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.     

利用模板匹配技术检测米尺度岩石断层黏滑实验中的声发射事件

声发射观测是岩石摩擦实验中研究强震孕育机制的重要观测手段之一.传统的声发射观测以触发记录为主,难以捕捉并提取微弱的声发射信号.近年来天然地震学中发展的模板匹配技术可以在连续波形记录中识别出微弱的事件,大大提升事件检测能力.本文发展了适用于多通道连续记录的声发射信号的模板匹配技术,并将该技术应用于分析米尺度岩石断层黏滑失稳过程.研究结果显示:模板匹配技术识别出的声发射数量约为传统方法识别的5倍,相对完备震级降低约0.3.临近失稳,检测目录完备震级以上声发射率表现出幂律增加的特征,且声发射事件聚集在失稳破裂起始点附近.失稳后,检测目录完备震级以上声发射率表现出幂律衰减,其中失稳后早期表现出较慢的衰减速度.另外,失稳后早期声发射事件表现出随对数时间沿断层迁移的趋势.最后,通过与天然地震学观测的结果对比表明,声发射模板匹配识别技术是研究大尺度岩石摩擦滑动失稳过程中前震和余震时空演化规律的有效手段,该技术助于在方法上为室内实验和野外观测进行融合研究提供思路.     

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.     

On the disturbance due to a spherical distortional pulse in an elastic medium

Summary Theoretical expressions are derived for the displacement, velocity and stress in the time domain induced by an axially symmetric shearing stress applied at the inner surface of a spherical cavity in a hornogeneous, isotropic, elastic medium of infinite extent. Theoretical seismograms are computed for a step source and for three sources with exponential decay in time. A satisfactory time-dependence of the source can be obtained by combining the step source with one or more exponentially decaying sources.     

A Point-process Analysis of the Matsushiro Earthquake Swarm Sequence: The Effect of Water on Earthquake Occurrence

Temporal characteristics of the famous Matsushiro earthquake swarm were investigated quantitatively using point-process analysis. Analysis of the earthquake occurrence rate revealed not only the precise and interesting process of the swarm, but also the relation between pore water pressure and the strength of the epidemic effect, and the modified Omori-type temporal decay of earthquake activity. The occurrence rate function (t) for this swarm is represented well aswhere f(t) represents the contribution of the swarm driver, which was the erupting water from the deep in this case, and the second term represents an epidemic effect of the modified Omori type. Based on changes in the form of f(t), this two-year long swarm was divided into six periods and one short transitional epoch. The form of f(t) in each period revealed the detail of the water erupting process. In the final stage, f (t) decayed according to the modified Omori-formula form, while it decayed exponentially in the brief respite of the water eruption in the fourth period. When an exponential decay of swarm activity is observed, we have to be cautious of a sudden restart of the violent activity. The epidemic effect is stronger when the pressure of the pore water is higher. Even when the pressure is not high, the p value in the epidemic effect is small, when there is plenty of pore water. However, the epidemic effect produced about a quarter of the earthquakes even though there was not much pore water in the rocks.     

川滇地区7级大震前中强震震源机制变化

分析了 70年代以来 ,川滇地区发生的 8次 7级大震前 5年内 ,发生在大震孕震区和震源区内的中强震震源机制解时空分布。结果表明 ,最早中强震发生在大震震源区或其附近 ,其发震应力场与区域构造应力场一致 ,与大多数大震发震应力场一致或接近。大多数中强震震源破裂特征与大震明显不同。之后有多次中强震发生在距大震震源区较远的大震孕震区内其他地方 ,它们的发震应力场往往经历了与区域构造应力场和大震应力场一致与不一致的多次交替变化。大震前最后 1个中强震也发生在距大震震源区较近的地方 ,其发震应力场与大震发震应力场明显不一致 ,偏转了 30°～ 5 0° ,或更多 ,大多数也与区域构造应力场不一致 ,有的中强震发震断裂破裂特征与大震不一致。大震前中强震震源机制的变化 ,反映了大震孕育过程的不同阶段 ,区域构造应力场的时空调整变化和增强过程 ,以及由此引发的构造断裂异常活动 ,揭示出与大震发生有关的应力场和震源破裂特征信息     

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.     

Stress drop assessment of the August 8, 2017, Jiuzhaigou earthquake sequence and its tectonic implications*

By using a broadband Lg attenuation model developed for the Tibetan Plateau, we isolate source terms by removing attenuation and site effects from the observed Lg-wave displacement spectra of the M7.0 earthquake that occurred on August 8, 2017, in Jiuzhaigou, China, and its aftershock sequence. Thus, the source parameters, including the scalar seismic moment, corner frequency and stress drop, of these events can be further estimated. The estimated stress drops vary from 47.1 kPa to 7149.6 kPa, with a median value of 59.4 kPa and most values falling between 50 kPa and 75 kPa. The estimated stress drops show significant spatial variations. Lower stress drops were mainly found close to the mainshock and on the seismogenic fault plane with large coseismic slip. In contrast, the highest stress drop was 7.1 MPa for the mainshock, and relatively large stress drops were also found for aftershocks away from the major seismogenic fault and at depths deeper than the zone with large coseismic slip. By using a statistical method, we found self-similarity among some of the aftershocks with a nearly constant stress drop. In contrast, the stress drop increased with the seismic moment for other aftershocks. The amount of stress released during earthquakes is a fundamental characteristic of the earthquake rupture process. As such, the stress drop represents a key parameter for improving our understanding of earthquake source physics.     

STATISTICAL ANALYSIS OF AIRBORNE ELECTROMAGNETIC SIGNATURES OF DIFFERENT CONDUCTOR GROUPS IN TROPICAL TERRAIN OF KARNATAKA,INDIA*

AEM (airborne electromagnetic) anomalies caused by massive sulphide conductors and superficial conductors can be recognized with a statistical method, as shown by an analysis of Input AEM data from Karnataka State. The weathering in the survey area is of tropical type. Parameters, such as various amplitude ratios and time parameters (inverse of decay rate) for exponential and power-law decay were analyzed for sulphide bodies, conducting soil, superficial conductors, and cultural conductors. Time parameters τ₁ (exponential decay) is defined as ratio of time differences between the third and fourth channel to the logarithmic value of the relative amplitude of the two channels. Time parameter K₁ (power-law decay) is defined as ratio of the difference of the logarithmic values of the delay times of the third and fourth channels to the logarithmic value of the relative amplitude of the two channels. The two parameters have been useful in recognizing sulphide conductors. Also the first channel Input amplitude and logarithmic plot of the transients appear to be helpful in conductor identification. Channel ratios seem to be the least effective parameters of conductor identification. In the area studied both power-law and the conventional exponential decay were found equally suitable for approximating Input AEM transients.     

Temporal variations of shear-wave splitting in field and laboratory studies in China

Observations of shear-wave splitting at seismic stations above a swarm of small earthquakes on Hainan Island, China, and other examples world-wide, suggest that the time-delays of split shear-waves monitor the build up of stress before earthquakes and the stress release as earthquakes occur. Rock physics experiments on marble specimens also show variations of shear-wave time-delays with uniaxial pressure analogous to the field observations. The rock experiments show an abrupt decrease in time-delays immediately before fracturing occurs. Similar precursory behaviour has been observed before earthquakes elsewhere, and is believed to be important for two reasons. Precursory changes in shear-wave splitting could be used for short-term forecasting, but of greater importance may be the information such behaviour provides about the source processes in earthquake preparation zones.     

双层黏弹介质模型条件下地震应力扰动的时空特征

基于简化双层黏弹介质模型及其变形过程的时间属性,考虑震后地壳上层(黏滞性相对较弱)短时间的弹性扰动及后续较长时间内下层(黏滞性相对较强)黏性变形对上层应力扰动的联合作用,研究地震应力扰动的时空变化.结果显示,地震应力扰动在震后一段时间内逐渐增大,之后缓慢衰减.并且模型参数越接近真实地体,应力扰动增大及衰减过程就越缓慢,持续时间越长,即地震活动具有较长时期的时间"记忆"特性.因而,在地震活动及库仑应力变化研究中,长时期的应力扰动影响不宜忽略.地震应力扰动有一定的空间作用范围,在此范围内应力扰动随距离衰减明显,距离震源较近处衰减较快、距离震源较远处衰减相对较慢,此范围之外应力扰动可忽略不计.在本文所取与实际地壳相对较为接近的模型参数条件下,应力扰动的空间作用范围大体是震源线性破裂尺度的2.5倍.地震应力扰动随震级增大而逐渐增大,在本文模型参数条件下,震级大于6级之后,应力扰动随震级快速增加.地震破裂尺度测量误差所导致的应力扰动计算误差的相对大小,与时间无关,与破裂尺度(震级)及震源距有关,随震源距的增大而增大,但对应力扰动的总体变化趋势及变化范围影响甚微.当距离足够远时,应力扰动计算误差的相对大小趋于常数,就6、7、8级地震而言,最大应力扰动计算误差分别小于应力扰动值本身的22%、30%及38%.     

2015年尼泊尔MW7.9地震应力状态与余震空间分布规律

本文利用2015年尼泊尔M_W7.9地震断层面滑动位移分布的运动学反演结果,通过傅里叶变换法得到了主断层面上的两分量应力状态,并研究了余震的空间分布和断层面上应力状态之间的关系.发现滑动位移分布与应力状态分布都相对较为集中,大约70%的余震分布在应力变化为正的区域,而其余发生在应力降区域的余震,又大多发生在应力变化梯度较大的地区.为了得到一个更符合实际的滑动模型来解释余震的触发机制,我们计算了波数域中滑动位移和应力状态的傅里叶谱,发现此次地震的滑动位移和应力状态近似满足k^-3和k^-2衰减.我们利用简化的圆盘模型说明了非均匀应力变化下的衰减过程,计算了圆盘模型的有效半径r_e约等于0.7倍的圆盘半径.这就说明圆盘模型中应力增加的部分应该占整个圆盘破裂面积的51%.在本次尼泊尔M_W7.9地震实例中,断层面上应力状态为负的区域比滑动位移为正的区域有了明显地缩小.事实表明,余震可以发生在有滑动位移的区域,非均匀应力降模型比均匀应力降模型更加接近真实的震源破裂过程.     

雁列式断层组合变形过程中的声发射活动特征

着重研究拉张或挤压型雁列式断层标本变形过程中声发射(简称AE，下同)的时空演化特征. 结果表明，预置构造对AE空间分布格局具有较强的控制作用，随着差应力的增加，AE首先在两个端点附近丛集，之后向两端点连线附近集中，出现明显的破裂局部化现象，较大事件还通常集中于某一端点附近反复发生. 前期微破裂丛集图象指示后期宏观破裂的扩展方向及扩展范围. 拉张和挤压型雁列区宏观破裂方向分别与轴向应力方向垂直和平行. 雁列式断层标本变形过程中，破裂前的弱化阶段相对较长、弱化过程明显. 微破裂事件累积频次指数增长可能是系统失稳前的典型征兆之一，而雁列区宏观破裂之后，AE数量逐渐减少、应变释放相对减弱. 摩擦滑动过程中，大的粘滑失稳前未见有AE活动前兆性的增强过程.雁列区的b值变化在失稳前显示趋势性降低——快速回升这一典型的变化过程，b值降低一般发生在差应力增强过程之中，并有可能延续至弱化阶段，而快速回升则一般发生在破裂失稳前的短时期内. 对比研究表明，构造差异所导致的b值差异远大于b值随差应力的增加而产生的变化量，而对同一构造标本，力学状态的改变会导致AE序列时序特征的急剧变化，较高的加载速率对应较高的应变能释放及明显的低b值. 先期破碎带由于较低的破坏强度，其对差应力的微小变化具有特殊敏感性，从而成为源、兆分离、窗口或敏感点效应等地震活动性前兆现象的一种可能的原因.      

Spatio-temporal characteristics of acoustic emission during the deformation of rock samples with compressional and extensional en-echelon faults

Introduction By the experimental techniques of acoustic emission (AE) and surface strain observation, a lot of achievements on the study of deformation and fracture process of samples with en-echelon faults has been obtained (Bomblakis, 1973; LIU, et al, 1986; DU, et al, 1989; LI, et al, 1989; ZHANG, FANG, 1989; ZHANG, LI, 1989; MA, et al, 1995a, b; LI, et al, 2000). However, there still exist many problems. For example, most of discussion on the spatio-temporal evolution of microf…     

Can Damage Mechanics Explain Temporal Scaling Laws in Brittle Fracture and Seismicity?

Time delays associated with processes leading to a failure or stress relaxation in materials and earthquakes are studied in terms of continuum damage mechanics. Damage mechanics is a quasi-empirical approach that describes inelastic irreversible phenomena in the deformation of solids. When a rock sample is loaded, there is generally a time delay before the rock fails. This period is characterized by the occurrence and coalescence of microcracks which radiate acoustic signals of broad amplitudes. These acoustic emission events have been shown to exhibit power-law scaling as they increase in intensity prior to a rupture. In case of seismogenic processes in the Earth's brittle crust, all earthquakes are followed by an aftershock sequence. A universal feature of aftershocks is that their rate decays in time according to the modified Omori's law, a power-law decay. In this paper a model of continuum damage mechanics in which damage (microcracking) starts to develop when the applied stress exceeds a prescribed yield stress (a material parameter) is introduced to explain both laboratory experiments and systematic temporal variations in seismicity.     

2003年12月1日新疆昭苏6.1级地震序列特征分析

通过分析昭苏6．1级地震序列特征、震源与介质破裂特性、震源环境与历史地震活动类比及区域地震和前兆资料的异常变化等，对该震前的区域地震活动背景、震后的震情趋势等有了较为细致的了解和判定。主要结果有：①在2003年新疆境内中强地震成组活动的背景下，于现代地震活动很弱的昭苏盆地内发生了6．1级地震；②6．1级地震为主一余型地震序列，震后序列衰减正常；③主震震源机制解结果与现代区域构造应力场特征吻合，表明构造活动具有明显的继承性，并对地震活动有着控制作用；④震源区域构造应力释放充分，区域应力场处于相对稳定的调整状态。综合各类分析结果后认为，震区发生更大地震的可能性很小。     

Recurrence Interval Statistics of Cellular Automaton Seismicity Models

The recurrence interval statistics for regional seismicity follows a universal distribution function, independent of the tectonic setting or average rate of activity (Corral, 2004). The universal function is a modified gamma distribution with power-law scaling of recurrence intervals shorter than the average rate of activity and exponential decay for larger intervals. We employ the method of Corral (2004) to examine the recurrence statistics of a range of cellular automaton earthquake models. The majority of models has an exponential distribution of recurrence intervals, the same as that of a Poisson process. One model, the Olami-Feder-Christensen automaton, has recurrence statistics consistent with regional seismicity for a certain range of the conservation parameter of that model. For conservation parameters in this range, the event size statistics are also consistent with regional seismicity. Models whose dynamics are dominated by characteristic earthquakes do not appear to display universality of recurrence statistics.     

Probability of earthquake recurrence with nonuniform stress rates and time-dependent failure

Current methods for calculation of long-term probabilities for the recurrence of large earthquakes on specific fault segments are based upon models of the faulting process that implicitly assume constant stress rates during the interval separating earthquakes and instantaneous failure at a critical stress threshold. However, observations indicate that the process of stress recovery following an earthquake involves rate variations at all time scales in addition to stress steps caused by nearby earthquakes. Additionally, the existence of foreshocks, aftershocks and possible precursory processes suggest that there may be significant time dependence of the earthquake nucleation process. A method for determining the conditional probabilities for earthquake occurrence under conditions of irregular stressing is developed that could be useful at all time scales including those pertinent to short-and intermediate-term prediction. Used with models for earthquake occurrence at a stress threshold, the addition of variable stressing introduces a simple scaling of the conditional probabilities by stress level and stress rate. A model for the time-dependent nucleation of earthquake slip has been proposed recently that is based upon laboratory observations of fault strength. This failure criterion results in large but relatively short duration changes in the probability of earthquake recurrence particularly following stress steps. Applied to populations of earthquakes the models predicts a 1/t decay of seismicity following stress steps as observed for aftershocks and for frequency of foreshock-mainshock pairs. The model suggests that variations of seismicity rates of small earthquakes in the nucleation zone of the expected earthquake directly indicate variations in probability of recurrence of the large earthquake.     

1999年度中俄合作岩石破裂实验研究

１９９９年１１月～１２月,俄罗斯科学院圣彼得堡Ｉｏｆｆｅ技术物理研究所Ｖ．Ｓ．Ｋｕｋ－ｓｅｎｋｏ教授等专家应邀来华,与中国地震局地球物理研究所和中国科学院地球物理研究所合作,在震源物理实验室开展分米级尺度岩石破裂实验研究。实验中采用了ＳＤＡＥ－８型系统,进行声发射地定位和实时显示。实验采用了多种典型构造模拟和加－卸载方式,取得了丰富的成果。该实验开辟了国际间实质性合作的新途径。     

Hypocenter distribution and focal mechanism of AE events during two stress stage creep in Yugawara andesite

More than a thousand acoustic-emission (AE) hypocenters were determined in a cylindrical andesite specimen under two-stage uniaxial creep at stresses of 204 and 214 MPa. Strains were monitored for 6 peripheral points at the middle part of the cylindrical specimen's wall. The strain data indicate gradual increase of nonuniform deformation during steady creep and strong intensification of the nonuniformity during acceleration creep and, therefore, biased stress distribution within the specimen. The correlation between dilatant strain and AE hypocenters was investigated for whether or not tensile cracks emit AE. The region with high AE activity shows only a small dilatant strain. This negative correlation between AE and the dilatant region may eliminate tensile cracks as possible AE sources. A composite focal-mechanism solution of local AE events, covering a wide solid angle of the focal hemisphere, indicates that shear fractures emit AE waves. The direction of the compressional axis in this solution shows a significant deviation from that inferred from the applied external force, suggesting that the local stress field is governed by preexisting weak zones that are, presumably, produced by tensile cracks within the specimen. AE hypocenters tended to form clusters during steady creep under the constant compressional stress. During acceleration creep caused by a small step increase of the external stress, the preceding clusters disappeared while a new cluster appeared in an incipient fault plane. This suggests that changes in seismicity pattern such as migrations or quiescences of swarm—important clues for earthquake predictions—may be caused by an instantaneous change in the tectonic-stress levels.