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We analyze the problem of calculating high-frequency ground motions (>1 Hz) caused by earthquakes having arbitrary spatial variations of rupture velocity and slip velocity (or stress drop) over the fault. We approximate the elastic wave Green's functions by far-field body waves, which we calculate using geometric ray theory. However, we do not make the traditional Fraunhofer approximation, so our method may be used close to large faults. The method is confined to high frequencies (greater than about 1 Hz) due to the omisson of near-field terms, and must be used at source-observer distances less than a few source depths, due to the omission of surface waves. It is easily used in laterally varying velocity structures. Assuming a simple parameterization of the slip function, the computational problem collapses to the evaluation of a series of line integrals over the fault, with one line integral per each time ti in the observer seismogram. The path of integration corresponding to observation time ti consists of only those points on the fault which radiate body waves arriving at the observer at exactly time ti. This path is an isochron of the arrival time function. An isochron velocity may be defined that depends on rupture velocity and resembles the usual directivity function. Observed ground motions are directly dependent upon this isochron velocity. Ground velocity is proportional to isochron velocity and ground acceleration is proportional to isochron acceleration in dislocation models of rupture. Ground acceleration may also be related to spatial variations of slip velocity on the fault, using the square of isochron velocity as a constant of proportionality. We show two rupture models, one with variable slip velocity and the other with variable rupture velocity, that cause the same ground acceleration at a single observer. The computational method is shown to produce reasonably accurate synthetic seismograms, compared to a method using complete Green's functions, and requires about 0.5 per cent of the computer time. It may be very effective in calculating ground motions in the frequency band 1 to 10 Hz at observers within a few source depths of large earthquakes, where most of the high-frequency motions may be caused by direct P and S waves. We suggest a possible method for inverting ground motions for both slip velocity and rupture velocity over the fault.

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... Fracture models which introduced variable slip function and rupture velocity showed that changes in rise time and rupture velocity lead to high-frequency radiation (Madariaga, 1977;Madariaga, 1983). Later, seismologists used ray-theory to calculate high-frequency radiation from earthquakes having spatial variations of rupture velocity, slip velocity and stress drop (Bernard and Madariaga, 1984;Spudich and Frazer, 1984) and ...

... Theoretical studies of supershear rupture (Hamano, 1974;Andrews, 1976;Das and Aki, 1977) followed by experimental works on plastic polymer (Wu et al., 1972;Rosakis et al., 1999) Quite recently, the emergence of dense and large aperture seismic arrays has provided a new method to investigate the spatial and temporal behavior of seismic energy release during large earthquakes. This method, called back-projection, utilizes the time-reversal property of seismic waves to retrieve their sources and was introduced by Spudich and Frazer (1984). Following the successful application of the back-projection method to the 2004 Sumatra-Andaman earthquake by Ishii et al. (2005), the back-projection method has been applied to numerous earthquakes (Kiser and Ishii, 2011;Zhang and Ge, 2010;Ishii, 2011;Wang and Mori, 2011;. ...

... Although high-frequency energy is more diffuse (especially in the two last time windows, 3 − 5µs and 4 − 6µs), we also observed a net correlation between the estimated position of the rupture front and high-frequency radiation. This result is in agreement with most of the studies that addressed the issue of high-frequency radiation which proposed that high-frequency radiation is related to changes in rupture velocity due to fault stress or frictional heterogeneity, and predict high-frequency waves to be mainly generated in the vicinity of the rupture front (Haskell, 1964;Aki, 1967;Madariaga, 1977;Madariaga, 1983;Spudich and Frazer, 1984). Recent numerical studies (Thomas et al., 2017;Thomas and Bhat, 2018;Okubo et al., 2019) also demonstrated that high-frequency radiation was highly enhanced when co-seismic damage was implemented in their rupture propagation models. ...

Au cours de cette thèse, nous avons reproduit expérimentalement des séismes à l’échelle centimétrique dans des conditions de pression proches de la réalité. Les expériences réalisées nous ont permis d’explorer deux grandes thématiques : (i) l’origine du rayonnement haute-fréquence pendant la rupture dynamique et (ii) les signaux précurseurs pendant la phase de nucléation de la rupture dynamique. Nos résultats montrent que le rayonnement haute-fréquence est concomitant à la propagation du front de rupture et que deux paramètres induisent une augmentation du rayonnement haute-fréquence : l’état de contrainte initial et la vitesse de rupture. Les analyses microstructurales des échantillons de roches suggèrent que la production d’endommagement cosismique ou de particules de gouge contribue au rayonnement haute fréquence. L’étude des signaux précurseurs (i.e., précurseurs acoustiques) montre que la nucléation est un processus en très large majorité asismique. Ce très faible couplage pourrait expliquer le peu d’observations de séismes précurseurs à l’échelle des failles crustales. L’analyse temporelle des émissions acoustiques suggère que leur dynamique est principalement contrôlée par l’accélération du glissement pendant la phase de nucléation. La microtopographie et la microstructure des échantillons de roches montrent que le couplage est directement relié à la rugosité du plan de faille. Une augmentation des conditions de pression favorise l’occurrence de processus de déformation plastique ou de fusion partielle au cours de la rupture sismique, ce qui diminue la rugosité et donc le couplage.

... In his pioneering work (Madariaga 1977) showed that HF seismic radiation is emitted when abrupt changes in the dislocation rate or in the rupture velocity occur during faulting. A growing body of literature has then investigated the spectral behaviour of seismic sources in relation to the earthquake complexity (e.g., Savage 1966, Aki 1967, Brune 1970, Bernard and Madariaga 1984, Spudich and Frazer 1984, Somerville et al. 1999). Classic approaches have been routinely performed to determine the space-time history of a rupture process through the inversion of seismic waveforms. ...

... We detail different source models, starting from the simple approach first proposed by Haskell (1964) and arriving to more realistic models, where the spatial and temporal complex behavior of earthquake processes is taken into account (e.g., Hanks 1979, Andrews 1981, Herrero and Bernard 1994. A growing body of literature has investigated the spectral behaviour of seismic sources (e.g., Aki 1967, Brune 1970, Savage 1966, Bernard and Madariaga 1984, Spudich and Frazer 1984. Here, we briefly resume the characteristic parameters usually determined from the low-frequency content of seismic signals and we dedicate particular attention to the high-frequency content of seismic signals, whose features are at the core of our study. ...

... The high-frequency content in the spectrum is therefore controlled by those singularities at the starting and final edges of the fault (Madariaga 1977). Such behaviour can be easily interpreted through the concept of isochrone Madariaga 1984, Spudich andFrazer 1984), a high-frequency (ray-theory) approximation to calculate seismic radiation from earthquake ruptures. The isochrone is the set of source points ssen by the station as radiating seismic waves at the same instant of time on the fault plane ( Figure 1.11). ...

Many studies have attempted to illuminate rupture complexities of large earthquakes through the use of coherent imaging techniques such as back-projection (BP). Recently, Fukahata et al. (2013) suggested that, from a theoretical point of view, the BP image of the rupture is related to the slip motion on the fault. However, the quantitative relationship between the BP images and the physical properties of the earthquake rupture process still remains unclear.Our work aims at clarifying how BP images of the radiated wavefield can be used to infer spatial heterogeneities in slip and rupture velocity along the fault. We simulate different rupture processes using a line source model. For each rupture model, we calculate synthetic seismograms at three teleseismic arrays and we apply the BP technique to identify the sources of high-frequency (HF) radiation. This procedure allows for the comparison of the BP images with the originating rupture model, and thus the interpretation of HF emissions in terms of along-fault variation of the three kinematic parameters: rise time, final slip, rupture velocity. Our results show that the HF peaks retrieved from BP analysis are most closely associated with space-time heterogeneities of slip acceleration. We verify our findings on two major earthquakes that occurred 9 years apart on the strike-slip Swan Islands fault: the Mw 7.3 2009 and the Mw 7.5 2018 North of Hondurasearthquakes. Both events followed a simple linear geometry, making them suitable for comparison with our synthetic approach. Despite the simple geometry, both slip-rate functions are complex, with several subevents. Our preliminary results show that the BP image of HF emissions allows to estimate a rupture length and velocity which are compatible with other studies and that strong HF radiation corresponds to the areas of large variability of the moment-rate function. An outstanding question is whether one can use the BP image of the earthquake to retrieve the kinematic parameters along the fault. We build on the findings obtained in the synthetic examples by training a neural network model to directly predict the kinematic parameters along the fault, given an input BP image. We train the network on a large number of different synthetic rupture processes and their BP images, with the goal of identifying the statistical link between HF radiation and rupture kinematic parameters. Our results show that the neural network applied to the BP image of the earthquake is able to predict the values of rise time and rupture velocity along the fault, as well as thecentral position of the heterogeneity, but not the absolute slip values, to which the HF BP approach is relatively insensitive. Our work sheds some light on the gap currently existing between the theoretical description of the generation of HF radiation and the observations of HF emissions obtained by coherent imaging techniques, tackling possible courses of action and suggesting new perspectives.

... In discussion with our coupling method, a concept of the isochrones is effective (Benard & Madariaga 1984;Spudich & Frazer 1984;Bizzarri & Spudich 2008). The isochrones formula plays an important role in our research, but its derivation in a general setting seems to be not trivial. ...

... The eikonal equation is a nonlinear partial differential equation and needs some efforts to solve it numerically (Sethian 1999;Sethian & Mihari Popovici 1999;Spiral & Kimmel 2004). In Spudich & Frazer (1984), they assume that S-wave travel time is governed by the eikonal equation. In our purpose, however, we need the travel time of the rupture propagation on curved fault surfaces. ...

... In their analysis of the stopping phase at a barrier, the isochrones theory is effectively used. In Spudich & Frazer (1984) and Bizzarri & Spudich (2008), the substantially same isochrones formulae are used for analysis of high frequency ground motion. In Bizzarri Figure 2. Local coordinates of isochrones (left) and isochrones band (right). ...

We describe three achievements for a ground motion simulation. First, we propose a kinematic modelling in which rupture delay time is governed by an eikonal equation on a Riemannian manifold and develop a coupling method between the ground motion simulation and the eikonal solver. In general the rupture velocity distribution is not spatially uniform and the rupture propagation depends on a fault shape. So we derive the eikonal equation by considering the Riemannian metric of the fault surface and give a detailed discretization of its difference scheme to deal with a curved surface fault. Next, in order to explain the effect of spatially discontinuous non-uniformity of rupture velocity, we introduce an isochrones jumping intensity and obtain a new decomposed isochrones formula in general settings. It is known that the representation theorem with the Green's function can be rewritten into an expression with a contour integral by the isochrones theory. The new formula says that the known isochrones formula for ground velocity can be decomposed into a trend component and a disturbance component. The disturbance component consists of the isochrones jumping intensity. Finally, by applying our ground motion simulation coupled with the eikonal solver and the decomposed isochrones formula, we investigate some relations between the non-uniformity of the rupture velocity and pulse-like disturbance of the ground motion velocity. Our simulations show that the disturbance of velocity waveform corresponds with that of rate of change of isochrones band area. It turns out that the pulse-like disturbance of velocity waveform occurs when isochrones move across the region where rupture velocity varies discontinuously. Thus we can explain that the pulse-like disturbance of the ground motion velocity occurs when the isochrones jumping intensity has nonzero value. Moreover, as another example of application of our simulation and formula, we show a distinctive dependence of peak ground velocity upon parameters such as the rupture velocity and the distance between a fault and an observer.

... Fracture models which introduced variable slip function and rupture velocity showed that changes in rise time and rupture velocity lead to high-frequency radiation (Madariaga, 1977(Madariaga, , 1983. Later, seismologists used ray-theory to calculate high-frequency radiation from earthquakes having spatial variations of rupture velocity, slip velocity, and stress drop (Bernard & Madariaga, 1984;Spudich & Frazer, 1984) and predicted that the starting and stopping phases of earthquakes to be responsible of high-frequency radiation. A good illustration of this phenomena is the 17th of January 1984 Northridge earthquake (Mw 6.7) for which Hartzell et al. (1996) identified the initiation of the rupture and its stopping to be concurrent with highfrequency radiation. ...

... Quite recently, the emergence of dense and large aperture seismic arrays has provided a new method to investigate the spatial and temporal behavior of seismic energy release during large earthquakes. This method, called back-projection, utilizes the time-reversal property of seismic waves to retrieve their sources and was introduced by Spudich and Frazer (1984). Following the successful application of the backprojection method to the 2004 Sumatra-Andaman earthquake by Ishii et al. (2005), the back-projection method has been applied to numerous earthquakes (Kiser & Ishii, 2011, Okuwaki et al., 2014, Zhang & Ge, 2010, Ishii, 2011, Wang & Mori, 2011. ...

... The correlation between the spatial and temporal evolution of high-frequency sources and the propagation of the rupture front provides concrete experimental evidence that highfrequency waves are concurrent with the propagation phase of the rupture front and that high-frequency radiation is emitted close to or behind the rupture tip. This result is in agreement with most of the studies that addressed the issue of high-frequency radiation which proposed that high-frequency radiation is related to changes in rupture velocity due to fault stress or frictional heterogeneity, and predict high-frequency waves to be mainly generated in the vicinity of the rupture front (Aki, 1967;Haskell, 1964;Madariaga, 1977Madariaga, , 1983Spudich & Frazer, 1984). Recent numerical studies (Okubo et al., 2018;Thomas et al., 2017;Thomas & Bhat, 2018) also demonstrated that high-frequency radiation was highly enhanced when coseismic damage was implemented in their rupture propagation models. ...

Plain Language Summary
Over geological time scales, partially or fully locked tectonic plates accumulate stress and strain. The stress and the strain build up on discontinuities that we call “faults.” Natural faults exist either inside a tectonic plate or at the boundary between two distinct tectonic plates. When the stress accumulated on a fault exceeds the strength of the fault, the accumulated stress and strain, which can be interpreted in term of accumulated energy, are suddenly released. This natural phenomenon is called an “earthquake.” During an earthquake, part of the energy is released in the form of seismic waves. Those seismic waves are responsible for the ground shaking. High‐frequency waves usually cause most of the damage. To better understand the physical parameters that influence the generation of high‐frequency waves, we experimentally reproduced microearthquakes and used them as a proxy to study real earthquakes. Our results showed that the higher the pressure acting on the fault when an earthquake is generated, the higher the amount of high‐frequency radiations. Moreover, our observations underlined that, during an earthquake, high‐frequency waves are released in specific areas on the fault. Thus, these results might be of relevance to improve seismic hazard assessment.

... Vallée and Satriano 2014). Using the ray theory and the concept of isochrones (that is, the locus of energy emissions arriving at a station simultaneously), (Spudich and Frazer 1984) demonstrated that the PGA is proportional to the temporal changes of isochrone velocity, which depends on spatial variations of rupture velocity and slip velocity function. Deploying kinematic rupture simulations, ( Schmedes and Archuleta 2008) showed that for a strike slip homogeneous rupture, the strongest changes of isochrones velocity is at a 'critical point', which remains at an almost constant position on the top fault boundary for all near fault stations. ...

... The parameters used for simulations are summarized in Table 2-2 (simulation A). We use the concept of isochrones to extract the part of the rupture that produces the PGA ( Spudich and Frazer 1984). Isochrones are all the points on the fault that radiate elastic waves such that the waves arrive at a given station at the same time. ...

... In the case of the far- field approximation (Equation (2-4), the isochrone at the PGA time is simply the rupture front at the PGA time (Figure 2-3 a, b, c and d). (Spudich and Frazer 1984) demonstrated that ground acceleration is proportional to the variations of isochrones velocity. In the far-field approximation, ground motion is then proportional to the variations of rupture velocity. ...

Accumulated data of strong ground motions have been providing us very important knowledge about rupture processes of earthquakes, propagation-path, site-amplification effects on ground motion, the relation between ground motion and damage... However, most of the ground motion databases used in the development of ground motion prediction models are primarily comprised of accelerograms produced by small and moderate earthquakes. Hence, as magnitude increases, the sets of ground motions become sparse. Ground motion databases are poorly sampled for short source-to-site distance ranges (‘Near-fault’ ranges). However, the strongest ground shaking generally occurs close to earthquake fault rupture. Countries of moderate to high seismicity for which major faults can break in the vicinity of its major cities are facing a major seismic risk, but the lack of earthquake recordings makes it difficult to predict ground motion. Strong motion simulations may then be used instead. One of the current challenges for seismologists is the development of reliable methods for simulating near-fault ground motion taking into account the lack of knowledge about the characteristics of a potential rupture. This thesis is divided into 2 parts. Part 1 focuses on better understanding the seismic rupture process and its relation with the near-fault ground motion. The mechanisms of peak ground motion generating are investigated for homogeneous as well as for heterogeneous ruptures. A quantitative sensitivity analysis of the ground motion to the source kinematic parameters is presented, for sites located in the vicinity of the fault rupture, as well as far from the rupture. A second chapter is dedicated to a major near-fault source effect: the directivity effect. This phenomenon happens when the rupture propagates towards a site of interest, with a rupture speed close to the shear-wave speed (Vs); the waves propagating towards the site adds up constructively and generates a large amplitude wave called the pulse. The features of this pulse are of interest for the earthquake engineering community. In this chapter, a simple equation is presented that relates the period of the pulse to the geometric configuration of the rupture and the site of interest, and to the source parameters.Part 2 is dedicated to better estimate the seismic hazard in Lebanon by simulating the strong ground motion at sites near the main fault (the Yammouneh fault). Lebanon is located in an active tectonic environment where the seismic hazard is considered moderate to high. Historically, destructive earthquakes occurred in the past, the last one dates back to 1202. However, strong motion has never been recorded in Lebanon till now due to the presently infrequent large-magnitude seismicity, and therefore facing an alarming note of potential new ruptures. The Yammouneh fault is a large strike-slip fault crossing Lebanon, making all its regions located within 25km away from the fault. At first, the crustal structure tomography of Lebanon, in terms of Vs, is performed using the ambient noise, in order to characterise the wave propagation from the rupture to the ground surface. To our knowledge, this is the first study of the 3D Vs tomography in Lebanon. Afterwards, a hybrid approach is presented to simulate broadband near-fault ground motion . At low-frequencies (≤1Hz), potential ruptures of M7 are simulated (as defined in the previous chapters), and the generated slip rate functions are convolved with the Green’s functions computed for the propagation medium defined by the Vs tomography. The ground-motion is complemented by a high-frequency content (up to 10Hz), using a stochastic model calibrated by near-fault recordings and accounting for the presence of the directivity pulse. The computed peak ground acceleration is compared to the design acceleration in Lebanon.

... Vallée and Satriano 2014). Using ray theory and the isochrones concept, (i.e. the simultaneous arrival of locus of energy emissions at a station,) Spudich and Frazer (1984) and Bernard and Madariaga (1984) demonstrated that the PGA is proportional to the temporal changes of isochrone velocity, dependant on spatial variations of rupture velocity and slip velocity function. By deploying kinematic rupture simulations Schmedes and Archuleta (2008) showed that for a homogeneous strike slip rupture, the strongest changes of isochrones velocity is at a 'critical point' that maintains an almost constant position on the top fault boundary for all near-fault stations. ...

... The slip, the rupture speed and the rise time are generally constant along the rupture (r ¼ 0), except at fault boundaries (due to applied tapering). The source parameters observed for this first analysis are depicted in Table 1 (rupture scenario A) and the results in Fig. 2. We use the isochrones concept to extract the part of the rupture that produces the PGA (Spudich and Frazer 1984;Bernard and Madariaga 1984). Isochrones are all the points on the fault that radiate elastic waves, such that the waves arrive at a given station at the same time. ...

... (5)], the isochrone at the PGA time is the rupture front at the PGA time ( Fig. 2a-d). Spudich and Frazer (1984) demonstrated that ground acceleration is proportional to the variations of isochronic velocity. In the farfield approximation, ground motion is proportional to the variations of rupture velocity. ...

Empirical ground motion prediction equations are calibrated from past earthquake seismic recordings. Although they are often used to predict Peak Ground Acceleration (PGA) and its variability, the use of these equations to predict near-fault PGA remains questionable due to the scarcity of near-fault recordings for large earthquakes (e.g. Mai Encyclopedia of complexity and systems science (pp. 4435–4474). New York: Springer. https://doi.org/10.1007/978-0-387-30440-3_263. 2009). The simulation of strong ground motion offers an attractive alternative for the assessment of near-fault seismic hazards, but the a priori choice of the source parameters used to describe the fault rupture process remains a complex issue. In order to better understand the effects of rupture parameters on surface ground motion and to capture the key source ingredients that impact ground motion variability, we simulated ground motions produced by various M7 strike-slip rupture earthquake scenarios on vertical faults. We computed ground motion up to 5 Hz using the far-field approximation as well as at the near-field stations located at 5 km, 25 km and 70 km from the fault (assuming a visco-elastic medium). The kinematic rupture parameters are modeled using a statistical rupture model generator as proposed by Song et al. Bulletin of the Seismological Society of America,99(4), 2564–2571 (2014). Our work demonstrates that PGA is mostly generated by abrupt changes in the rupture propagation (e.g. stopping phases at the fault boundaries or strong heterogeneities of rupture speed along the fault). We observed that PGA is mostly controlled by average rupture speed and average stress drop (in the far-field), and to a lesser extent by the standard deviation of the rupture speed. It is worth noting that for the set of stations in study, the correlation between source parameters and spatial correlation length does not affect average PGA and related variability significantly.

... Isochrones are then used to provide simple geometric illustrations of how directivity varies between dipping dip-slip and vertical strike-slip faults. Bernard and Madariaga (1984) and Spudich and Frazer (1984; developed the isochrone integration method to compute near-source ground motions for finite-fault rupture models. Isochrones are all the positions on a fault that contribute seismic energy that arrives at a specific receiver at the same time. ...

... Then, all seismic radiation from a fault can be described with rupture and healing isochrones. Ground velocities (v) and accelerations (a) produced by rupture or healing of each point on a fault can be calculated from (Spudich and Frazer, 1984;Zeng et al., 1991;Smedes and Archuleta, 2008) ...

... Variations of T r on the fault surface associated with supershear rupture velocities, or regions on the fault where rupture jumps discontinuously can cause large or singular values of c, called critical points by Farra et al. (1986). Spudich and Frazer (1984) showed that the reciprocal of c, isochrone slowness is equivalent to the seismic directivity function in the twodimensional case. Thus, by definition, critical points produce rupture directivity, and as is shown with simulations later, need not be associated strictly with forward rupture directivity, but can occur for any site located normal to a portion of a fault plane where rupture velocities are supershear. ...

This progressive sequence of ground motion surprises suggests that the current state ofknowledge in strong motion seismology is probably not adequate to make unequivocal strong ground motion predictions. However, with these caveats in mind, strong ground motion estimation provides substantial value by reducing risks associated with earthquakes and engineered structures. We present the current state of earthquake ground motion estimation. We start with seismic source characterization, because this is the most important and challenging part of the problem. To better understand the challenges of developing ground motion prediction equations (GMPE) using strong motion data, we present the physical factors that influence strong ground shaking. New calculations are presented to illustrate potential pitfalls and identify key issues relevant to ground motion estimation and future ground motion research and applications. Particular attention is devoted to probabilistic implications of all aspects of ground motion estimation.

... The complexity of the source manifests itself in terms of short-period seismic waves, at frequencies higher than the corner frequency of far field waveform spectra f c (e.g. Madariaga 1977, Spudich and Frazer 1984, Ruiz et al. 2011 which is controlled by the source duration. At these higher frequencies, the classic inversion techniques are no longer adequate, both because of computational limitations and by our lack of knowledge of the Earth's structure at those frequencies. ...

... The role played by depth phases is amplified by the simple horizontal line source geometry chosen here, where all the fault points lay at the same depth. In a 1D line source, the locus of points on the fault from which radiation arrives at the observation point at the same time, also called the isochrones (Bernard and Madariaga, 1984;Spudich and Frazer, 1984), are in fact just a single point. In this setting, the recording stations -see‖ at a given time the energy emitted from just a single point, where depth phases emerge from the same depth. ...

The retrieval of earthquake finite-fault kinematic parameters after the occurrence of an earthquake is a crucial task in observational seismology. Routinely-used source inversion techniques are challenged by limited data coverage and computational effort, and are subject to a variety of assumptions and constraints that restrict the range of possible solutions. Back-projection (BP) imaging techniques do not need prior knowledge of the rupture extent and propagation, and can track the high-frequency (HF) radiation emitted during the rupture process. While classic source inversion methods work at lower frequencies and return an image of the slip over the fault, the BP method highlights fault areas radiating HF seismic energy. Patterns in the HF radiation are attributable to the spatial and temporal complexity of the rupture process (e.g. slip heterogeneities, changes in rupture speed and in slip velocity). However, the quantitative link between the BP image of an earthquake and its rupture kinematics remains unclear. Our work aims at reducing the gap between the theoretical studies on the generation of HF radiation due to earthquake complexity and the observation of HF emissions in BP images. To do so, we proceed in two stages, in each case analyzing synthetic rupture scenarios where the rupture process is fully known. We first investigate the influence that spatial heterogeneities in slip and rupture velocity have on the rupture process and its radiated wave field using the BP technique. We simulate two different rupture processes using a 1D line source model: a homogeneous process, where the kinematic parameters are constant along the line, and a heterogeneous process, where we introduce a central segment along the line that has a step change in kinematics. For each rupture model, we calculate synthetic seismograms at three teleseismic arrays and apply the BP technique to reveal how HF emissions are influenced by the three kinematic parameters controlling the synthetic model: the rise time, final slip, and rupture velocity. Our results show that the HF peaks retrieved from BP analysis are better associated with space-time heterogeneities of slip acceleration. We then build on these findings by testing whether one can retrieve the kinematic rupture parameters along the fault using information from the BP image alone. We apply a machine learning, convolutional neural network (CNN) approach to the BP images of a large set of simulated 1D rupture processes to assess the ability of the network to retrieve, from the progression of HF emissions in space and time, the kinematic parameters of the rupture. These rupture simulations include along-strike heterogeneities whose size is variable and within which the parameters of rise-time, final slip, and rupture velocity change from the surrounding rupture. We show that the CNN trained on 40 000 pairs of BP images and kinematic parameters returns excellent predictions of the rise time and the rupture velocity along the fault, as well as good predictions of the central location and length of the heterogeneous segment. Our results also show that the network is insensitive towards the final slip value, as expected from theoretical results.

... Introducing variable slip function and rupture velocity, fracture models showed that changes in rise time and rupture velocity lead to high-frequency radiation (Madariaga, 1977, Madariaga, 1983. Later, seismologists used ray-theory to calculate high-frequency radiation from earthquakes having spatial variations of rupture velocity, slip velocity and stress drop (Bernard and Madariaga, 1984, Spudich andFrazer, 1984) and predicted starting and stopping phases of earthquakes to be responsible of high-frequency radiation. A good illustration is the January 17th 1984 Northridge earthquake (Mw 6.7) for which Hartzell at al [1996] identified the initiation of the rupture and its stopping to be concurrent with high-frequency radiation. ...

... The correlation between the spatial and temporal evolution of high-frequency sources and the propagation of the rupture front provides concrete good experimental evidence that high-frequency waves are concurrent with the propagation phase of the rupture front and that high-frequency radiation is emitted close to or behind the rupture tip. This result is in agreement with most of the studies that addressed the issue of high-frequency radiation which proposed that high-frequency radiation is related to changes in rupture velocity due to fault stress or mechanical properties heterogeneity, and predict highfrequency waves to be mainly generated in the vicinity of the rupture front (Madariaga, 1977, Madariaga, 1983, Haskell, 1964, Aki, 1967, Spudich and Frazer, 1984. Recent numerical studies (Thomas et al., 2017, Thomas and Bhat, 2018, Okubo et al., 2018 also demonstrated that highfrequency radiation was highly enhanced when coseismic damage was implemented in their rupture propagation models. ...

We monitor dynamic rupture propagation during laboratory stick-slip experiments performed on saw-cut Westerly granite under upper crustal conditions (10-90 MPa). Spectral analysis of high-frequency acoustic waveforms provided evidences that energy radiation is enhanced with stress conditions and rupture velocity. Using acoustic recordings bandpass filtered to 400-800 kHz (7-14 mm wavelength) and highpass filtered above 800 kHz, we back projected high-frequency energy generated during rupture propagation. Our results show that the high-frequency radiation originates behind the rupture front during propagation and propagates at a speed close to that obtained by our rupture velocity inversion. From scaling arguments, we suggest that the origin of high-frequency radiation lies either in the fast dynamic stress-drop in the breakdown zone together with off-fault co-seismic damage propagating behind the rupture tip. The application of the back projection method at the laboratory scale provides new ways to locally investigate physical mechanisms that control high-frequency radiation.

... The reference depth for projection for the main shock equal to 10.9 km is used for differential travel time to source depth. The array imaged the features of rupture process towards South-eastand North-west trend from the epicenter along the line connecting the ruptured fault (Bernard and Madariaga 1984;Spudich and Frazer 1984). Due to larger aperture, the European seismic array presents the rupture process in sharper images of both main events. ...

This study highlights the 2D rupture propagation of 2005 M7.6 Kashmir earthquake. Beamforming and multi signals classification (MUSIC) back projection techniques are applied on the recorded seismic waves of teleseismic seismometers to model the rupture propagation of 2005 M7.6 Kashmir earthquake. These techniques are robust enable to model propagated rupture extents and rupture phases as the event data is available with the teleseismic seismometers. The imaging of the 2005 M7.6 Kashmir earthquake source propagation with beamforming technique shows three phases of rupture propagation of 10, 10, and 25 s with the velocity of 0, 2.2, and 1.9 km/s respectively. While resolving the rupture propagation with MUSIC, it is observed that rupture propagated 26 and 14 s covering distance of 50 and 31 km respectively. Both techniques confirm the bidirectional rupture propagation with various velocities and time. Both the rupture speed and duration vary due to both applied technique resolutions. Our findings are quite consistent with the published slip models of the Kashmir earthquake. The modeling of 2005 Kashmir earthquake propagation provides key information about the mechanics of Bagh-Balakot Thrust, western Himalaya side.

... The 20-30 s slips remain as strike-slip ruptures, but their nodal planes are rotated about ∼10° clockwise at a ∼233° azimuth (Figure 3). If this geometric variation holds true, the fault rotation can serve as a restraining bend (Bruhat et al., 2016), which may have caused a sudden deceleration of the rupture and generated stopping phases, radiating strong high-frequency seismic energy (Bernard & Madariaga, 1984;Madariaga, 1977;Okuwaki & Yagi, 2018;Spudich & Frazer, 1984). ...

Plain Language Summary
On 14 August 2021, a devastating magnitude 7.2 earthquake struck Southern Haiti, causing over 2,000 casualties and severe infrastructure damage. Southern Haiti situates in between the Caribbean and North American plates, where they converge obliquely at the boundary. The relative motion displaces the plates horizontally and accumulates stress along a major left‐lateral fault network. The oblique plate motion also causes an uplift of the region due to the boundary‐normal compression. Therefore, earthquakes in the region rupture in complex ways. However, the physical relations between the tectonic regime and the earthquake rupture development are poorly understood, posing challenges to local risk management. Here we use global seismic records to resolve the rupture history of the 2021 Haiti earthquake. We find the earthquake composed of two distinct rupture episodes: a reverse faulting subevent near the epicenter and a strike‐slip faulting subevent further west. Both subevents ruptured faults that deviate away from the left‐lateral geometry of the Enriquillo‐Plantain Garden fault zone. Our results show that the complex tectonic setting of the convergence boundary is imprinted in a segmented fault network with various distinct faulting styles, which may have been influenced by the local small‐scale plate fragmentation.

... The situation is different from the case where the stress conditions are homogeneous on the fault. Given homogeneous stress conditions, once the rupture goes to the supershear velocity, the supershear rupture front dominates the rupture propagation and causes dramatic differences in ground motions compared to a pure subshear rupture [Spudich and Frazer, 1984;Aagaard and Heaton, 2004;Archuleta, 2004, 2005;Bernard and Baumont, 2005]. Local supershear rupture velocities associated with large stress drops have also been observed in other dynamic models [e.g., Day, 1982b;Olsen et al., 1997;Peyrat et al., 2001]. ...

1] We present two spontaneous rupture models of the 2004 M w 6.0 Parkfield earthquake constrained by near-source ground motions. We start with a stress drop distribution calculated from a kinematic slip distribution. Using a linear slip-weakening friction law, we utilize trial and error to obtain both the stress conditions and frictional parameters on the fault that produce synthetics consistent with records. The material contrast across the San Andreas Fault is incorporated using different one-dimensional velocity structures on each side of the fault. An approximately constant S parameter of 0.3 and a uniform slip-weakening distance of 0.15 m are used in the dynamic models. In our preferred dynamic model, consistent with the ground motion and GPS, the slip is bounded by seismicity streaks at 5 and 10 km depths, confirming a locked zone at depth. The stress drop is approximately 10 MPa in the hypocentral region and about 2 MPa elsewhere. The material contrast across the fault causes significant normal stress variations ($1 MPa), leading to a larger strength drop to the southeast than to the northwest. The main rupture front propagates at nearly a constant subshear rupture velocity $3 km/s in both directions. The total radiated energy determined from the preferred dynamic model is 1.1 Â 10 13 J, seismic moment is 1.0 Â 10 18 Nm, and fracture energy is 3.0 Â 10 13 J. The limited number of aftershocks in the slipped area suggests the important role of stress on the distribution of seismicity in the locked zone.

... It is known that intense high-frequency waves can be radiated as a result of a rapid change of rupture-front velocity, slip velocity or both (e.g. Madariaga 1977;Spudich & Frazer 1984;. The multiple energy burst spots located around the rupture front correspond to fluctuations in the rupture propagation rate. ...

The 26 May 2019 MW 8.0 Peru intraslab earthquake ruptured the subducting Nazca plate where the dip angle of the slab increases sharply and the strike angle rotates clockwise from the epicentre to north. To obtain a detailed seismic source model of the 2019 Peru earthquake, including not only the rupture evolution but also the spatiotemporal distribution of focal mechanisms, we performed comprehensive seismic waveform analyses using both a newly developed flexible finite-fault teleseismic waveform inversion method and a back-projection method. The source model revealed a complex rupture process involving a back-propagating rupture. The initial rupture propagated downdip from the hypocentre, then unilaterally northward along the strike of the slab. Following a large slip occurring ∼50–100 km north of the hypocentre, the rupture propagated bilaterally both further northward and back southward. The spatial distribution of focal mechanisms shows that the direction of T-axis azimuth gradually rotated clockwise from the epicentre northward, corresponding to the clockwise rotation of the strike of the subducting Nazca plate, and the large-slip area corresponds to the high-curvature area of the slab iso-depth lines. Our results show that the complex rupture process, including the focal-mechanism transition, of the Peru earthquake was related to the slab geometry of the subducting Nazca plate.

... The concept of isochrones is utilized in this section to associate specific characteristics of the fault rupture process with pulse-like ground strains and rotations observed in the near-fault region. An isochrone is the locus of points on a fault from which the seismic waves all arrive at a selected station at the same time (Bernard & Madariaga 1984;Spudich & Frazer 1984;Schmedes & Archuleta 2008). Assuming that all significant seismic radiation consists of direct S-wave arrivals, the long-period ground-motion pulses at near-fault stations may conveniently be associated with specific parts of the fault by plotting the S-wave isochrones (Mavroeidis & Papageorgiou 2010b;O'Connell et al. 2012). ...

Previous studies have demonstrated that finite-fault simulations of actual or hypothetical earthquakes using deterministic, physics-based simulation techniques constitute an effective tool for characterizing near-fault ground strains and rotations in the low-frequency range. The characteristics of these motions are further investigated in this study by performing forward ground-motion simulations of three well-documented strike-slip earthquakes (i.e. 2004 Mw 6.0 Parkfield, 1979 Mw 6.5 Imperial Valley, 1999 Mw 7.5 Izmit) using models of the seismic source and crustal structure available in the literature. Time histories of ground strains and rotations are numerically generated at near-fault stations and at a dense grid of observation points extending over the causative fault. This is achieved by finite differencing translational motions simulated at very closely spaced stations using a kinematic modeling approach. The simulation results show that the three strike-slip earthquakes produce large-amplitude pulse-like shear strain and torsion in the forward direction of rupture propagation. The time histories of specific components of displacement gradient, strain, and rotation at near-fault stations can be estimated from those of ground velocities using a phase velocity, whereas peak ground torsions in the near-fault region can be reasonably estimated from peak horizontal ground velocities using a scaling factor. However, both the phase velocity and the scaling factor exhibit significant variability in the near-fault region of the considered earthquakes. The concept of isochrones is also utilized to associate fault rupture characteristics with near-fault ground strains and rotations. The results indicate that the seismic energy radiated from the high-isochrone-velocity region of the fault—which encompasses areas of large slip locally driven by high stress drop—arrives at a near-fault station in a short time interval that coincides with the time window of the large-amplitude pulse-like shear strain and torsion.

... We consider radially isotropic rupture front expansion with constant rupture velocity of 0.8 times shear wave velocity for all scenarios. Constant rupture speed, however diminishes the high-frequency radiations from the source (Spudich and Frazer 1984;Mai Fig. 9 Bias along epicentral distance for simulated peak ground velocity (PGV), peak ground acceleration (PGA), and spectral accelerations at period 1 s (Sa(T = 1 s)) with respect to the corresponding recorded data for the Mw 5.4 [Event-HM1] and the Mw 4.6 [Event-HM2]. Note: The plots corresponds to horizontal components of ground motion obtained as the geometric mean of East-West and North-South components 2009), but facilitates quantifying the basin effects more effectively. ...

This study presents broadband ground motions for the Indo-Gangetic basin, a large sedimentary basin in India, for potential future great (Mw 8.5) Himalayan earthquakes. We use a recently developed 3D earth structure model of the basin as an input to simulate low-frequency ground motion (0–0.5 Hz). These ground motions are further combined with high-frequency scattering waveforms by using a hybrid approach, thus yielding broadband ground motions (0–10 Hz). We calibrate the 3D model and scattering parameters by comparing the simulated ground motions against available recorded data for two past earthquakes in Himalaya. Our approach accounts for the physics of interaction between the scattered seismic waves with deep basin sediments. Our results indicate that the ground motion intensities exhibit frequency-dependent amplification at various basin depths. We also observe that in the event of a great earthquake, the ground motion intensities are larger at deep basin sites near the source and exhibit an attenuating trend over distance similar to the ground motion models. The extreme ground motion simulations performed in our study reveal that the national building codes may not provide safe recommendations at deep basin sites, especially in the near field region. The period-dependent vertical-to-horizontal spectral ratio deviates from the code-recommended constant 2/3 at least up to 6 s at these sites.

... when the rupture front hits the fault edges) and local fault slip acceleration (e.g., refs. [21][22][23] ). We use a kinematic representation of the rupture process. ...

An unusually damaging Mw 4.9 earthquake occurred on November 11, 2019 in the south east of France within the lower Rhône river valley, an industrial region that hosts several operating nuclear power plants. The hypocentre of this event occurred at an exceptionally shallow depth of about 1 km. Here we use far-field seismological observations to demonstrate that the rupture properties are consistent with those commonly observed for large deeper earthquakes. In the absence of strong motion sensors in the fault vicinity, we perform numerical predictions of the ground acceleration on a virtual array of near-fault stations. These predictions are in agreement with independent quantitative estimations of ground acceleration from in-situ observations of displaced objects. Both numerical and in-situ analyses converge toward estimates of an exceptional level of ground acceleration in the fault vicinity, that locally exceeded gravity, and explain the unexpectedly significant damage.

... Theoretical and numerical studies indeed show that near-fault ground acceleration is primarily controlled by local processes on the fault, occurring near the recording site, including local stress drop, strong spatial variations of the rupture velocity (in particular when the rupture front hits the fault edges) and local fault slip acceleration (e.g. ref. [21][22][23] ). We use a kinematic representation of the rupture process. ...

On November 11, 2019, an unusually damaging Mw4.9 earthquake occurred in the south east of France within the lower Rhône river valley, an industrial region hosting several operating nuclear power plants. This event is exceptional considering its very shallow depth (<1 km). Based on farfield seismological observations, we demonstrate that the rupture properties are consistent with the ones commonly observed for large deeper earthquakes, implying that the near-surface faulting generated strong high-frequency seismic waves. In the absence of strong motion sensors in the fault vicinity, we perform numerical predictions of the ground acceleration on a virtual array of near-fault stations, that matches with the locations of independent quantitative estimations from in-situ observations of displaced objects (natural and anthropic). Both numerical and in-situ analyses converge toward an exceptional level of ground acceleration in the fault vicinity, exceeding gravity, and at the origin of the damage. This dramatically changes the perception of the impacts of superficial moderate earthquakes on seismic hazard assessment.

... The variation in earthquake rupture speed is an important parameter that affects the ground motion (Madariaga, 1983) and is one of the sources of high-frequency radiation (Madariaga, 1977;Spudich and Frazer, 1984;Vallée, Landès, Shapiro, and Klinger, 2008). Some theoretical analyses suggest that supershear ruptures emit diminished high-frequency radiation (Burridge, 1973;Andrews, 1976). ...

We analyze the ground motions of supershear ruptures in the Burridge-Andrews (BAM) and free-surface-induced (FSI) mechanisms. BAM supershear ruptures require higher initial shear stress than FSI supershear ruptures, thus cause faster rupture speed. The supershear slip pulse initiates at the free surface or upper boundary of a fault for FSI and at the hypocenter depth in the in-plane direction for BAM in homogeneous models. Both BAM and FSI supershear ruptures can generate Mach waves and the Rayleigh wave field. The BAM and the faster FSI supershear ruptures can make stronger shear Mach waves. The shear Mach waves show a larger amplitude of negative component for a faster FSI supershear rupture. The FSI supershear ruptures make the strongest dilatational wave next to the fault while it is at a distance away from the fault for the BAM supershear ruptures. The different shapes of supershear rupture front, locations of supershear triggered and the rupture speeds contribute to these contrasts. And we find that the BAM and the faster FSI supershear ruptures can trigger more high-frequency radiation due to the sharper and stronger shear Mach waves. Finally, the comparisons between the simulated seismograms and the records of PS10 indicate that the 2002 Denali earthquake may represent a supershear one with the FSI mechanism and a slow supershear rupture speed.

... To show the effect of magnitude on results, Figure 9 compares the three-component velocity waveforms of three PBSs scenarios at two sites, Atatürk Airport (in the following: Airport) and Burgazada (Prince Island) located in the western and eastern part of Istanbul, respectively. Furthermore, to highlight the relevance of hypocenter and slip distribution for a given M w , in Figure 10 the FD effect, it is useful to compare the isochrone lines that connect the locus of points on the fault characterized by the same arrival time (Bernard and Madariaga, 1984;Spudich and Frazer, 1984). The latter, referred also to as isochrone time, can be computed for a given station and a single point on the fault, as the sum of two contributions: (1) the time that the rupture front takes to reach a point on the fault (rupture time) and (2) the time that the seismic wave takes to reach a station from that point (travel time). ...

In this article, the outcomes of a research cooperation between Politecnico di Milano, Italy, and Munich RE, Germany, aiming to improve ground-motion estimation in the Istanbul area through 3D physics-based numerical simulations (PBSs), are illustrated. To this end, 66 PBSs were run, considering earthquake scenarios of magnitude ranging from Mw 7 to 7.4 along the North Anatolian fault (NAF; Turkey), offshore Istanbul. The present article focuses on the detailed introduction of the simulated scenarios comprising: (1) the setup of the 3D numerical model, (2) the validation of the model with recordings of a recent earthquake, (3) the PBSs results, (4) a parametric study on the effect of different features of the seismic source, and (5) a comparison with well-established ground-motion prediction equations to highlight the main differences resulting from the use of a standard empirical approach as opposed to physics-based “source-to-site” numerical simulations. As a main outcome of this study, we observed as, for magnitude Mw 7 and 7.2, PBSs are in agreement with empirical prediction models whereas, for magnitude Mw 7.4, PBSs provide higher ground-motion estimates, as a consequence of directivity effects, amplified by the specific geometry of the portion of the NAF facing Istanbul.

... A key observation in the current study is that heterogeneous rupture propagation, even when the initial stress pattern is homogeneous outside of the nucleation zone, can lead to highly heterogenous ground motion patterns in the near field. This result is not surprising, as seismologists have long known (e.g., Spudich & Frazer, 1984) that spatial variations in rupture propagation and slip amplitude both can lead to bursts of seismic radiation. However, it is worth emphasizing that the heterogeneous ground motion patterns seen in Figures 4 and 11, as well as the differences between the ground motions within each figure, are due to the different geometric relationships for how the dynamic rupture develops and propagates. ...

The free surface is shown to be one of the key factors that may promote supershear rupture propagation on strike‐slip faults even if its initial shear stress is not larger enough as predicted by the Burridge‐Andrews mechanism. However, previous study has shown the free surface‐induced supershear rupture may be unsustained, which turns to sub‐Rayleigh rupture itself as the rupture propagates. We study the near‐field ground motion of sustained and unsustained supershear ruptures using the finite difference method based on the dynamic rupture processes of three vertical strike‐slip faults with the same initial stresses and different hypocenter depths. Both the unsustained supershear ruptures with shallower hypocenter depths show the sub‐Rayleigh characteristics in the peak ground velocity distribution, that is, strong amplitude is noticed beyond the end of the fault without any observed Mach cone. We observe that the arrival time differences between the maximum fault‐perpendicular and fault‐parallel velocity reveal the Mach cone clearly for the sustained supershear rupture. A distinct supershear phase in the high‐frequency seismograms is observed. We also compare the 70° dip strike‐slip case in which unsustained supershear rupture may still present sub‐Rayleigh characteristics in the near‐field ground motion, and the break of symmetry perturbs the normal stress as rupture propagates, which has a key influence in the rupture propagation and ground motion. Our work provides a new insight to understand the supershear rupture in nature earthquakes and the relationship between the ground motion and the rupture process on the fault.

... The density of the measure, can be interpreted as the intensity of isolines (isochrons) of the function on the set Such an object is not new to seismologists (see (Spudich and Frazer, 1984)). Its importance stems from the fact that the analytical properties of the density are intimately related to the HF asymptotics of ...

... Manifestations of this complexity include rapid accelerations and decelerations of the rupture front, slip heterogeneity (Chester & Chester 2000;Dieterich & Smith 2009;Dunham et al. 2011b;Shi & Day 2013), resistance to slip (Dieterich & Smith 2009;Fang & Dunham 2013), supershear transitions (Bruhat et al. 2016), variability in moment release (Zielke et al. 2017), in nucleation processes (Harbord et al. 2017;Ozawa et al. 2019), and inelastic deformation (Hirakawa & Ma 2018). Such rupture behavior is also of particular interest to earthquake engineers when modeling building response, since rupture variability produces high frequency waves, and subsequent ground motion (Haskell 1964;Spudich & Frazer 1984;Dunham et al. 2011b;Shi & Day 2013). Irregularities in fault geometry provide a simple explanation for commonly observed spatial and temporal variations of fault slip (Andrews 1980). ...

Field studies have characterized natural faults as rough, nonplanar surfaces at all scales. Fault roughness induces local stress perturbations during slip, which dramatically affect rupture behavior, resulting in slip heterogeneity. However, the relation between fault roughness and slip heterogeneity remains a key knowledge gap between current numerical and field studies. In this study, we analyze numerical simulations of earthquake rupture to determine how roughness influences final slip. Using a rupture catalog containing thousands of dynamic rupture simulations on band-limited self-similar fractal fault profiles with varying roughness and background shear stress levels, we quantify how fault roughness affects the spectral characteristics of the resulting slip distribution. We find that slip distributions become increasingly more self-affine, that is, containing more short wavelength fluctuations as compared to the self-similar fault profiles, as roughness increases. We also find that, at very short wavelengths (<1km), the fractal dimension of the slip distributions dramatically changes with increasing roughness, background shear stress, and rupture speed (sub-Rayleigh vs. supershear). The existence of a critical wavelength around 1 km, under which more short wavelengths are either preserved or created, suggests the role of rupture process and dynamic effects, together with fault geometry, in controlling the final slip distributions. The same spectral analysis is performed on high-resolution coseismic surface slip distributions from a catalog of real strike-slip earthquakes. Compared to numerical simulations, all earthquakes feature slip distributions that are much more self-affine than the slip distributions from numerical simulations. A different critical wavelength, here around 5-6 km, appears, potentially informing about a critical asperity length. While we show here that the relation between fault roughness and slip is much more complex than expected, this study is a first attempt at using statistical analyses of numerical simulations on rough faults to investigate observed coseismic slip distributions.

... We set up a 600-m × 600-m subvertical fault (dip of 84 • ) with 5-m × 5-m uniform grid cells, centered around the hypocenter of the MSH (47.421073 • , 9.319911 • , depth 4.332 km) (Diehl et al., 2017). Using the P and S wave velocity model from Diehl et al. (2017), and fixing the same constant rupture velocity in both along-dip and along-strike directions (i.e., circular rupture), the P, Sn, and Se RSTFs are back-projected over the rupture isochrones, that is, points on the fault plane that are "seen" as rupturing at the same time from the relative station position (Bernard & Madariaga, 1984;Spudich & Frazer, 1984). Without any a priori information on the moment rate on the fault, the RSTFs are uniformly distributed (back-projected) on the corresponding isochrones, and the resulting moment rate map (IBP image) is the spatial average on the fault of the back-projected RSTFs. ...

Plain Language Summary
In most models and analyses, small earthquakes (i.e., magnitude less than 4) are considered either point sources or homogeneous “penny‐shaped” surfaces. While these assumptions may be valid, details of earthquake ruptures are more complex. Here we study a magnitude 3 induced earthquake that occurred in the St. Gallen geothermal reservoir (NE Switzerland) in 2013. We image the earthquake rupture by refocusing on the fault the seismic energy recorded at six sensors located within 15 km from the source. Our results show that a detailed description of the rupture process of such a small earthquake can indeed be obtained: The rupture propagates from the hypocenter in NNE direction for 150 m, with an average velocity of 2 km/s, breaking into a less active portion of the fault, where no earthquake was previously recorded. The proposed method could be routinely applied during geothermal reservoir operations to allow rapid assessment of fault structures involved in the reservoir creation process.

... Now we will try to informally explain the HF asymptotics found above. In our models, the source function near the frontal surface had a singularity of the object is not new to seismologists (see [Spudich and Frazer, 1984] ...

This paper discusses local features in the source function that generate in the far zone simultaneously (a) a quadratic decay of the source spectrum and (b) the loss of radiation directivity at high frequencies. A. Gusev drew attention to this problem and suggested that a positive solution can be obtained for earthquake rupture front with a rather complex ("lace") structure. Below we give a theoretical solution of the problem and show that the front structure can be simple enough, but not smooth.

... Yagi et al. (2012a) suggested that this algorithm provides higher resolution images of seismic energy release than the conventional back-projection methods that produced more blurred images; hence it gives the more robust distribution of the high-frequency radiation with the detailed motion of rupture front and frequency dependence. It is known that the high-frequency radiation reflects the complex rupture process (e.g., Spudich and Frazer 1984). The HBP method also differs from waveform inversion in that it uses a deconvolution process to mitigate the effect of subsurface structure. ...

Seismological observations provide essential input parameters for numerical tsunami simulations. Here, we present source mechanism parameters, finite-fault source rupture models and numerical tsunami simulation results for the destructive October 28, 2012 Haida Gwaii-Canada (Mw 7.7) and September 16, 2015 Illapel-Chile (Mw 8.3) earthquakes and resulting tsunamis. These two earthquakes were controlled by active tectonic features along the subduction zones that had developed in response to the convergent movements of lithospheric plates. The faulting geometry (strike, dip, and rake angles), focal depth, fault dimensions, average and maximum slip values on the fault planes and seismic moments of the earthquakes are estimated by analyzing teleseismic long-period P- and SH-waves and broadband P-waveforms and using waveform inversion and hybrid back-projection methods. The obtained slip models of the earthquakes reveal heterogeneous slip distributions on fault planes with long source durations (~ 80 s and 150 s) and low stress drop values (10–15 bars). Numerical simulations of tsunami wave propagation are further performed using the uniform and non-uniform slip models and nonlinear long-wave equations in spherical coordinates. The shape and arrival times of leading tsunami waves are adequately constrained particularly with the heterogeneous slip distribution models. The general characteristics of synthetic tsunami waveforms (e.g., amplitude, shape, arrival time) calculated using the non-uniform slip model, are more consistent with the observed tsunami records than those of a uniform slip model. It is further seen that simulation results using preliminary and fast slip models for both earthquakes give only approximate early tsunami estimates; tsunami wave heights and arrival times to the coasts are mostly not well simulated. The results indicate that tsunami simulations based on finite-fault source slip models likely contribute to the determination of tsunamigenic coastal regions by revealing locations, arrival times, amplitudes, and directions of tsunami waves within a close approximation to observed records off-shore and far from the source region. They provide sufficient information to facilitate tsunami warning and mitigation challenges after the destructive earthquakes. We further suggest that joint inversions of GPS, tsunami, teleseismic and strong ground motion records and higher resolution bathymetry data are needed in order to obtain better correlations between observed and synthetic tsunami data, especially for the later arriving waves.

... A simple yet powerful method for understanding the general properties of seismic radiation from classical dislocation models was proposed by Bernard and Madariaga (1984) and Spudich and Frazer (1984). The method was recently extended to study radiation from supershear ruptures by Bernard and Baumont (2005). ...

Earthquakes are due to fast internal deformation in the Earth, mainly to sudden slip on active faults. We study the origin of seismic sources from first principles, starting with a classical point force. Then, we introduce general moment tensor sources as the result of internal inelastic deformation in the Earth; and we discuss seismic wave radiation and energy balance. Once these basic results are established, we study seismic radiation from extended source models. The effect of finiteness on seismic radiation is to produce focusing along directions normal to the fault and directivity in the direction of rupture propagation. We conclude by introducing some general features of dynamic source models like stress and slip velocity concentration, energy release rate, energy balance, and the general properties of seismic radiation from a circular fault.

... If we are to interpret the backprojection results in terms of earthquake dynamics, understanding rigorously their relation is critical. Theoretical studies indicate that the high-frequency seismic waves can be excited during abrupt changes in rupture velocity (Bernard & Madariaga, 1984;Madariaga, 1977;Spudich & Frazer, 1984) caused either by the arrest of the rupture (Madariaga, 1976) or by kinks of the fault geometry (Madariaga et al., 2006). Moreover, Fukahata et al. (2014) propose that the backprojection images are equivalent to either slip or slip rate on the fault, provided that the Green's functions from the sources to the receivers are incoherent delta functions. ...

We develop a methodology that combines compressive sensing back-projection (CS-BP) and source spectral analysis of teleseismic P waves to provide metrics relevant to earthquake dynamics of large events. We improve the CS-BP method by an auto-adaptive source grid refinement as well as a reference source adjustment technique to gain better spatial and temporal resolution of the locations of the radiated bursts. We also use a two-step source spectral analysis based on i) simple theoretical Green's functions that include depth phases and water reverberations and on ii) empirical P-wave Green's functions. Furthermore, we propose a source spectrogram methodology that provides the temporal evolution of dynamic parameters such as radiated energy and falloff rates. Bridging back-projection and spectrogram analysis provides a spatial and temporal evolution of these dynamic source parameters. We apply our technique to the recent 2015 Mw 8.3 megathrust Illapel earthquake (Chile). The results from both techniques are consistent and reveal a depth-varying seismic radiation that is also found in other megathrust earthquakes. The low frequency content of the seismic radiation is located in the shallow part of the megathrust, propagating unilaterally from the hypocenter towards the trench while most of the high frequency content comes from the downdip part of the fault. Interpretation of multiple rupture stages in the radiation is also supported by the temporal variations of radiated energy and falloff rates. Finally, we discuss the possible mechanisms, either from pre-stress, fault geometry, and/or frictional properties to explain our observables. Our methodology is an attempt to bridge kinematic observations with earthquake dynamics.

... As explained by Halldorsson et al. [37], the subevent time histories are subsequently summed up at the station, appropriately lagged in time accounting for the time it takes the rupture front to reach the subevent and for the travel time of the seismic radiation from the subevent to the station. For a particular station, the arrival time of the seismic radiation emitted by each subevent is estimated using the concept of isochrones [93,67,40]. The isochrones are computed based on the rupture times obtained from the low-frequency simulation and the travel times obtained using the fault-to-station geometry and the 1-D velocity model summarized in Table 1. ...

The 1995 MW 6.4 Aigion earthquake is one of the largest and most destructive seismic events that have occurred in Greece over the past few decades. The ground shaking in the near-fault region was recorded by a strong-motion accelerograph in the city of Aigion, at a distance of about 16 km from the epicenter. The recorded horizontal ground acceleration exceeded 0.5 g, whereas the horizontal components of ground velocity exhibited pulse-like motions of large amplitude. These ground-motion characteristics have been attributed to forward rupture directivity combined with the effects of soil and topography. In this article, broadband synthetic ground motions are generated at selected locations and at a dense grid of observation points extending over the causative fault of the 1995 Aigion earthquake using a hybrid deterministic-stochastic method. The low-frequency components of the synthetic ground motion are simulated using the discrete wavenumber method and the generalized transmission and reflection coefficient technique, whereas the high-frequency components of the synthetic ground motion are generated using the stochastic modeling approach and the specific barrier model. The two independently derived ground-motion components are then combined using matched filtering at a crossover frequency of 2 Hz to generate broadband ground-motion time histories and response spectra. The effects of soil and topography on the simulated ground motion in the city of Aigion are also investigated through site response analysis. In addition, the strong motion recorded at Aigion is corrected for crustal anisotropy using the cross-correlation technique, thus further enhancing the alignment of recorded and synthetic ground-motion time histories. Finally, the synthetic ground motions are compared with ground-motion estimates obtained from observed geotechnical damage, USGS ShakeMaps, and ground-motion prediction equations.

... The saturation effect for scaling large magnitudes associated with initial P wave data can be explained in terms of the concept of isochrones, which define a set of points on a fault plane whose radiation arrives at a given station at a certain time (Spudich and Frazer, 1984;Bernard and Madariaga, 1984). The fault area, which generates high frequency seismic radiation, might be encompassed by the initial P wave isochrones in a short time. ...

We propose a method that employs the squared displacement integral (ID2) to estimate earthquake magnitudes in real time for use in earthquake early warning (EEW) systems. Moreover, using τc and Pd for comparison, we establish formulas for estimating the moment magnitudes of these three parameters based on the selected aftershocks (4.0 ≤ Ms ≤ 6.5) of the 2008 Wenchuan earthquake. In this comparison, the proposed ID2 method displays the highest accuracy. Furthermore, we investigate the applicability of the initial parameters to large earthquakes by estimating the magnitude of the Wenchuan Ms 8.0 mainshock using a 3-s time window. Although these three parameters all display problems with saturation, the proposed ID2 parameter is relatively accurate. The evolutionary estimation of ID2 as a function of the time window shows that the estimation equation established with ID2Ref determined from the first 8-s of P wave data can be directly applicable to predicate the magnitudes of 8.0. Therefore, the proposed ID2 parameter provides a robust estimator of earthquake moment magnitudes and can be used for EEW purposes.

We image the rupture process of the 2021 Mw 7.4 Maduo, Tibet earthquake using slowness-enhanced back-projection and joint finite fault inversion, which combines teleseismic broadband body waves, long-period (166-333 s) seismic waves, and 3D ground displacements from radar satellites. The results reveal a left-lateral strike-slip rupture, propagating bilaterally on a 160-km-long north-dipping sub-vertical fault system that bifurcates near its east end. About 80% of the total seismic moment occurs on the asperities shallower than 10 km, with a peak slip of 5.7 m. To simultaneously match the observed long-period seismic waves and static displacements, notable deep slip is required, despite a tradeoff with the rigidity of the shallow crust. This coseismic deep slip within the ductile middle crust could result from strain localization and dynamic weakening. Local crustal structure and synthetic long-period Earth response for Tibet earthquakes thus deserve further investigation. The WNW branch ruptures ~75 km at ~2.7 km/s, while the ESE branch ruptures ~85 km at ~3 km/s, though super-shear rupture propagation possibly occurs during the ESE propagation from 12 s to 20 s. Synthetic back-projection tests confirm overall sub-shear rupture speeds and reveal a previously undocumented limitation caused by the signal interference between two bilateral branches. The stress analysis on the forks of the fault demonstrates that the pre-compression inclination, rupture speed, and branching angle could explain the branching behavior on the eastern fork.

Backprojection (BP) methods are widely used for estimating the fault rupture processes; however, they are inherently susceptible to noise. Hence, noise suppression is an important research target. In this paper, we develop a fault rupture imaging method by combining beamforming‐based BP and MUltiple Signal Classification (MUSIC), which realizes artificial noise suppression at a high spatial resolution. The stations are grouped into arrays according to the SH wave radiation coefficients, and MUSIC analysis is performed on each array. The MUSIC spectral images of these arrays are binarized and then multiplied by the BP images. Spatial filtering is also applied to the images based on the possible range of the rupture velocity and rise time. When tested using synthetic test data, the proposed method worked as expected. We then applied this method to the 2016 Kumamoto earthquake by interpolating the travel times from the observed travel times of relocated hypocenters using a 3‐D velocity structure model. In the area of large slip and slip rate approximately 30–50 km from the hypocenter on the Futagawa fault, the spatiotemporal evolution of the fault ruptures and waveform inversion results were generally in harmony. The distributions of the low‐ and high‐frequency seismic radiations are complementary, as is understood in the context of fault rupture physics. This method can aid in understanding and modeling the details of seismic radiation sources, enabling the accurate prediction of strong ground motion even in near‐fault areas.

We constructed an integrated rupture model of the 2021 Mw 7.1 Fukushima earthquake, an intraplate earthquake, by resolving both its spatiotemporal distribution of slip-rate and high-frequency (∼1 Hz) radiations. We analyzed near-field seismic observations using a novel finite-fault inversion method that allows automatic parameterization and teleseismic data from multiple arrays using the MUSIC Backprojection (BP) method that enhances imaging resolution. The inverted slip distribution obtained from waveforms filtered in the frequency band of 0.02–0.2 Hz showed that the kinematic rupture propagated along both the strike (∼35 km) and dip directions (∼85 km), and that the large-slip area was located southwest to the hypocenter with a maximum slip of ∼1.03 m. Overall, no obvious frequency-dependent rupture behaviors occurred during the rupture process due to the deep nucleation of the Fukushima earthquake on a heterogeneous fault where sizes of asperities do not monotonically increase with depth, which sheds light on understanding the rupture dynamics of intraplate earthquakes in subduction zones. Both the slip inversion and BP revealed the general rupture feature of this earthquake with southwestward and up-dip directivity. A comparison of BPs between multiple arrays indicates that the source-receiver geometry and the directivity effect of an earthquake may cause critical discrepancies in BPs of different arrays. From the temporal change of stress around the hypocenter of the 2021 Fukushima earthquake due to the 2011 Tohoku-Oki Mw 9.1 earthquake, the long-term dominance of viscoelastic relaxation increased the Coulomb failure function (CFF) by 0.3–0.7 MPa, indicating that the occurrence of the Fukushima earthquake has been likely promoted by the postseismic deformation due to the Tohoku-Oki earthquake.

The 2008 Mw 7.9 Wenchuan earthquake, one of the largest continental intraplate events instrumentally recorded, struck the central part of Sichuan Province in southwestern China causing great destruction and loss of life but also providing a wealth of seismological data, geodetic measurements, and tectonic observations. The Wenchuan earthquake ruptured two northwest‐dipping imbricate oblique reverse faults along the middle segment of the Longmenshan fault zone—a northeast‐trending thrust belt located at the boundary between the Tibetan Plateau and the Sichuan Basin. In this study, a hybrid approach that combines deterministic modeling at low frequencies with stochastic modeling at high frequencies is used to simulate broadband ground motions at 52 strong‐motion stations and 506 geodetic sites in the vicinity of the causative fault. The low‐frequency components of the synthetic ground motion are simulated using an extended kinematic source model embedded in a layered medium, whereas the high‐frequency components are generated using a stochastic finite‐fault model. The two independently derived ground‐motion components are then combined using matched filtering at a crossover frequency of 0.8 Hz to generate broadband ground‐motion time histories and response spectra. The temporal and spectral characteristics of the synthetic and recorded ground motions at the 52 strong‐motion stations are compared and the effect of soil nonlinearity on the simulated ground motions is investigated through 1‐D nonlinear site response analysis. Finally, the simulated permanent ground displacements at the 506 geodetic sites are evaluated against geodetic observations and the peak amplitudes of the synthetic ground motions at the same locations are compared with predictions of empirical ground‐motion models.

The back projection method is a tremendously powerful technique for investigating the time dependent earthquake source, but its physical interpretation is elusive. We investigate how earthquake rupture heterogeneity and directivity can affect back‐projection results (imaged location and beam power) using synthetic earthquake models. Rather than attempting to model the dynamics of any specific real earthquake, we use idealized kinematic rupture models, with constant or varying rupture velocity, peak slip rate, and fault‐local strike orientation along unilateral or bilateral rupturing faults, and perform back‐projection with the resultant synthetic seismograms. Our experiments show back‐projection can track only heterogeneous rupture processes; homogeneous rupture is not resolved in our synthetic experiments. The amplitude of beam power does not necessarily correlate with the amplitude of any specific rupture parameter (e.g., slip rate or rupture velocity) at the back‐projected location. Rather, it depends on the spatial heterogeneity around the back‐projected rupture front, and is affected by the rupture directivity. A shorter characteristic wavelength of the source heterogeneity or rupture directivity toward the array results in strong beam power in higher frequency. We derive an equation based on Doppler theory to relate the wavelength of heterogeneity with synthetic seismogram frequency. This theoretical relation can explain the frequency‐ and array‐dependent back‐projection results not only in our synthetic experiments but also to analyze the 2019 M7.6 bilaterally rupturing New Ireland earthquake. Our study provides a novel perspective to physically interpret back‐projection results and to retrieve information about earthquake rupture characteristics.

Most earthquake ruptures propagate at speeds below the shear wave velocity within the crust, but in some rare cases, ruptures reach supershear speeds. The physics underlying the transition of natural subshear earthquakes to supershear ones is currently not fully understood. Most observational studies of supershear earthquakes have focused on determining which fault segments sustain fully grown supershear ruptures. Experimentally cross-validated numerical models have identified some of the key ingredients required to trigger a transition to supershear speed. However, the conditions for such a transition in nature are still unclear, including the precise location of this transition. In this work, we provide theoretical and numerical insights to identify the precise location of such a transition in nature. We use fracture mechanics arguments with multiple numerical models to identify the signature of supershear transition in coseismic off-fault damage. We then cross-validate this signature with high-resolution observations of fault zone width and early aftershock distributions. We confirm that the location of the transition from subshear to supershear speed is characterized by a decrease in the width of the coseismic off-fault damage zone. We thus help refine the precise location of such a transition for natural supershear earthquakes.

A devastating magnitude 7.2 earthquake struck Southern Haiti on 14 August 2021. The earthquake caused severe damages and over 2000 casualties. Resolving the earthquake rupture process can provide critical insights into hazard mitigation. Here we use integrated seismological analyses to obtain the rupture history of the 2021 earthquake. We find the earthquake first broke a blind thrust fault and then jumped to a disconnected strike-slip fault. Neither of the fault configurations aligns with the left-lateral tectonic boundary between the Caribbean and North American plates. The complex multi-fault rupture may result from the oblique plate convergence in the region that the initial thrust rupture is due to the boundary-normal compression and the following strike-slip faulting originates from the Gonâve microplate block movement, orienting towards the SW-NE direction. The complex rupture development of the earthquake suggests that the regional deformation is accommodated by a network of segmented faults with diverse faulting conditions.

The retrieval of earthquake finite-fault kinematic parameters after the occurrence of an earthquake is a crucial task in observational seismology. Routinely-used source inversion techniques are challenged by limited data coverage and computational effort, and are subject to a variety of assumptions and constraints that restrict the range of possible solutions. Back-projection (BP) imaging techniques do not need prior knowledge of the rupture extent and propagation, and can track the high-frequency (HF) radiation emitted during the rupture process. While classic source inversion methods work at lower frequencies and return an image of the slip over the fault, the BP method underlines fault areas radiating HF seismic energy. HF radiation is attributed to the spatial and temporal complexity of the rupture process (e.g., slip heterogeneities, changes in rupture speed and in slip velocity). However, the quantitative link between the BP image of an earthquake and its rupture kinematics remains unclear. Our work aims at reducing the gap between the theoretical studies on the generation of HF radiation due to earthquake complexity and the observation of HF emissions in BP images. To do so, we proceed in two stages, in each case analyzing synthetic rupture scenarios where the rupture process is fully known. We first investigate the influence that spatial heterogeneities in slip and rupture velocity have on the rupture process and its radiated wave field using the BP technique. We simulate different rupture processes using a 1D line source model. For each rupture model, we calculate synthetic seismograms at three teleseismic arrays and apply the BP technique to identify the sources of HF radiation. This procedure allows us to compare the BP images with the causative rupture, and thus to interpret HF emissions in terms of along-fault variation of the three kinematic parameters controlling the synthetic model: rise time, final slip, rupture velocity. Our results show that the HF peaks retrieved from BP analysis are better associated with space-time heterogeneities of slip acceleration. We then build on these findings by testing whether one can retrieve the kinematic rupture parameters along the fault using information from the BP image alone. We apply a machine learning, convolutional neural network (CNN) approach to the BP images of a large set of simulated 1D rupture processes to assess the ability of the network to retrieve from the progression of HF emissions in space and time the kinematic parameters of the rupture. These rupture simulations include along-strike heterogeneities whose size is variable and within which the parameters of rise-time, final slip, and rupture velocity change from the surrounding rupture. We show that the CNN trained on 40,000 pairs of BP images and kinematic parameters returns excellent predictions of the rise time and the rupture velocity along the fault, as well as good predictions of the central location and length of the heterogeneous segment. Our results also show that the network is insensitive towards the final slip value, as expected from a theoretical standpoint.
https://eartharxiv.org/repository/view/2528/

Teleseismic P, SH, and SV first motions and SH to SV amplitude ratios recorded at eight teleseismic receivers from the 1949 magnitude 7.1 Olympia, Washington, earthquake in combination with data from three stations at regional distances were utilized in a grid testing routine to constrain focal mechanism. Identification of the pP phase places the event at 54 km depth. Distinct pulses, assumed to be source effects, are observed in the far-field waveforms. Analysis of these pulses for directivity made possible discrimination between the fault and auxiliary planes. The plane taken to represent the fault surface strikes east-west ± 15°, dips 45° ± 15° to the north, and has nearly pure left-lateral slip. The preferred source model has an eastward propagation of 40 km. Surface reflections of successive source pulses suggest an upward component of propagation of 5 km. Bounds on the earthquake location and rupture of the 13 April event were determined using depth and source mechanism constraints from the teleseismic study and characteristics of local strong ground motion recordings. The 9-sec S-instrument trigger time seen in the Seattle acceleration recordings places the event at least 60 km from Seattle. Strong motion velocity at the Olympia Highway Test Laboratory is characterized by an impulsive and rectilinear S wave. The low amplitude of the vertical component of initial S motion suggests that either the epicenter is within 5 km of the Olympia Highway Test Laboratory for a pure incident SV wave or located along an azimuth of N159° if the wave is SH. The combined constraint of minimum distance from Seattle and the S polarization angle implied by the teleseismic data focal mechanism places the initiation of rupture 5 to 10 km north to north-northwest of the Olympia Highway Test Laboratory at 47.13°N, 122.95°W. This is approximately 20 km west of previously determined epicenters. The T axis, gently dipping to the southeast, supports other evidence that the Juan de Fuca plate dips to the southeast in a zone between segments of the plate north and south of the event's location. The fault plane's slip is taken to indicate that subduction is still active beneath Washington and that motion of the two segments is probably independent.

Specific Barrier Method (SBM) is a method used for ground motion generation from a finite fault surface. It is based on a regular distribution of rupturing circular subevents located on the fault plane and random arrival times of the waves generated by those cracks. This approach does not consider the whole rupture kinematics, i.e. the rupture propagation from the hypocenter to the subevents, and leaves parts of the fault unbroken (barriers). In this paper, we propose a modified version of SBM for generation of synthetic ground motions from a fault surface. In this version, we modify the Probability Density Function (PDF) for the arrival time of the waves coming from different parts of the fault in order to better account for the fault kinematics and the distance between fault point and receiver. In this way, we can simulate the middle part of the acceleration spectrum (i.e. between 0.1 and 7 Hz) with more accuracy. Moreover, a new arrangement for locating cracks throughout the fault plane is proposed to add flexibility to the model and enable it to make the part of the spectrum with frequency larger than 7 Hz more like what happens in nature. In such an arrangement, called ‘geometry packing’ in this paper, the size of circles varies within a chosen specific allowable range, while the circles cover all over the fault plane without any overlaps. To validate the proposed modified SBM technique, the synthetic Fourier spectra are compared with recordings of the 2008 Mw6.9 Iwate-Miyagi (Japan) earthquake. Finally, we present some parametric studies to show how different features of the proposed PDFs affect the results from the SBM approach.

We conduct a study on 3D seismic b values in southwest Taiwan and provide new insight into the characteristics of seismogenic structures through the heterogeneity of stress distribution and the variations in stress states in a seismic cycle. Our results indicate that, prior to the 2010 Jiasian earthquake and the 2016 Meinong earthquake, significantly low seismic b values occurred in the upper crust. For depths greater than 14 km, stress heterogeneity was more significant. We also propose a new scenario with a block rotation model derived from GPS data to explain how the rupture directions of the 2010 Jiasian and 2016 Meinong earthquakes are associated with the development of new fault systems in the mid-crust region.

We studied the low- and high-frequency source processes of the 2017 Jiuzhaigou earthquake, and focused on when and where the high-frequency (2.0–10.0 Hz) seismic energy was radiated in relation to the low-frequency (0.02–0.5 Hz) seismic energy. The low-frequency source processes were jointly inverted using teleseismic P and SH broadband waves combined with near-field strong motion data. A three-dimensional model considering real topography was used to compute the near-field Green's function. The high-frequency wave radiation was inversed using the envelope of acceleration seismograms of near-field records and the empirical Green's function by the differential evolution algorithm. The results showed the 2017 Jiuzhaigou earthquake was a typical strike-slip earthquake with a seismic moment of about 6.9 × 10 ¹⁸ N m and maximum slip of about 1.5 m. During the 2 s following the initial rupture, the earthquake radiated large high-frequency waves but small low-frequency waves. Then, the rupture broke the high stress drop area in the up-dip direction of the hypocenter, generating an obvious asperity located just above the hypocenter. About 60% of the seismic moment of the earthquake was released during about 2–6 s after initiation of the rupture. The high-frequency radiation areas were mostly located on the lower periphery of the asperity, in accordance with the complementary relationship between the distribution of high-frequency waves and zones of large slip, which might be attributable to both the initiation of the rupture and the breaking of the high stress drop area. Parts of the high-frequency radiation areas overlapped the lower part of the asperity, where the high-frequency radiation occurred about 2 s earlier than the low-frequency radiation. The stress drop of the 2017 Jiuzhaigou earthquake was considerably lower than average for global intraplate earthquakes, but similar to that of the 2013 Lushan earthquake.

Waveform backprojection (BP) is a key technique of earthquake-source imaging, which has been widely used for extracting information of earthquake source evolution that cannot be obtained by kinematic source inversion. The technique enjoys considerable popularity, owing to the simplicity of its implementation and the robustness of its processing, but the physical meaning of BP images has remained elusive. In this study, we reviewed the mathematical representation of BP and hybrid BP (HBP) methods, following the pioneering work of Fukahata et al. (2014), to clarify the physical implications of BP images. We found that signal intensity in BP and HBP images is scaled with the amplitude of the Green’s function that corresponds to a unit-step slip, which results in the signal intensity being depth dependent. We propose variants of BP and HBP, which we call kinematic BP and HBP, respectively, to relate the BP signal intensity to slip motion of an earthquake by modifying the normalizing factors used in the original BP and HBP methods. The original BP and HBP images remain useful for assessing the spatiotemporal strength of the wave radiation, which scales with the amplitude of the Green’s function, whereas the kinematic BP and HBP methods are suitable for imaging the slip motion that is responsible for the high-frequency radiation produced during the source-rupture process.

Earthquake early warning (EEW) system is one of the most useful tools to mitigate seismic hazards. Although EEW approaches have already been developed worldwide, the issues of improving the accuracy and applicability were still controversial. On the basis of the existing measurable parameters related to the earthquake magnitude, we proposed a method in terms of the squared integral displacement (ID2) to anticipate the magnitude in real-time for EEW purpose that can be used for large earthquakes. With the main-shock and aftershocks of the 2008 Ms 8.0 Wenchuan earthquake, we investigated the regression relationships between the proposed ID2 and earthquake magnitude, based on which the magnitude predication formula was derived. Then the predicated accuracy from ID2 was compared with that from τc and Pd. By comparing the differences between the estimated and reported magnitudes, the proposed ID2 method was of the highest accuracy among the investigated 3 parameters. Next we analysed the ground motion characteristics in the frequency domain and established the correlations between the Fourier amplitude and spectral intensity of the initial P wave with the magnitude estimation differences, based on which a revision method of reducing such differences was put forward and the saturation effects of predicting large magnitudes were well mitigated simultaneously.

Slip inversion is a widely used analysis method using seismic and/or geodetic data, which determines the spatial and temporal distribution of fault slip, namely, a slip model. This chapter reviews the history, formulation, application, and extension of slip inversion. Constituent elements of slip inversion problem are data preparation, model parameterization, and calculation of synthetic data. For each of these elements, we introduce typical treatments and provide total mathematical formulations. Frequently used data are far-field broadband seismograms, near-field strong motions, and various geodetic data. Both linear and nonlinear parameterizations are possible to represent slip models. While Green's functions for layered structure have been used for synthetic wave calculation, 3-D Green's function and empirical Green's functions are adequate for more complex structures. Different combinations of these constituents can lead to significant differences between slip models of the same event. We then discuss issues associated with the inversion method. Various optimization schemes are applicable to find the best estimates of parameters and their uncertainties. Sometimes, an inversion problem requires additional smoothing constraints for regularization because it tends to be partly underdetermined. The weight of these constraints can be determined objectively using Bayesian modeling. Another topic related to the method is the frequency characteristics of a slip distribution and slip inversion in the frequency domain. As an example of well-studied earthquake, we compare various slip models of the 1999 Chi-Chi, Taiwan, earthquake published by different research groups and find that there are common characteristics among the models. As another method to reveal the spatiotemporal complexity of earthquake rupture, I introduce the back-projection method and compare it with slip inversion, using some results for the 2011 Tohoku-Oki, Japan, earthquake. We also review derivative studies based on slip models. There are some common characteristics in many slip models, such as slip pulse, complementary distributions of aftershocks and large slips, and rupture directivity. Slip models can provide clues to find governing laws and properties of earthquake dynamic rupture directly or indirectly. Attempts to scale the complexity of earthquake ruptures have just started and will be important both for understanding the physics of earthquakes and for reliable strong-motion predictions.

The complexity induced in seismic body waves by rapid or discontinuous spatial changes of elastic moduli and density can be efficiently modeled in radially symmetrical structure by a collection of closely related ray-, integral transform-, and modal-based solutions of the elastic equations of motion. These modeling techniques can be extended to include viscoelastic attenuation, simple point-source representations of earthquake faulting, weakly varying three-dimensional (3D) structure and elastic anisotropy, and some effects of scattering. The effects of stronger and smaller-scale 2D and 3D heterogeneity can be modeled by numerical and hybrid analytic/numerical methods.

Geometric discontinuities within fault systems known as geometric barriers contribute to irregular rupture evolutions during earthquakes. We applied a hybrid backprojection method to high-frequency teleseismic P-waveforms to investigate the role of geometric barriers in the rupture propagation during the MW 7.9 2008 Wenchuan, China, earthquake. We found that sources of high-frequency waves were concentrated near the intersections of a northwesttrending cross-cutting fault with the dominant northeast-trending fault system and in areas around steps between fault segments of the dominant fault system. We recognized these areas as geometric barriers to rupture propagation. Our analysis of the high-frequency waves associated with the geometric discontinuities within the fault system showed that geometric barriers can decelerate or stop rupture propagation, but can also accelerate rupture when the rupture front crosses a geometric barrier and instigates rupture in an adjacent fault segment. Our result suggests that geometric discontinuities within fault systems can cause earthquake rupture propagation that is more complex than that of faults of simpler geometry associated with subduction zone megathrust earthquakes. © The Author(s) 2017. Published by Oxford University Press on behalf of The Royal Astronomical Society.

This chapter introduces the principle of the back-projection method in the frame of the seismic wave propagation. This provides the solid basis for the technique. To further validate the method, synthesized waveforms are generated to apply the back-projection the procedure. The resultant imaging supports the argument that the relative back-projection method can mitigate the “swimming” artifacts in the traditional back-projection imaging results.

The interpretation of refraction profiles that traverse laterally varying velocity structures has been hindered by lack of a practical algorithm for computing synthetic seismograms to compare with observations. However, a modification of zero-order asymptotic ray theory to incorporate the amplitudes of rays that turn in a velocity gradient, as well as reflected waves, allows the computation of high-frequency synthetic seismograms for laterally varying velocity structures. The method is general in that synthetic seismograms may be computed for any structure through which rays can be traced. We have modeled seismic refraction data from the Imperial Valley, California, by applying this method. A major feature of our model is a sedimentary column that thickens from -4 km at the Salton Sea to -5.5 km at the United States- Mexico border. To approximate the observed amplitude behavior, a dipping velocity discontinuity was modeled near 13-km depth, but velocity gradients were found to be more appropriate than sharp boundaries in the rest of the model.

By comparing synthetic particle velocities with the near-source strong motion data, a faulting model for the 1979 Imperial Valley earthquake was constructed. The calculation of the synthetic seismograms takes into account the vertical inhomogeneity of the elastic parameters in the Imperial Valley and the spatial variation of the slip rate parameters on the fault plane. The faulting model has the following principal features: Faulting occurred on the Imperial fault and on the Brawley fault, rupture on the Brawley fault being triggered by rupture on the Imperial fault; The Imperial Fault is a plane 35 km long and 13 km wide with a strike of 323 degree , measured clockwise from north, and a dip of 80 degree NE; Faulting on the Imperial fault is primarily right-lateral strike slip with a small component of normal dip slip in the sediments at its northern end; The rupture velocity on the Imperial fault is highly variable; Although the slip on the Brawley fault contributes only about 4% of the total moment, it greatly affects the ground motion at nearby stations; and The total seismic moment is 6. 7 multiplied by 10**1**8 N m where the Imperial fault contributes 6. 4 multiplied by 10**1** 8 N m and the Brawley fault contributes 2. 7 multiplied by 10**1**7 N m.

An explicit expression is derived for the body force to be applied in the absence of a dislocation, which produces radiation identical to that of the dislocation. This equivalent force depends only upon the source and the elastic properties of the medium in the immediate vicinity of the source and not upon the proximity of any reflecting surfaces. The theory is developed for dislocations in an anisotropic inhomogeneous medium; in the examples isotropy is assumed. For displacement dislocation faults, the double couple is an exact equivalent body force.

We construct a theoretical three-dimensional kinematical model of shallow-focus earthquake faulting in order to investigate the ratio of the P- and S-wave corner frequencies of the far-field elastic radiation. We attempt to incorporate in this model all of the important gross kinematical features which would arise if ordinary mechanical friction should be the dominant traction resisting fault motion. These features include a self-similar nucleation at a single point, a subsonic spreading of rupture away from that point, and a termination of faulting by smooth deceleration. We show that the ratio of the P-wave corner frequency to the S-wave corner frequency for any model which has these features will be less than unity at all points on the focal sphere.

We study high-frequency radiation from a dislocation model of rupture propagation at the earthquake source. We demonstrate that in this case all the radiation emanates from the rupture front and, by a change of variables, that at any instant of time the high-frequency waves reaching an observer come from a line on the fault plane that we call isochrone. An asymptotic approximation to near-source velocity and acceleration is obtained that involves a simple integration along the isochrones for every time step. It is shown that wave front discontinuities (critical or stopping phases) are radiated every time an isochrone becomes tangent to a barrier. This leads to what we call the critical ray approximation which is given in a closed form. The previous results are compared with discrete wavenumber synthetics obtained by Bouchon (1982) for the Gilroy 6 recording of the Coyote Lake earthquake of 1980. The fit between the asymptotic and full numerical method is extremely good. The critical ray approximation permits the identification of different phases in Bouchon's synthetics and the prediction of the behavior of the signal in the vicinity of their arrival time.

In a medium consisting of elastic layers with irregular interfaces, Kirchhoff-Helmholtz (KH) theory can be extended to synthesize the motion due to various generalized rays. An exact elastic form of the KH integral is first derived, then various asymptotic approximations are used to convert this integral into one which can be rapidly evaluated to give the motion of a single generalized ray. A new method for overcoming the problem of a caustic on the reflector becomes apparent when the KH integral is regarded as a member of a larger family of equivalent 1-fold integrals all of which are derivable from the same multifold path integral. For velocity models that are independent of one spatial direction (strike) a method is given for approximately converting 2-D results into 3-D results. -from Authors

The deHoop-Knopoff representation theorem, which relates observed seismic waves to a displacement discontinuity s defined on a surface S, is posed so that seismograms may be directly inverted for estimates of s and of the spatial and temporal resolution of s. Solutions s can be constructed either by parametrizing the fault surface as a number of point double couples or by representing s as an expansion of orthogonal functions. Of the infinite number of possible solutions satisfying a single seismic-data set, methods for constructing particular solutions (e.g., best fitting, or closest to a desired solution) are given. Application of inverse theory to the deHoop-Knopoff representation theorem leads to a convenient way to include various types of seismic data-e.g., long- and short-period teleseismic, near field accelerogram, and geodetic-into a single inversion.

The 1979 Imperial Valley, California, earthquake (Ms -- 6.9) was recorded on the El Centre differential array, a 213-m-long linear array of 5 three-component digital accelerometers 5.6 km from the nearest tectonic surface rupture. Although absolute time was not recorded on the array elements, a relative time base was established using the main shock hypocentral P wave and the P and S waves from a later aftershock. A cross-correlation technique was used to measure the difference in arrival times of individual seismic waves in a moving 0.6 to 1.2 sec window at each array element, which would then be converted into the wave's slowness (1/velocity) along the array. When applied to the main shock vertical and horizontal accelerograms, results from both components of motion indicated that the early arriving energy came from a source to the south of the array, and the source of the energy moved rapidly to the north of the array during the strong shaking. The ground motions at the array elements were well correlated for about the first 11 sec of motion. These observations suggest that we have observed the initiation of rupture south of the array and its subsequent propagation along the fault to a position north of the array in about 10 sec, and that the energy was radiated from a fairly compact region around the rupture front. If the observed vertical and horizontal ground motions are assumed to be caused by P and S waves, respectively, then the observed slownesses show irregularities which can be interpreted as implying that the observed high-frequency ground motions originated at irregularly distributed regions on the fault surface, or that the rupture velocity was variable, or both. One possible interpretation of the data suggests that the rupture proceeded at near P-wave velocity over a 7-km-long section of fault. Average rupture velocities of about 2.7 to 3.2 km/sec at 8 km depth are consistent with the data, and 2.8 km/sec is weakly preferred under the assump- tion that rupture propagates at a fixed fraction of the shear velocity. The large vertical pulse, which had a peak acceleration of 1.7 g at E06, was emitted from the portion of the fault extending 25 to 30 km northwest of the hypocenter near Meloland overpass, and not from the point on the fault closest to the differential array. Nothing can be said about fault behavior southeast of the hypocenter.

Propagation of plane strain shear cracks is calculated numerically by using finite difference equations with second-order accuracy. The rupture model, in which stress drops gradually as slip increases, combines two different rupture criteria: (1) slip begins at a finite stress level; (2) finite energy is absorbed per unit area as the crack advantages. Solutions for this model are nonsingular. In some cases there may be a transition from rupture velocity less than Rayleigh velocity greater than shear wave velocity. The locus of this transition is surveyed in the parameter space of fracture energy, upper yield stress, and crack length. A solution for this model can be represented as a convolution of a singular solution having abrupt stress drop with a 'rupture distribution function. The convolution eliminates the singularity and spreads out the rupture front in space-time. If the solution for abrupt stress drop has an inverse square root singularity at the crack tip, as it does for sub-Rayleigh rupture velocity, then the rupture velocity of the convolved solution is independent of the rupture distribution function and depends only on the fracture energy and crack length. On the other hand, a crack with abrupt stress dropt propagating faster than the shear wave velocity has a lower-order singularity. A supershear rupture front must necessarily be spread out in space-time if a finite fracture energy is absorbed as stress drops.

A spherically symmetric source of sound, situated within a solid homogeneous isotropic elastic sphere, emits a short P-pulse of small amplitude. Scalar potentials for the P and SV disturbances are shown to exist and to satisfy wave equations with velocities alpha and beta, respectively. At the free spherical surface the theory of characteristics yields the approximate shape for both P and SV pulses immediately after reflexion (PP and PS), and these approximations with Kirchhoff's formula give the pulse shapes of PP and PS near the geometrical acoustic arrival time at any point not too near the cusps of the caustics of PP and PS reflexions. This method gives rise to certain integrals analogous to those leading to stationary phase approximations but involving the distribution delta', the derivative of the Dirac delta-'function', instead of the exponential function. The main result is the calculation of the pulse shape for the field point near the caustics and near the axis. For an incident step pulse figures exhibit the transition in the pulse shape as the field point crosses a caustic or approaches the axis. This calculation involves simple, though laborious, numerical manipulation of complete elliptic integrals. These results are quite general and would apply to any axially symmetric reflexion problem where the pulse length is much shorter than the principal radii of curvature of the reflecting surface. Finally, these formulae are applied to the reflexion in a sphere. The method adopted in this paper has the advantage of directness, and at each stage of the calculation the physical significance of all variables and expressions is easily seen.

We analyze three-dimensional finite difference solutions for a simple shear-crack model of faulting to determine the effects of fault length and width on the earthquake slip function. The fault model is dynamic, with only rupture velocity, fault dimensions, and dynamic stress-drop prescribed. The numerical solutions are accurate for frequencies up to 5 Hz, and are combined with asymptotic results for shear cracks in order to characterize the slip function at higher frequencies. Near the hypocenter, the slip velocity exhibits a square root singularity whose intensity increases with hypocentral distance. At distances greater than the fault width, w, growth of the velocity intensity ceases, and the slip function becomes nearly invariant with distance along the fault length. Close-form expressions are developed for the dependence of static slip, slip rise time, and slip velocity intensity on fault geometry. The numerical results imply that uniform-dislocation kinematic earthquake models in which slip is represented by a ramp time-function will under-predict high-frequency ground motion relative to low-frequency ground motion. A further implication of the numerical solutions is that the nature of the in-elastic processes at the advancing edge of a long fault will depend on fault width, but will be independent of rupture length.

The computation of theoretical seismograms for models in which the elastic parameters and density vary only with depth (in a plane, cylindrical or spherical geometry) reduces to the solution of an ordinary differential equation plus the evaluation of inverse transformations. In principle, the problem is straightforward. In practice, many techniques and approximations can be used at each stage and many combinations and variants are possible. In this paper, we discuss a new method of evaluating the inverse transforms. Any method can be used to solve the differential equation and we only discuss a few analytic approximations to illustrate the new method. The inverse transformations are a frequency and wavenumber integral. Essentially four techniques can be used to evaluate these depending on the order of integration and whether the wavenumber integral is distorted from the real axis. Three of these have been widely used, but the technique of evaluating the frequency integral first and keeping the wavenumber real is new. In this paper, we discuss some of the advantages of this combination.

A new three-dimensional earth modeling is proposed as a framework to obtain more detailed and accurate information about the earth's interior. We start with a layered medium of classic seismology but divide each layer into many blocks and assign a parameter to each block which describes the velocity fluctuation from the average for the layer. Our data are the teleseismic P travel time residuals observed at an array of seismographs distributed on the surface above the earth's volume we are modeling. By isolating various sources of errors and biases we arrive at a system of equations to determine the model parameters. The solution was obtained by the use of generalized inverse and stochastic inverse methods. Our method also gives a lower limit of the true rms slowness fluctuation in the earth under the assumption of ray theory. Using P wave residual data from the Norwegian Seismic Array (Norsar), we have obtained the map of velocity anomalies at various depths up to a depth of 126 km. The rms slowness fl

Frequency domain calculation of extended source seismograms

- Spudich