Dynamic faulting under rate-dependent friction

We discuss the effects of rate-dependent friction on the propagation of seismic rupture on active faults. Several physicists using Burridge and Knopoff's box and spring model of faulting have proposed that fault complexity may arise from the spontaneous development of a self-similar stress distribution on the fault plane. If this model proves to be correct, it has important consequences for the origin of the complexity of seismic sources. In order to test these ideas on a more realistic earthquake model, we developed a new boundary integral equation method for studying rupture propagation along an antiplane fault in the presence of nonlinear rate-dependent friction. We study rupture dynamics of models with single and twin asperities. In our models, asperities are places on the fault with a higher value of prestress. Othewise all fault parameters are homogeneous. We show that for models with such asperities, a slip velocity weakening friction leads to the propagation of supersonic healing phases and to the spontaneous arrest of fracture if the prestress outside the asperities is low enough. For models with asperities, we can also observe narrow slip velocity pulses, qualitatively similar to the so-called Heaton pulses observed in some earthquake accelerograms. We also observe a complex distribution of stress after the rupture that depends on details of the initial distribution of asperities and on the details of the friction law.     

基于速率-状态摩擦定律的地震二维准动力学数值模拟:浅层正应力变化对断层演化的影响

本文基于结合速率-状态摩擦定律（RSF）的二维准动力学数值模型,以半空间垂直走滑断层为研究对象,通过比较两种正应力随深度变化模型的模拟结果,研究了浅层正应力变化对断层演化参数、地震孕育过程、震后滑移传播等方面的影响.结果显示,我们的数值模型在给定模型参数和约束条件下,能够完整模拟出地震周期中震间、震前、同震以及震后多个特征阶段.常数正应力模型下,动态破裂在浅层速率强化区停止,而在浅层变化正应力模型下动态破裂可以传播至自由表面,导致浅层更高的最大滑移速率和同震滑移量.两种模型下的地震矩、地震周期、平均应力降和震后滑移传播等差别不明显.两种滑移模型的傅氏振幅谱与理论K^-2模型傅氏振幅谱均符合较好,且浅层变化正应力模型下的拐角波数值高于常数正应力模型,说明两种模型均符合地震同震滑移模型的运动学特征,并且浅层变化正应力模型下最终应该产生高于常数正应力模型的高频强地面运动水平.我们认为选用不同的模型参数对最终结果存在显著影响,应当根据具体问题来选择模型参数,这样才能在保证结果准确前提下有效提高计算效率.     

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.     

Comparison of source parameters estimated in the frequency and time domains for seismic events at the Rudna copper mine,Poland

Source parameters estimated in the frequency domain for 100 selected seismic events from the Rudna copper mine, with moment magnitude ranging from 1.4 to 3.6, were collected to study their scaling relations and to compare them with the parameters estimated in the time domain. The apparent stress and static stress drop, corrected for the limited bandwidth recording, increase slightly in a similar manner with increasing seismic moment. The ratio of apparent stress to static stress drop, a measure of radiation efficiency, is practically constant and its mean value is close to 0.1. For 37 seismic events, with moment magnitude between 1.9 and 3.4, source parameters were estimated in the time domain from relative source time functions, that displayed unilateral rupture propagation, and their rupture velocity could be estimated. It ranges from 0.23 to 0.80 of shear wave velocity and is almost independent of seismic moment. The fault length, estimated from the average source pulse width and rupture velocity, is clearly dependent on seismic moment and is smaller than the source radius estimated from the corner frequency on the average by about 25 percent. There is no correlation between the values of static stress drop estimated in the frequency and time domains, but the time domain stress drop is in general similar to that estimated in the frequency domain. The apparent stress increases with increasing rupture velocity, and the ratio of apparent stress to static stress drop seems also to depend on rupture velocity.     

2011年M_W9.0东日本大地震动力学破裂过程的数值模拟

力学上,地震可以看作在应力场作用下由于断层带介质的突然损伤或软化导致的断层带失稳事件.本文基于这个地震动力学模型,利用一种可以模拟断层大位错的有限元方法,研究了2011年MW9.0东日本大地震(Tohoku-Oki)的动力学破裂过程.比较了无障碍体和具有不同刚度障碍体的断层带模型产生的断层位移、位错和应力降.主要结果表明,障碍体的存在并不明显地改变障碍体区域的初始构造应力场.对有障碍体情形,准静态结果显示断层上盘最大逆冲位移和最大剪切位错分别为51m和58m,均发生在海底表面海沟处,与无障碍体的结果(最大剪切位错约55m)相比差别不大;下盘最大倾向位移(-10m)并不与上盘最大值出现在同一位置,而是在障碍体处.障碍体处剪应力降(约11 MPa)大于周围非障碍体区域.障碍体处正应力降的最大值约为3 MPa.模拟结果似乎不支持海山是导致本次地震异乎寻常大位错的原因,而倾向于断层带剪切刚度在地震过程中极度损伤或软化.     

Slip-Weakening Models of the 2011 Tohoku-Oki Earthquake and Constraints on Stress Drop and Fracture Energy

We present 2D dynamic rupture models of the 2011 Tohoku-Oki earthquake based on linear slip-weakening friction. We use different types of available observations to constrain our model parameters. The distribution of stress drop is determined by the final slip distribution from slip inversions. As three groups of along-dip slip distribution are suggested by different slip inversions, we present three slip-weakening models. In each model, we assume uniform critical slip distance eastward from the hypocenter, but several asperities with smaller critical slip distance westward from the hypocenter. The values of critical slip distance are constrained by the ratio of deep to shallow high-frequency slip-rate power inferred from back projection source imaging. Our slip-weakening models are consistent with the final slip, slip rate, rupture velocity and high-frequency power ratio inferred for this earthquake. The average static stress drop calculated from the models is in the range of 4.5–7 MPa, though large spatial variations of static stress drop exist. To prevent high-frequency radiation in the region eastward from the hypocenter, the fracture energy needed there is in the order of 10 MJ/m², and the average up-dip rupture speed cannot exceed 2 km/s. The radiation efficiency calculated from our models is higher than that inferred from seismic data, suggesting the role of additional dissipation processes. We find that the structure of the subduction wedge contributes significantly to the up-dip rupture propagation and the resulting large slip at shallow depth.     

A crack-like rupture model for the 19 September 1985 Michoacan,Mexico, earthquake

Evidence supporting a smooth crack-like rupture process of the Michoacan earthquake of 1985 is obtained from a major earthquake for the first time. Digital strong motion data from three stations (Caleta de Campos, La Villita, and La Union), recording near-field radiation from the fault, show unusually simple ramped displacements and permanent offsets previously only seen in theoretical models. The recording of low frequency (0 to 1 Hz) near-field waves together with the apparently smooth rupture favors a crack-like model to a step or Haskell-type dislocation model under the constraint of the slip distribution obtained by previous studies. A crack-like rupture, characterized by an approximated dynamic slip function and systematic decrease in slip duration away from the point of rupture nucleation, produces the best fit to the simple ramped displacements observed. Spatially varying rupture duration controls several important aspects of the synthetic seismograms, including the variation in displacement rise times between components of motion observed at Caleta de Campos. Ground motion observed at Caleta de Campos can be explained remarkably well with a smoothly propagating crack model. However, data from La Villita and La Union suggest a more complex rupture process than the simple crack-like model for the south-eastern portion of the fault.     

关于“用单元降刚法探索中国大陆强震远距离跳迁及主体活动区域转移”的讨论——横向各向同性“杀伤单元”才是更好的途径

《地球物理学报》第55卷第1期的"用单元降刚法探索中国大陆强震远距离跳迁及主体活动区域转移"一文提出用减小剪切模量的方法模拟地震断层错动的效应,这种处理虽然可以模拟压扭性走滑断层错动时的剪应力降低,但会导致垂直于断层的正应力也剧烈变化,因此这种简单的减小弹性模量的方法并不合理.本文探讨了一种更好的方法--横向各向同性"杀伤单元",利用该方法模拟断层滑动效应得到了较为合理的结果,与Okada解析解吻合较好,能够正确反映断层滑动时的应力变化.     

3-D Simulation of Earthquake Generation Cycles and Evolution of Fault Constitutitve Properties

--The earthquake generation cycle consists of tectonic loading, quasi-static rupture nucleation, dynamic rupture propagation and stop, and subsequent stress redistribution and fault restrengthening. From a macroscopic point of view, the entire process of earthquake generation cycles should be consistently described by a coupled nonlinear system of a slip-response function, a fault constitutive law and a driving force. On the basis of such a general idea, we constructed a realistic 3-D simulation model for earthquake generation cycles at a transcurrent plate boundary by combining the viscoelastic slip-response function derived for a two-layered elastic-viscoelastic structure model, the slip- and time-dependent fault constitutive law that has an inherent mechanism of fault restrengthening, and the steady relative plate motion as a driving force into a single closed system. With this model we numerically simulated the earthquake generation cycles repeated in a seismogenic region on a plate interface, and examined space-time changes in shear stress, slip deficits and fault constitutive properties during one complete cycle in detail. The occurrence of unstable dynamic slip brings about decrease both in fault strength and shear stress to a constant residual level. After the arrest of dynamic slip, the breakdown strength drop j†_p of fault is restored rapidly and the process of stress accumulation resumes in the seismogenic region. On the other hand, the restoration of the critical weakening displacement D_c proceeds gradually with time through the interseismic period. The restoration of D_c can be regarded as the macroscopic manifestation of the microscopic recovery process of fractal fault surface structure. Through numerical simulation with a multi-segmented fault model, we examined the effects of viscoelastic fault-to-fault interaction. The effect of transient viscoelastic stress transfer through the asthenosphere is significant as well as the direct effect of elastic stress transfer, and it possibly explains the time lag of the sequential occurrence of large events along a plate boundary.     

断层自发破裂传播的数值模拟:速率状态摩擦定律本构参数的影响

地震往往受控于滑动面的摩擦性质,这种摩擦性质可以由速率状态摩擦定律较好地描述.速率状态摩擦定律中的本构参数a和b与动态摩擦系数相关,从而影响着同震位移与剪切应力的时空演化.本文在前人工作的基础上,采用三维边界积分方程法模拟速率状态摩擦定律控制下均匀全空间中平面断层的自发破裂传播过程,并详细讨论了a和b对滑动速率、剪切应力和破裂传播速度的影响.数值结果表明a和b的不同取值将导致不同的破裂行为,b－a的值越大,断层越不稳定,这种不稳定性有利于裂纹的产生与扩展.但滑动速率的时空分布不只依赖b－a,而且还与a和b的具体取值有关,断层面上滑动速率峰值与剪切破裂强度均随着a的减小而增大,随着b的增大而增大.相关结果有助于加深对断层自发破裂传播的认识.     

基于非结构化网格的边界积分方程方法的断层自发破裂模拟

地震自发破裂模拟是震源动力学研究的重要内容,了解复杂的断层动力学破裂过程对深入认识震源特征和解释运动学反演结果具有重要意义.基于边界积分方程方法的破裂模拟已经被广泛使用,大多采用的是平面断层模型的结构化网格划分.由于实际的断层往往具有较为复杂的几何特征,为了更为灵活地刻画断层几何复杂性,我们建立断层模型的三角形网格离散方案,通过精确的解析解形式来计算断层各个单元之间的应力格林函数,联立滑动弱化摩擦准则和非奇异边界积分方程,对断层的自发破裂过程进行了模拟.在简单的平面断层模型下,将计算结果与前人的结果进行了对比,验证了方法的正确性与有效性.对于几种常见的复杂断层模型,例如弯折、阶跃、含障碍体断层等,我们模拟了其破裂过程并对计算结果进行了比较与分析.模拟结果表明,非结构化网格划分的边界积分方程方法能够很好地模拟平面矩形断层或由其组成的规则断层,同时也能成功地模拟具有复杂几何形状的不规则断层上的动力学破裂过程.本研究的结果显示了边界积分方程方法在模拟复杂断层系统的动力学破裂问题上具有较广阔的应用前景.     

Testing the Validity of Simulated Strong Ground Motion from the Dynamic Rupture of a Finite Fault, by Using Empirical Equations

This paper is concerned with testing the validity of the ground motions estimated by combining a boundary integral equation method to simulate dynamic rupture along finite faults with a finite difference method to compute the subsequent wave propagation. The validation exercise is conducted by comparing the calculated ground motions at about 100 hypothetical stations surrounding the pure strike-slip and pure reverse faults with those estimated by recent ground motion estimation equations derived by regression analysis of observed strong-motion data. The validity of the ground motions with respect to their amplitude, frequency content and duration is examined. It is found that the numerical simulation method adopted leads to ground motions that are mainly compatible with the magnitude and distance dependence modelled by empirical equations but that the choice of a low stress drop leads to ground motions that are smaller than generally observed. In addition, the scatter in the simulated ground motions, for which a laterally homogeneous crust and standard rock site were used, is of the same order as the scatter in observed motions therefore, close to the fault, variations in source propagation likely contribute a significant proportion of the scatter in observed motions in comparison with travel-path and site effects.     

流体饱和多孔隙介质弹性波方程边界元解法研究

基于流体饱和多孔隙各向同性介质模型,本文首先推导了流体饱和多孔隙介质中弹性波传播的频率域系统动力方程及边界积分方程,然后给出了流体饱和多孔隙介质弹性波方程的基本解,最后,利用本文给出的边界元方法对流体饱和多孔隙各向同性介质中的弹性波传播进行了数值模拟.结果表明：不论是从固相位移,还是液相位移的地震合成记录都能看到明显的慢速P波,本文提出的流体饱和多孔隙介质弹性波边界元法是有效可行的.     

基于边界积分方程方法的弯折断层破裂传播过程控制因素分析

本文利用边界积分方程方法,以基于三角形网格的全空间格林函数及离散积分核计算为基础,进行了最常见的弯折断层的破裂传播过程模拟.为了去除边界积分方程方法中格林函数计算存在的高度奇异性,研究采用分部积分等方法对动力学方程进行了重整化和离散化处理.地震力学过程可以被视为断层由静摩擦转为动摩擦的过程,对于震源破裂过程的动力学模拟,摩擦准则起着重要作用,本研究采用常用的滑动弱化摩擦准则.计算引入Courant-Friedrich-Lewy比值来表达场点的影响,并控制计算的收敛性和稳定性.通过与典型算例的比对,检验了方法的正确性和有效性.地震破裂能否穿越断层弯折部位继续传播是震源动力学研究的重要内容,基于此,本文建立了多种理论弯折断层模型,模拟了断层弯折对地震破裂传播的控制作用,并通过改变断层周边初始应力场、断层弯折角度大小以及滑动弱化距离大小等来分析各个因素对破裂传播的影响.模拟结果表明:断层面上初始破裂区域内外的应力越高,破裂越容易越过断层弯折部位继续传播;初始破裂区域半径越大,或滑动弱化距离越小,破裂也越容易发生,并越过弯折部位继续传播.同样的初始条件,断层弯折角度越大,断层弯折作为障碍体,对破裂传播的阻碍作用越显著.小的弯折角,其破裂传播过程与平面断层差别不明显,基本仍以椭圆方式对称向两侧传播.     

饱和多孔介质三维时域黏弹性人工边界与动力反应分析的显式有限元法

本文基于Biot的饱和多孔介质本构方程,考察具有辐射阻尼的外行球面波,推导了饱和多孔介质三维黏弹性人工边界的法向和切向边界方程;在已有的饱和多孔介质二维显式有限元数值计算方法基础上,提出该理论的三维方法,并开发了实现该三维方法的有限元程序.算例表明饱和多孔介质三维时域黏弹性人工边界与动力反应分析的显式有限元法具有较好的精度和稳定性.     

基于动摩擦应力下调滑移弱化机制，利用地表破裂同震位移约束2001年昆仑山口西MS8.1地震破裂传播速度

利用2001年昆仑山口西MS8.1地震现场观测所提供的地表破裂同震位移数据,使用简单滑移弱化破裂模型,估算了发震主断层上的破裂传播速度. 该模型中考虑了断层破裂时动摩擦过程中应力上调和下调机制对地震波辐射能量分配的影响. 对比Bouchon和Valleacute;e有关昆仑山口西地震主断层破裂传播速度超过剪切波速度,甚至达到P波速度的结果, 采用动摩擦应力下调时的滑移弱化模型 (分数应力降模型),结果表明,伴随较高的地震波辐射效率,主断层的平均破裂传播速度等于或小于瑞利波速度,这与许力生和陈运泰的体波反演结果,以及陈学忠震源应力场估算的结果是一致的. 最后,联系到由地表破裂现象所反映出的断层力学特征,如与视应力相关的分数应力降 (动摩擦应力下调), 基于滑移弱化模型, 讨论了可能的震源破裂机制.      

Toward a physical undenstanding of earthquake scaling relations

In seismological literature, there exist two competing theories (the so-calledW model andL model) treating earthquake scaling relations between mean slip and rupture dimension and between seismic moment and rupture dimension. The core of arguments differentiating the two theories is whether the mean slip should scale with the rupture width or with the rupture length for large earthquakes. In this paper, we apply the elastic theory of dislocation to clarify the controversy. Several static dislocation models are used to simulate strike-slip earthquakes. Our results show that the mean slip scales linearly with the rupture width for small earthquakes with a rupture length smaller than the thickness of the seismogenic layer. However, for large earthquakes with a rupture length larger than the thickness of the seismogenic layer, our models show a more complicated scaling relation between mean slip and rupture dimension. When the rupture length is smaller than a cross-over length, the mean slip scales nearly linearly with the rupture length. When the rupture length is larger than a cross-over length, the mean slip approaches asymptotically a constant value and scales approximately with the rupture width. The cross-over length is a function of the rupture width and is about 75 km for earthquakes with a saturated rupture width of 15 km. We compare our theoretical predictions with observed source parameters of some large strike-slip earthquakes, and they match up well. Our results also suggest that when large earthquakes have a fixed aspect ratio of rupture length to rupture width (which seems to be the case for most subduction earthquakes) the mean slip scales with the rupture dimension in the same way as small earthquakes.     

2013年MS7.0芦山地震的动力学破裂过程及其影响因素

2013年4月20日在四川芦山发生了M S7.0地震,震源运动学反演结果给出了此次地震的破裂过程和同震滑动分布.为了更好地理解造成芦山地震破裂过程的力学原因,本文综合野外地质调查、余震定位、深地震反射剖面等结果,构建芦山地震铲型断层模型,以震源运动学反演结果为约束,将震源参数与震源附近的构造应力场结合,建立断层面上滑动量和牵引力的时空分布关系,通过试错法给定震源动力学计算参数模拟芦山地震破裂传播的可能情况,进而分析讨论不同动力学计算参数对芦山地震破裂过程和同震滑动分布的影响.结果显示,初始应力是决定断层是否发生错动的关键;临界滑动弱化位移D c对破裂滑动速率有着很大的影响;成核区半径和初始应力主要影响破裂成核的快慢;局部不均匀破裂强度主要影响破裂行为和断层最终滑动量分布.利用边界积分方程法可以有效计算芦山地震铲型断层模型的动力学破裂过程,再现此次地震的主要特征.通过探究动力学参数对破裂过程影响,可解释运动学反演结果所揭示的破裂特征的力学原因,对于深入了解地震震源过程的物理本质和预测未来可能发生的地震的主要特征有着重要的参考意义.     

Improvement in the Fault Boundary Conditions for a Staggered Grid Finite-difference Method

The staggered grid finite-difference method is a powerful tool in seismology and is commonly used to study earthquake source dynamics. In the staggered grid finite-difference method stress and particle velocity components are calculated at different grid points, and a faulting problem is a mixed boundary problem, therefore different implementations of fault boundary conditions have been proposed. Viriuex and Madariaga (1982) chose the shear stress grid as the fault surface, however, this method has several problems: (1) Fault slip leakage outside the fault, and (2) the stress bump beyond the crack tip caused by S waves is not well resolved. Madariaga et al. (1998) solved the latter problem via thick fault implementation, but the former problem remains and causes a new issue; displacement discontinuity across the slip is not well modeled because of the artificial thickness of the fault. In the present study we improve the implementation of the fault boundary conditions in the staggered grid finite-difference method by using a fictitious surface to satisfy the fault boundary conditions. In our implementation, velocity (or displacement) grids are set on the fault plane, stress grids are shifted half grid spacing from the fault and stress on the fictitious surface in the rupture zone is given such that the interpolated stress on the fault is equal to the frictional stress. Within the area which does not rupture, stress on the fictitious surface is given a condition of no discontinuity of the velocity (or displacement). Fault normal displacement (or velocity) is given such that the normal stress on the fault is continuous across the fault. Artificial viscous damping is introduced on the fault to avoid vibration caused by onset of the slip. Our implementation has five advantages over previous versions: (1) No leakage of the slip prior to rupture and (2) a zero thickness fault, (3) stress on the fault is reliably calculated, (4) our implementation is suitable for the study of fault constitutive laws, as slip is defined as the difference between displacement on the plane of z = + 0 and that of z = − 0, and (5) cessation of slip is achieved correctly.