Statistical Variability and Tokunaga Branching of Aftershock Sequences Utilizing BASS Model Simulations

Aftershock statistics provide a wealth of data that can be used to better understand earthquake physics. Aftershocks satisfy scale-invariant Gutenberg–Richter (GR) frequency–magnitude statistics. They also satisfy Omori’s law for power-law seismicity rate decay and Båth’s law for maximum-magnitude scaling. The branching aftershock sequence (BASS) model, which is the scale-invariant limit of the epidemic-type aftershock sequence model (ETAS), uses these scaling laws to generate synthetic aftershock sequences. One objective of this paper is to show that the branching process in these models satisfies Tokunaga branching statistics. Tokunaga branching statistics were originally developed for drainage networks and have been subsequently shown to be valid in many other applications associated with complex phenomena. Specifically, these are characteristic of a universality class in statistical physics associated with diffusion-limited aggregation. We first present a deterministic version of the BASS model and show that it satisfies the Tokunaga side-branching statistics. We then show that a fully stochastic BASS simulation gives similar results. We also study foreshock statistics using our BASS simulations. We show that the frequency–magnitude statistics in BASS simulations scale as the exponential of the magnitude difference between the mainshock and the foreshock, inverse GR scaling. We also show that the rate of foreshock occurrence in BASS simulations decays inversely with the time difference between foreshock and mainshock, an inverse Omori scaling. Both inverse scaling laws have been previously introduced empirically to explain observed foreshock statistics. Observations have demonstrated both of these scaling relations to be valid, consistent with our simulations. ETAS simulations, in general, do not generate Båth’s law and do not generate inverse GR scaling.     

序列衰减与余震激发研究进展及应用成果

着重于序列衰减与余震激发,系统介绍了修改的大森公式、ETAS模型及BASS模型的最新理论研究进展及应用成果。修改的大森公式是迄今为止对序列衰减的最好描述,据此可对序列衰减特征进行定量表述。大森公式本质上是一种典型的现象统计模型,但由于其参数少、计算简单,并且确实能够反映序列衰减的总体特征,因而在实际中应用广泛。ETAS模型考虑了具有统计自相似特征的次级余震激发问题,次级余震激发强度与父地震强度有关,这在物理过程方面对大森公式进行了大大的拓展。由于考虑了次级余震的激发问题,ETAS模型不但在余震序列研究方面比修改的大森公式有了明显的进展,而且在诸如平静检测、余震群集剔除、背景地震活动评估、外因触发地震活动检测等方面也有诸多应用。BASS模型遵循修改的Bath定律,而ETAS模型遵循的是与父地震震级有关的相似率,这是BASS模型与ETAS模型的最大区别,因而相对ETAS模型而言,BASS模型是一种完全自相似的理想化模型,但目前基于BASS模型的应用研究尚不多见     

2008年和2014年于田地震对周边断层发震概率的影响

大震后区域静态库仑应力变化直接影响地震活动性速率的变化、主震断层外余震和即将失稳断层的发震概率的变化.利用滑移速率和状态相依赖的摩擦定律,结合2008年3月21日于田地震前后的地震活动性水平,定量计算了2008年于田地震后该地区周边断层发震概率的变化,着重解释了2014年于田地震发震的可能根源.此外,本文还对库仑应力明显变化的周边三条断层进行了发震概率的定量计算.贡嘎错断裂中段、贡嘎错断裂西南段和康西瓦断裂中段分别经历了发震概率先降后升、先升后升和先降后降两个阶段,充分显示了库仑应力的细微变化造成的周边断层的危险性的变化.这三条断裂发生7.0级以上地震的发震概率超越95%均需要500年左右;贡嘎错断裂西南段发生中强地震的可能性较大,而康西瓦断裂中段活跃度较低.     

地震的"序列归属"问题与ETAS模型——以唐山序列为例

历史上发生过强震地区的余震活动可能持续较长时间,而余震序列在何时可被看作正常的"背景地震活动",即"序列归属"问题在地球动力学和地震物理中有重要意义.时-空"传染型余震序列"(ETAS)模型可分离"背景"地震和"丛集"地震,并用概率形式表示作为相应事件的可能性,为考察此问题提供了可能.本文以1976年唐山M_S7.8地震序列为例,对唐山地区1970年以来的M_L4.0以上地震进行了时-空ETAS模型拟合,并以2010年以来发生的3次M_S4.0以上地震为例讨论了它们的"序列归属"问题.研究结果显示,3次M_S4.0以上地震的背景地震概率分别为0.72、0.88和0.76,表明它们作为1976年唐山M_S7.8的余震的可能性较低,更可能为背景地震.     

基于经验模型和物理模型研究2013 M_S7.0芦山地震余震序列

基于中国地震台网中心2013 MS7.0芦山地震余震数据我们首先确定了余震空间分布范围并根据G-R关系计算了主震后半小时内的完备震级Mc=3.5,并且得到了ML≥3.5和ML≥3.0的地震在2001年至芦山地震前的背景场地震发生率.通过Omori-Ustu经验定律和两种Dieterich模型对芦山地震余震发生率的拟合,我们发现阶梯型Dieterich模型只能模拟p=1的情况,从而造成了模拟曲线与观测数据的差别;前人研究表明震后滑移同样是产生余震的原因,如果假设余震序列由主震静态剪应力Δτ和震后滑移共同作用所产生,我们数值模拟得到的对数型Dieterich模型能够较好地推断余震发生率R随时间t增加而衰减的趋势,能够从物理机制上解释MS7.0芦山地震余震序列衰减指数大于1这一现象.通过对数型Dieterich模型的拟合并结合Andrews的方法,我们还得到MS7.0芦山地震Aσ约为0.155 MPa,ta约为8.4年,这一值与前人研究结果十分接近.     

Forecasting aftershock activity: 1. Adaptive estimates based on the Omori and Gutenberg–Richter laws

The method for forecasting the intensity of the aftershock processes after strong earthquakes in different magnitude intervals is considered. The method is based on the joint use of the time model of the aftershock process and the Gutenberg–Richter law. The time model serves for estimating the intensity of the aftershock flow with a magnitude larger than or equal to the magnitude of completeness. The Gutenberg–Richter law is used for magnitude scaling. The suggested approach implements successive refinement of the parameters of both components of the method, which is the main novelty distinguishing it from the previous ones. This approach, to a significant extent, takes into account the variations in the parameters of the frequency–magnitude distribution, which often show themselves by the decreasing fraction of stronger aftershocks with time. Testing the method on eight aftershock sequences in the regions with different patterns of seismicity demonstrates the high probability of successful forecasts. The suggested technique can be employed in seismological monitoring centers for forecasting the aftershock activity of a strong earthquake based on the results of operational processing.     

Aftershock sequence simulations using synthetic earthquakes and rate-state seismicity formulation

We use an efficient earthquake simulator that incorporates rate-state constitutive properties and uses boundary element method to discretize the fault surfaces, to generate the synthetic earthquakes in the fault system. Rate-and-state seismicity equation is subsequently employed to calculate the seismicity rate in a region of interest using the Coulomb stress transfer from the main shocks in the fault system. The Coulomb stress transfer is obtained by resolving the induced stresses due to the fault patch slips onto the optimal-oriented fault planes. The example results show that immediately after a main shock the aftershocks are concentrated in the vicinity of the rupture area due to positive stress transfers and then disperse away into the surrounding region toward the background rate distribution. The number of aftershocks near the rupture region is found to decay with time as Omori aftershock decay law predicts. The example results demonstrate that the rate-and-state fault system earthquake simulator and the seismicity equations based on the rate-state friction nucleation of earthquake are well posited to characterize the aftershock distribution in regional assessments of earthquake probabilities.     

Aftershock Statistics of the 1999 Chi–Chi, Taiwan Earthquake and the Concept of Omori Times

In this paper we consider the statistics of the aftershock sequence of the m = 7.65 20 September 1999 Chi–Chi, Taiwan earthquake. We first consider the frequency-magnitude statistics. We find good agreement with Gutenberg–Richter scaling but find that the aftershock level is anomalously high. This level is quantified using the difference in magnitude between the main shock and the largest inferred aftershock $ {{\Updelta}}m^{ *}. $ Typically, $ {{\Updelta}}m^{ *} $ is in the range 0.8–1.5, but for the Chi–Chi earthquake the value is $ {{\Updelta}}m^{ *} $  = 0.03. We suggest that this may be due to an aseismic slow-earthquake component of rupture. We next consider the decay rate of aftershock activity following the earthquake. The rates are well approximated by the modified Omori’s law. We show that the distribution of interoccurrence times between aftershocks follow a nonhomogeneous Poisson process. We introduce the concept of Omori times to study the merging of the aftershock activity with the background seismicity. The Omori time is defined to be the mean interoccurrence time over a fixed number of aftershocks.     

基于时-空ETAS模型给出的川滇地区背景地震活动和强震潜在危险区

利用基于时-空传染型余震序列（Epidemic Type Aftershock Sequence, 简称ETAS）模型的随机除丛法,重新审视了2008年5月12日汶川M_S8.0地震前可能存在的长期地震活动异常,研究了川滇地区背景地震活动特征,并评估了当前的强震危险状态.对川滇地区1970年以来的M_L3.0以上的背景地震和丛集地震活动的研究结果表明,该地区地震丛集特征明显、时空分布很不均匀、地震序列常有前震事件.直接将概率值作为地震计数的权重,对地震丛集率空间分布图像分析表明,汶川M_S8.0地震前,龙门山断裂带中南段存在着长期、大范围的地震丛集率低值区,震前该段处于应力闭锁状态.对川滇地区地震丛集率低值区内背景地震与全部地震的累积次数、b值和新定义的Δb等统计参量的分析表明,龙日坝与龙门山断裂带具有地震活动的关联性,川滇地区当前的强震潜在危险区可能是巧家地区和汶川M_S8.0地震破裂尚未穿越的龙门山断裂带南段.此外,还发现b值倾向于反映局部应力场变化,而Δb能较为敏感地给出更大范围应力场的相对变化.     

A Damage Mechanics Model for Aftershocks

— All earthquakes are followed by an aftershock sequence. A universal feature of aftershock sequences is that they decay in time according to the modified Omori’s law, a power-law decay. In this paper we consider the applicability of damage mechanics to earthquake aftershocks. The damage variable introduced in damage mechanics quantifies the deviation of a brittle solid from linear elasticity. We draw an analogy between the metastable behavior of a stressed brittle solid and the metastable behavior of a superheated liquid. The nucleation of microcracks is analogous to the nucleation of bubbles in the superheated liquid. In this paper we obtain a solution for the evolution of damage after the instantaneous application of a constant strain to a rod. We show that the subsequent stress relaxation can reproduce the modified Omori’s law. It is argued that the aftershocks themselves cause random fluctuations similar to the thermal fluctuations associated with phase transitions.     

Preliminary study on aftershock decay rate of the 2013 M s7.0 Lushan earthquake

We obtained a catalog of early aftershocks of the 2013 Lushan earthquake by examining waveform from a nearby station MDS which is 30.2 km far away from the epicenter, and then we analyzed the relation between aftershock rate and time. We used time-window ratio method to identify aftershocks from continuous waveform data and compare the result with the catalog provided by China Earthquake Networks Center (CENC). As expected, a significant amount of earthquakes is missing in CENC catalog in the 24 h after the main shock. Moreover, we observed a steady seismicity rate of aftershocks nearly in the first 10,000 s before an obvious power-law decay of aftershock activity. We consider this distinct early stage which does not fit the Omori law with a constant p (p ~ 1) value as early aftershock deficiency (EAD), as proposed by previous studies. Our study suggests that the main shock rupture process is different from aftershocks’ processes, and EAD can vary in different cases as compared to earthquakes of strike-slip mechanism in California.     

唐山老震区地震活动特征

整理校正唐山长时间地震序列数据,用基于时空ETAS模型的随机除丛法,分析1976年唐山长时间地震序列的活动特点,发现唐山老震区1992年后进入余震活动的晚期阶段,目前地震活动未完全恢复到背景地震的水平。用频度和应变能两种不同的异常指标分析唐山老震区作为“余震窗口”对中强震的指示意义,发现唐山余震窗口的早期阶段（1980-1986年）,频次作异常指标映震效果较好,晚期阶段（1992-2010年）,应变能作为异常指标映震效果更好。     

山东乳山震群遗漏地震检测及震源区活动特征分析

对震群活动特征的深入研究可以为区域地震危险性分析和地震活动趋势判断提供有效的依据,但受台网布局和震群位置的影响,地震目录中往往会遗漏一些地震,而地震目录的完整性将会影响震群活动特征分析的可靠性.因此,本文利用基于GPU加速的模板匹配方法对山东乳山震群2014年5月至2015年6月期间固定台站和流动台阵记录的连续波形进行...     

Development of an aftershock occurrence model calibrated for Turkey and the resulting likelihoods

This paper presents the calibration of Omori's aftershock occurrence rate model for Turkey and the resulting likelihoods. Aftershock occurrence rate models are used for estimating the probability of an aftershock that exceeds a specific magnitude threshold within a time interval after the mainshock. Critical decisions on the post-earthquake safety of structures directly depend on the aftershock hazard estimated using the occurrence model. It is customary to calibrate models in a region-specific manner. These models depend on rate parameters(a, b, c and p) related to the seismicity characteristics of the investigated region. In this study, the available well-recorded aftershock sequences for a set of Mw ≥ 5.9 mainshock events that were observed in Turkey until 2012 are considered to develop the aftershock occurrence model. Mean estimates of the model parameters identified for Turkey are a =-1.90, b = 1.11, c = 0.05 and p = 1.20. Based on the developed model, aftershock likelihoods are computed for a range of different time intervals and mainshock magnitudes. Also, the sensitivity of aftershock probabilities to the model parameters is investigated. Aftershock occurrence probabilities estimated using the model are expected to be useful for post-earthquake safety evaluations in Turkey.     

Spatio-temporal characteristics of aftershocks and seismogenic structure of the 2011 MW9.0 Tohoku earthquake,Japan

The Tohoku megathrust earthquake, which occurred on March 11, 2011 and had an epicenter that was 70 km east of Tohoku, Japan, resulted in an estimated ten's of billions of dollars in damage and a death toll of more than 15 thousand lives, yet few studies have documented key spatio-temporal seismogenic characteristics. Specifically, the temporal decay of aftershock activity, the number of strong aftershocks (with magnitudes greater than or equal to 7.0), the magnitude of the greatest aftershock, and area of possible aftershocks. Forecasted results from this study are based on Gutenberg-Richter's relation, Bath's law, Omori's law, and Well's relation of rupture scale utilizing the magnitude and statistical parameters of earthquakes in USA and China (Landers, Northridge, Hector Mine, San Simeon and Wenchuan earthquakes). The number of strong aftershocks, the parameters of Gutenberg-Richter's relation, and the modified form of Omori's law are confirmed based on the aftershock sequence data from the M_W9.0 Tohoku earthquake. Moreover, for a large earthquake, the seismogenic structure could be a fault, a fault system, or an intersection of several faults. The seismogenic structure of the earthquake suggests that the event occurred on a thrust fault near the Japan trench within the overriding plate that subsequently triggered three or more active faults producing large aftershocks.     

2019年6月17日四川长宁MS6.0地震早期序列参数分析及强余震概率预测

2019年6月17日四川长宁发生M_S6.0地震, 之后发生了一系列的强余震, 为更好地分析此次地震的序列特征以及强余震的可预测属性, 采用国际上对复杂序列拟合相对较好的“传染型余震序列”(ETAS)模型以及基于Reseanberg-Jones(R-J)模型发展的Omi-R-J模型, 通过连续滑动、 拟合和余震发生率预测, 对地震序列的模型参数稳定性、 预测结果进行了比较研究, 并利用N-test、 T-test检验方法对预测结果进行了效能评估。 结果表明, 相比于其他中强震序列参数, 此次长宁M_S6.0地震序列参数中反映激发能力的αETAS较其他序列明显偏小, 而反映衰减能力的pORJ值和应力累积水平的bORJ值相对较小, 与此次余震序列丰富、 持续时间相对较长相吻合; ETAS和Omi-R-J模型对于复杂序列在[3.0, 3.5, 4.0]三个震级档的强余震仍具有一定的预测能力; 总体的“每个地震的信息增益”(IGPE)计算结果显示, ETAS模型略优于Omi-R-J模型, 前者或更适合复杂地震序列的余震预测。     

Modeling and forecasting the aftershock processes in Kamchatka

The aftershock processes that occurred in 1990?C2008 on the Kamchatka Peninsula and in the adjacent water area are numerically modeled with the aim of forecasting the aftershock activity. The aftershocks are identified by the Molchan-Dmitrieva algorithm followed by the analysis of spatiotemporal distribution of the earthquakes, which gives the final aftershock sequences. The simulations are based on the relaxation and triggering models of the aftershock processes. The studied models adequately describe and reliably simulate the temporal behavior of the aftershock sequences. An attempt is made to forecast the aftershock processes in Kamchatka using the ETAS model. It is shown that forecasting based on the data observed during the preceding time intervals is quite accurate. This approach can be applied in the centers of seismological monitoring for estimating the aftershock activity within the first days after a strong earthquake.     

Statistical Analysis of Triggered Seismicity in the Kresna Region of SW Bulgaria (1904) and the Umbria-Marche Region of Central Italy (1997)

A version of the restricted trigger model is used to analyse the temporal behaviour of some aftershock sequences. The conditional intensity function of the model is similar to that of the Epidemic Type Aftershock-Sequence (ETAS) model with the restriction that only the aftershocks of magnitude bigger than or equal to some threshold M_tr can trigger secondary events. For this reason we have named the model Restricted Epidemic Type Aftershock-Sequence (RETAS) model. Varying the triggering threshold we examine the variants of the RETAS model which range from the Modified Omori Formula (MOF) to the ETAS model, including such models as limit cases. In this way we have a quite large set of models in which to seek the model that fits best an aftershock sequence bringing out the specific features of the seismotectonic region struck by the crisis. We have applied the RETAS model to the analysis of two aftershock sequences: The first is formed by the events which followed the strong earthquake of M=7.8 which occurred in Kresna, SW Bulgaria, in 1904. The second includes three main shocks and a large swarm of minor shocks following the quake of 26 September 1997 in the Umbria-Marche region, central Italy. The MOF provides the best fit to the sequence in Kresna; that leads to the thought that just the stress field changes due to the very strong main shock generate the whole sequence. On the contrary, the complex behaviour of the seismic sequence in Umbria-Marche appears when we make the threshold magnitude vary. Setting the cut-off magnitude M₀=2.9 the best fit is provided by the ETAS model, while if we raise the threshold magnitude M₀=3.6 and set M_tr=5.0, the RETAS model turns out to be the best model. In fact, observing the time distribution of this reduced data set, it appears more evident that especially the strong secondary events are followed by a cluster of aftershocks.     

Statistical evaluation and probabilistic modeling of aftershock sequences of Iranian plateau

The present research focuses on the statistical evaluation of Iranian plateau aftershocks from an engineering perspective and presents probabilistic models applicable for generating random earthquake scenarios. Accordingly, a comprehensive earthquake data catalog including the period from 1964 to 2016 is prepared. Data are declustered into 37 separate mainshock-aftershock sequences by considering the completeness moment magnitude of the database. The well-known modified Omori occurrence rate formula is adopted to determine the recurrence time of the events, considering the effect of secondary aftershocks. In addition to computing the probability density functions of the parameters of the Omori formula, the joint probability distribution of the aftershock occurrence versus magnitude and occurrence time is obtained for modeling their magnitude sequences. The obtained results are applicable for producing randomly generated mainshock-aftershock scenarios.     

A Recent Application of the ETAS Model and a Proposed Method for Prediction of Strong Aftershocks

In this paper, 28 aftershock sequences are selected, which are distributed in different areas including north China, southwest of China, northwest of China, Taiwan area, Turkey and Greece. In order to investigate the characteristics of these sequences along with different temporal and spatial coordinates, each sequence has been divided into dozens of segments called ``sub-sequences''. The ETAS (Epidemic Type Aftershock Sequences) model is applied to each ``sub-sequence'', and therefore the vectors of parameters of ETAS could be evaluated. Another model named LR (Logistic Regression) model is used to seek the correlate relation between the parameters of ETAS applied to every earthquake ``sub-sequence'' and seismicity. All the analyses and estimations imply that the characteristic of decay of aftershock sequences in different temporal and spatial domains seems to be characterized by the parameters of the ETAS model applied to some aftershock sequences or ``sub-sequences'', and there are some proportional correlate relations between the evaluation of LR model and the occurrence probability of the succeeding strong seismic energy release.