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.     

基于广义帕累托分布的马尼拉海沟俯冲带地震危险性估计

选取马尼拉海沟俯冲带作为潜源区, 基于广义帕累托分布, 通过对一定时段内超过某一阈值的震级数据进行拟合, 建立该潜源区地震危险性估计模型, 估计强震重现水平和震级上限, 并对估计结果的不确定性进行了分析, 得到马尼拉海沟俯冲带震级上限为9.0级, 10 a、 50 a、 100 a、 200 a马尼拉海沟俯冲带的震级重现水平期望值分别为7.1级、 7.6级、 7.7级、 7.9级。     

Trimmed L-moments (1,0) for the generalized Pareto distribution

Abstract

Statistical analysis of extremes is often used for predicting the higher return-period events. In this paper, the trimmed L-moments with one smallest value trimmed—TL-moments (1,0)—are introduced as an alternative way to estimate floods for high return periods. The TL-moments (1,0) have an ability to reduce the undesirable influence that a small value in the statistical sample might have on a large return period. The main objective of this study is to derive the TL-moments (1,0) for the generalized Pareto (GPA) distribution. The performance of the TL-moments (1,0) was compared with L-moments through Monte Carlo simulation based on the streamflow data of northern Peninsular Malaysia. The result shows that, for some cases, the use of TL-moments (1,0) is a better option as compared to L-moments in modelling those series.

Citation Ahmad, U.N., Shabri, A. & Zakaria, Z.A. (2011) Trimmed L-moments (1,0) for the generalized Pareto distribution. Hydrol.Sci. J. 56(6), 1053–1060.     

基于广义帕累托分布的地震震级分布尾部特征分析

极值理论在地震危险性分析中有着重要应用, 发震震级超过某一阈值的超出量分布可以近似为广义帕累托分布. 基于广义帕累托分布给出了若干地震活动性参数的估计公式, 包括强震震级分布、 地震复发周期和重现水平、 期望重现震级、 地震危险性概率和潜在震级上限等; 以云南地区震级资料为基础数据, 讨论了阈值选取、 模型拟合诊断和参数估计; 在此基础上计算了该地区的地震活动性参数. 结果表明, 广义帕累托分布较好地刻画了强震震级分布, 通过超阈值(POT)模型计算的复发周期与实际复发间隔统计基本一致, 高分位数估计在一定阈值范围内表现稳定, 为工程抗震中潜在震级上限的确定提供了一种途径.     

Selection between the generalized Pareto and kappa distributions in peaks-over-threshold hydrological frequency modelling

Hydrologists use the generalized Pareto (GP) distribution in peaks-over-threshold (POT) modelling of extremes. A model with similar uses is the two-parameter kappa (KAP) distribution. KAP has had fewer hydrological applications than GP, but some studies have shown it to merit wider use. The problem of choosing between GP and KAP arises quite often in frequency analyses. This study, by comparing some discrimination methods between these two models, aims to show which method(s) is (are) recommended. Three specific methods are considered: one uses the Anderson-Darling goodness-of-fit (GoF) statistic, another uses the ratio of maximized likelihood (closely related to the Akaike information criterion and the Bayesian information criterion), and the third employs a normality transformation followed by application of the Shapiro-Wilk statistic. We show this last method to be the most recommendable, due to its advantages with sizes typically encountered in hydrology. We apply the simulation results to some flood POT datasets.

EDITOR D. Koutsoyiannis; ASSOCIATE EDITOR E. Volpi     

基于广义帕累托分布的地震震级分布尾部特征分析

极值理论在地震危险性分析中有着重要应用,发震震级超过某一阈值的超出量分布可以近似为广义帕累托分布.基于广义帕累托分布给出了若干地震活动性参数的估计公式,包括强震震级分布、地震复发周期和重现水平、期望重现震级、地震危险性概率和潜在震级上限等;以云南地区震级资料为基础数据,讨论了阈值选取、模型拟合诊断和参数估计;在此基础上计算了该地区的地震活动性参数.结果表明,广义帕累托分布较好地刻画了强震震级分布,通过超阈值(POT)模型计算的复发周期与实际复发间隔统计基本一致,高分位数估计在一定阈值范围内表现稳定,为工程抗震中潜在震级上限的确定提供了一种途径.     

中国地震活动度分布特征

讨论了中国地区地震活动度S的分布特征。     

Source Parameters of Weak Crustal Earthquakes of the Vrancea Region from Short-period Waveform Inversion

High-frequency records of nine low magnitude shallow earthquakes of the Vrancea (Romania) seismic region are inverted for the seismic moment tensor (MT). An approach is suggested regarding how to obtain at least a rough estimate of the MT when the information on the structure of the crust is poor. Here simple 1-D layered models are used in the Greens function synthesis despite the fact that the structure of the region is undoubtedly very complex. Different 1-D models were used for different source-station paths to approximate lateral variations. Record of a station located on a ray path which crosses a structure differing substantially from a 1-D model may however bias the retrieved MT essentially. Therefore, we did not collectively process all records, but subsets of stations separately. We check the consistency of the MTs resulting from these individual bootstrap solutions, and reject those which differ substantially, assuming that this is due to the oversimplification of the forward modeling. Thereafter, the averaged moment tensor yielded by the consistent subsets is accepted. Moreover, the distribution of the T, N and P axes from the moment tensors used for averaging provides a rough estimate of their reliability.Following this simplistic procedure, we found seven acceptably constrained solutions among nine events processed. Their P axes are compared with the general trend of the stress in the area: three comply with it, the others should be attributed to the complex stress field active in the region.Acknowledgment This research has been made possible by MURST (40% and 60%), by UNESCO-IGCP project 414 Realistic Modelling of Seismic Input for Megacities and Large Urban Areas and NATO SfP 972266. Partial support issued from the Grant Agency of Czech Rep. (Grant 205/02/0383) and from the National Agency for Science, Technology and Innovation of Romania (Grant 6185 GR/2000).     

Distribution of apparent stress in western China

Introduction Seismological data (mainly focal mechanism data) has shown to be capable of providing important constraint on the geodynamical models of the eastern Asia in the perspective of stress orientation, stress state (compressional, tensional, or shear), and the source of stress (HUAN, et al, 1979; Xu, et al, 1992). One of the outstanding problems left is the stress level. In principle, within the framework of elasticity, it is impossible to obtain the information of the absolute stress…     

Study of the Non-Uniform Distribution of Earthquake Risk Within Potential Sources

Potential sources are aggregates of probable future epicenters.In this area,for source models currently,in common use for seismic risk analysis in China,the mean area of each potential source is about 3000-4000 km2.It is assumed that seismic risk has a uniform distribution within the range of each potential source,but studies have shown that the uniform distribution model to a large extent may give an underestimation of the seismic risk.In this paper,the relative distribution of historical epicenters in space within potential sources is discussed,a method is proposed to quantitatively describe the non-uniform distribution of strong earthquakes within potential sources,and some preliminary results are given.By using the results of this paper,seismic risk analysis and seismic zonation can be made more scientific and more reasonable.     

Linear combinations of order statistics to estimate the position and scale parameters of the Generalized Pareto distribution

 Ad hoc techniques for estimating the position and the scale parameters of the Generalized Pareto distribution are introduced. The estimators proposed are simple linear combinations of the order statistics: they provide valuable estimates of the parameters of interest, both when the shape parameter is known and when it is unknown (this latter case being of great relevance in practical applications), and show a good performance as well when the sample size is small. The procedures are tested on simulated data, and comparisons with other techniques are shown.     

广义极值分布在地震危险性分析中的应用

利用广义极值分布研究地震最大发震震级的规律,给出了广义极值分布下地震危险性分析的相关公式与方法,并对台湾地区历史地震资料进行极值统计分析,发现最大震级超过7级的地震理论发震次数与实际发震次数完全一致,在此基础上预测了未来几年发震的危险性.     

钢筋混凝土框架抗震结构地震能量分布及耗散研究

基于能量平衡原理,对多层钢筋混凝土框架结构的地震输入能量的分布及耗散规律进行了研究。选用8条天然地震波和2条人工波,运用Perform-3D软件,对多层钢筋混凝土框架结构模型在7度罕遇地震作用下的弹塑性能量进行数值仿真计算。计算了钢混框架结构在不同地震波下的地震总输入能量、滞回耗能、阻尼耗能以及滞回耗能占总耗能的比例时程,分析了地震能量在各分量中的分布及分配规律;分析了阻尼比和延性比对地震输入能量的影响,确定了滞回耗能随阻尼比和延性比的变化规律;研究了钢筋混凝土框架结构梁柱构造和竖向侧移刚度变化对地震输入能及其分量的影响,确定了多层钢筋混凝土框架结构滞回耗能沿竖向的分布规律及沿横向在框架构件中的分配,研究了框架结构存在薄弱层情况下的滞回耗能的分布规律。揭示了多自由度钢筋混凝土框架结构地震输入能量及其分布规律,可为基于能量平衡原理的抗震设计理论在工程实际中的运用提供有益的参考。     

An Anisotropic Attenuation Law of Macroseismic Intensity Performed on Virtual Intensity Distribution of Seismogenic Zones

An anisotropic attenuation law, based on an anisotropic characterization of intensity distribution for seismogenic zones, is proposed. This approach, that distinguishes itself for its consistency to the observed data, initially reconfigured by filtering procedures, is particularly suitable for seismic hazard evaluation.     

基于三参量威布尔分布的潜在震源区强震危险性估计

本文提出了一种基于三参量威布尔分布模型估计潜在震源区强震危险性的方法。选择日本东海-南海地震带为潜在震源区,分别基于强震发震时间间隔服从二参量和三参量的威布尔分布,估计该区强震危险性,结果表明三参量威布尔分布的拟合效果优于二参量威布尔分布。选择马尼拉海沟俯冲带为潜在震源区,基于三参量威布尔分布估计该区强震危险性,结果显示未来10、30和50年该区强震（M≧7.5）复发概率分别为62%、82%和89%,最短发震时间间隔估计为1.70年。     

Maximum product of spacings estimators for the generalized pareto and log-logistic distributions

The maximum product of spacings (MPS) method is discussed from the standpoint of information theory. MPS parameter and quantile estimates for the generalized Pareto distribution and the two parameter log-logistic distribution are compared with the maximum likelihood(ML) and probability weighted moment (PWM) estimates.     

Source parameters of the 2004 Kaliningrad earthquakes

An analysis of source parameters of the two unexpected earthquakes from the Kaliningrad (Russia) area is presented. The earthquakes occurred on 21 September 2004 at 11:05:01 and 13:32:31 UT, respectively. The first event was located at the latitude φ = 54.924°N and the longitude λ = 20.120°E, with a depth h = 16 km, and the second event at φ = 54.876°N, λ = 20.120°E and h = 20 km. Magnitudes Mw of the two events were very similar: 5.1 and 5.2. The magnitude values reported by various international data centers have been meaningfully different. The reason is the presence of high-frequency components in Z velocity component of the S wavefield. They were observed along the direction defined by two stations, BLEU in Sweden and SUW in Poland, located in opposite sides of the source. Along the direction perpendicular to it, the effects are relatively very small. The high-frequency waves are understood to mean components in the 6–8 Hz band for event 1 and 2-4 Hz for event 2. The effects in question are also clearly visible on displacement spectrograms. The magnitude values calculated at such stations from S-wave amplitudes or from seismic spectra are clearly overestimated and are close to 6. Therefore, we made a careful selection of channels in order to determine the spectral parameters and, on this basis, the source parameters. The size of the source is relatively small, of about 2 km. The closest seismic station is at 100 source radii from the source. One can clearly see the effect of the TT zone which markedly reduces the seismic moment value for seismic stations laying on the opposite sides of the source. Both events have very similar spatial distributions of the source parameters: magnitude, seismic moment and radius.     

2000年上半年全球地震活动

2000年上半年全球共发生MW≥6．5地震14次。其中MW≥7．0地震7次；浅源地震10次；中源地震3次；深源地震1次。浅源大地震发生的区域集中于印度尼西亚和南印度洋地区。6月份发生的地震所释放的地震矩占上半年全球所有地震释放的地震矩的一大半。     

2000年下半年全球地震活动

2000年下半年全球共发生Mw≥6．5地震27次，其中23次为浅源地震，深源地震4次。发生Mw≥7．0地震5次，最大震级达Mw8．0。11月16日和17日，在俾斯麦群岛和所罗门群岛交界地区，发生了3次Mw≥7．8地震。2000年，全球地震活动处于1976年以来的较高水平。     

2002年上半年全球地震活动

2002年上半年，全球发生MW≥6．5地震16次，其中13次浅源地震，2次中源地震，1次深源地震；发生MW≥7．0地震6次。上半年MW≥7．0地震的地震矩释放是1976年以来的偏低水平。中国及邻区地震活动比较活跃：3月3日，在兴都库什地区发生了1次MW7．3中源地震；6月28日，在中国吉林汪清发生了1次MW7．3深源地震。统计检验表明，地震矩释放表现出持久性质。