确定方法

有限差分方法是波场数值模拟的一个重要方法,交错网格差分格式比规则网格差分格式稳定性更好,但方法本身都存在因网格化而形成的数值频散效应,这会降低波场模拟的精度与分辨率.为了缓解有限差分算子的数值频散效应,精确求解空间偏导数,本文把求解波动方程的线性化方法推广到用于求解弹性波方程交错网格有限差分系数;同时应用最大最小准则作为模拟退火(SA)优化算法求解差分系数的数值频散误差判定标准来求解有限差分系数.通过上述两种方法,分别利用均匀各向同性介质和复杂构造模型进行了数值正演模拟和数值频散分析,并与传统泰勒展开算法、最小二乘算法进行比较,验证了线性化方法和模拟退火方法都能有效压制数值频散,并比较了各个算法的特点.     

Parametric uncertainty assessment of hydrological models: coupling UNEEC-P and a fuzzy general regression neural network

Due to the complicated nature of environmental processes, consideration of uncertainty is an important part of environmental modelling. In this paper, a new variant of the machine learning-based method for residual estimation and parametric model uncertainty is presented. This method is based on the UNEEC-P (UNcertainty Estimation based on local Errors and Clustering – Parameter) method, but instead of multilayer perceptron uses a “fuzzified” version of the general regression neural network (GRNN). Two hydrological models are chosen and the proposed method is used to evaluate their parametric uncertainty. The approach can be classified as a hybrid uncertainty estimation method, and is compared to the group method of data handling (GMDH) and ordinary kriging with linear external drift (OKLED) methods. It is shown that, in terms of inherent complexity, measured by Akaike information criterion (AIC), the proposed fuzzy GRNN method has advantages over other techniques, while its accuracy is comparable. Statistical metrics on verification datasets demonstrate the capability and appropriate efficiency of the proposed method to estimate the uncertainty of environmental models.     

用于弹性波模拟的交错网格预处理和多次校正的Chebyshev谱元法

本文基于弹性波动方程,从其弱形式出发,利用Galerkin变分原理,通过对方程进行空间和时间上的离散,在空间域中引入预条件共轭梯度的逐元算法,在时间域中引入时间积分的交错网格预处理/多次校正算法,发展了弹性波模拟的Chebyshev谱元算法。针对均匀固体介质和具有倾斜分层的分区均匀固体介质模型,通过与有限差分算法结果相比较验证其精度的可信性,同时利用该算法模拟了弹性波在具有水平分层的任意起伏自由表面模型中的传播,并分析了其传播特点。研究表明,我们提出的交错网格预处理/多次校正算法的Chebyshev谱元算法,保留了有限元法的优势,并且采用了具有最优张量乘积技术的元到元的算法,能够处理带有起伏自由表面的复杂介质模型,它具有比有限元法收敛快,计算效率较高等优点,特别适合于复杂结构和复杂介质中的弹性波传播的数值模拟。     

An operational modal analysis method in frequency and spatial domain

A frequency and spatial domain decomposition method （FSDD） for operational modal analysis （OMA） is presented in this paper, which is an extension of the complex mode indicator function （CMIF） method for experimental modal analysis （EMA）. The theoretical background of the FSDD method is clarified, Singular value decomposition is adopted to separate the signal space from the noise space. Finally, an enhanced power spectrum density （PSD） is proposed to obtain more accurate modal parameters by curve fitting in the frequency domain. Moreover, a simulation case and an application case are used to validate this method.     

The four dimensional variational data assimilation with multiple regularization parameters as a weak constraint (Tikh-4D-Var) and its preliminary application on typhoon initialization

Traditional variational data assimilation (VDA) with only one regularization parameter constraint cannot produce optimal error tuning for all observations. In this paper, a new data assimilation method of “four dimensional variational data assimilation (4D-Var) with multiple regularization parameters as a weak constraint (Tikh-4D-Var)” is proposed by imposing different regularization parameters for different observations. Meanwhile, a new multiple regularization parameters selection method, which is suitable for actual high-dimensional data assimilation system, is proposed based on the posterior information of 4D-Var system. Compared with the traditional single regularization parameter selection method, computation of the proposed multiple regularization parameters selection method is smaller. Based on WRF3.3.1 4D-Var data assimilation system, initialization and simulation of typhoon Chaba (2010) with the new Tikh-4D-Var method are compared with its counterpart 4D-Var to demonstrate the effectiveness of the new method. Results show that the new Tikh-4D-Var method can accelerate the convergence with less iterations. Moreover, compared with 4D-Var method, the typhoon track, intensity (including center surface pressure and maximum wind speed) and structure prediction are obviously improved with Tikh-4D-Var method for 72-h prediction. In addition, the accuracy of the observation error variances can be reflected by the multiple regularization parameters.