Scaling of slip weakening distance with final slip during dynamic earthquake rupture

Transcription

1 Saling of slip weakening distane with final slip during dynami earthquake rupture Massimo Coo (1), Elisa Tinti (1), Chris Marone (1,2), and Alessio Piatanesi (1) (1) Istituto Nazionale di Geofisia e Vulanologia, Rome, Italy (2) Department of Geosienes, Pennsylvania State University, University Park, USA. Submitted to Speial Issue of PAGEOPH on Fault-zone Properties and Earthquake Rupture Dynamis 09 May 2008

2 Abstrat We disuss physial models for the harateristi slip weakening distane D of earthquake rupture with partiular fous on saling relations between D and other earthquake soure parameters. We use inversions of seismi data to investigate the breakdown proess, dynami weakening, and measurement of D. We disuss limitations of suh measurements. For studies of breakdown proesses and slip weakening it is important to analyze time intervals shorter than the slip duration and those for whih slip veloity is well resolved. We analyze the relationship between D and the parameters and Da, whih are defined as the slip at the peak slip veloity and the peak tration, respetively. We disuss approximations and limitations assoiated with inferring the ritial slip weakening distane from. Current methods and available seismi data introdue potential biases in estimates of D and its saling with seismi slip due to the limited frequeny bandwidth onsidered during typial kinemati inversions. Many published studies infer erroneous saling between D and final slip due to inherent limitations, impliit assumptions, and poor resolution of the seismi inversions. We suggest that physial interpretations of D based on its measurement for dynami earthquake rupture should be done with aution and the aid of aurate numerial simulations. Seismi data alone annot, in general, be used to infer physial proesses assoiated with D although the estimation of breakdown work is reliable. We emphasize that the parameters T a and peak slip veloity ontain the same dynami information as D and breakdown stress drop. This further demonstrates that inadequate resolution and limited frequeny bandwidth impede to onstrain dynami rupture parameters. Coo et al. Saling of dynami slip weakening distane p. 1

3 1. Introdution Understanding shear tration evolution during nuleation and dynami propagation of earthquakes is one of the major tasks for seismologists and Earth sientists. Earthquakes are the most important expression of faulting and knowledge of the proesses ontrolling dynami fault weakening during propagation of a seismi rupture is a ruial goal. This should be ahieved by olleting geologial and geophysial observations of natural faults, from laboratory experiments of frition and frature, and by modeling seismi data and earthquake rupture using theoretial models and numerial simulations. Dynami fault weakening an be fully desribed by the total shear tration evolution at a target point on the fault plane as a funtion of time or slip (Rie and Coo, 2006). Figure 1 shows an example of dynami tration, slip and slip veloity evolution as a funtion of time (panel a) and dynami shear tration as a funtion of slip (panel b) for a target point on the fault plane; the latter is ommonly alled the slip weakening urve. The dynami tration evolution of Figure 1 was obtained using a numerial proedure, disussed below, in whih the rupture history derived from a kinemati inversion of seismi reordings is used as a boundary ondition on the assumed fault plane (Ide and Takeo, 1997; Bouhon, 1997; Tinti et al., 2005a). We prefer to fous on dynami tration evolution and dynami rupture parameters that we derive from seismologial observations, whih allow us to onstrain the rupture history inluding final slip, rupture time, slip duration, and details of the soure time funtion. Dynami fault weakening is haraterized by the stress degradation near the propagating rupture front. We disuss models in whih shear stress drops from an upper yield value (τ y ) to a residual level (τ f ) in an extremely short time, alled the breakdown time (T b ), and over a harateristi slip, alled the slip weakening distane (D ). The spatial extent of the breakdown zone (X b ), defined as the region of shear stress degradation near the tip of a propagating rupture, depends on the slip weakening distane. Slip veloity reahes its peak in a time T a (see Tinti et al., 2005b), orresponding to the duration of positive slip aeleration. Generally, T a is shorter or equal to the breakdown time ( T T ), as shown in Figure 1. Constraining slip veloity a b and tration evolution via analysis of seismially radiated waveforms is an extremely Coo et al. Saling of dynami slip weakening distane p. 2

4 important but very diffiult task for modeling high frequeny radiation of the earthquake soure. The seismi slip duration (i.e., the rise time) is another important parameter haraterizing the rupture history and it is physially assoiated with healing mehanisms. In the literature, numerous studies represent dynami rupture propagation through either a rak-like rupture mode or a self-healing pulse propagation (e.g., Cohard and Madariaga, 1994, 1996; Zheng and Rie, 1998; Coo et al., 2004). The healing of slip, whih may ause short slip durations and/or slip-veloity pulse-mode rupture propagation, has been attributed to heterogeneity of initial-stress or strength on the fault plane (e.g., Beroza and Mikumo, 1996; Bizzarri et al., 2001) or to properties of the onstitutive law adopted to represent fault frition (e.g., Perrin et al., 1995; Beeler and Tullis, 1996). Self-healing ruptures have been doumented during rupture propagation between dissimilar materials and in other ases (Weertman, 1980; Andrews and Ben-Zion, 1997; Cohard and Rie, 2000). Seismologial models often assume a soure time funtion of finite duration (see for instane the slip veloity plotted in Figure 1a) and therefore they may be onsidered more onsistent with selfhealing slip rather than with rak-like models. Indeed, tration evolution shown in Figure 1 exhibits restrengthening assoiated with healing of slip. The purpose of this study is to eluidate the physial interpretation and seismi measurement of the harateristi slip weakening distane (D ) with partiular fous on the breakdown proess and dynami weakening. A key feature of our approah is the fous on the time sale of the breakdown proess. We analyze time intervals shorter than the slip duration and ensure that periods of large slip veloity are well resolved (e.g., Figure 1). The lass of shear tration evolution models for dynami fault weakening represented by Figure 1 are required to radiate seismi waves and to release the applied tetoni stress. Several stress parameters an be defined from the tration evolution shown in Figure 1: the strength exess (τ y - τ o ), the dynami stress drop (τ o - τ f ) and the breakdown stress drop (τ y - τ f ). Where τ o is the initial value of stress for a partiular position on the fault plane. The area below the slip weakening urve and above the residual stress level (τ f ) is traditionally identified with the frature energy (G) (see Palmer and Rie, 1973; Andrews, 1976; Rie et al., 2005), although reently, several authors have proposed that a similar quantity alled the breakdown work (W b ) is more Coo et al. Saling of dynami slip weakening distane p. 3

5 appropriate (Tinti et al., 2005a; Coo et al., 2006), at least for interpreting seismologial observations. The breakdown work is a more general definition of seismologial frature energy (Coo et al., 2006) and it is different from frature energy as defined in lassi frature mehanis (see Aberrombie and Rie, 2005; Coo and Tinti, 2008). The seismologial breakdown work aounts for 1) the portion of the mehanial work dissipated within the fault zone, inluding surfae area prodution, heat and other fators, 2) tration evolution in the pre-yield stress region, whih represents the energy lost during the initial slip-hardening phase (Figure 1), and 3) the effets of spatial and temporal variations in slip diretion. Beause it is a more realisti representation of the earthquake proess, we use the breakdown work as a proxy for seismologial frature energy in this study. The use of breakdown work also provides a means of studying spatial variations in seismologial frature energy beause it an be defined at eah point on the fault plane. Breakdown work represents the only measurable portion, through seismologial observations, of the energy absorbed on the rupture plane for frature and fritional dissipation (see Coo et al., 2006). Therefore, measuring breakdown work is quite important for understanding the earthquake energy balane and for onstraining the energy to be radiated as seismi waves. In this study, we fous on the physial interpretation and measurement of the ritial slip weakening distane D via modeling of seismologial data. Our approah requires knowledge of the rupture history in order to image slip and/or slip veloity evolution and to onstrain dynami tration evolution on the fault plane. The parameter D is ommonly measured from the same slip weakening urves (see Figure 1b) used to measure breakdown stress drop and breakdown work (or frature energy). Reently, Mikumo et al. (2003) and Fukuyama et al. (2003) have proposed an alternative approah that allows estimation of the slip weakening distane diretly from seismi observations. They proposed to measure the slip at the time of peak slip veloity, alled, and use this as a proxy for D. As shown in Figure 1a, differs from D and, as we will disuss later, their ratio depends on fault onstitutive properties (Tinti et al., 2004). However, has been onsidered as a reliable approximation of in some ases. Fukuyama et al. (2003) stated that this approximation works well for smoothly propagating ruptures. In this paper, we further disuss the validity of using alternative seismologial estimate of. D D as an Coo et al. Saling of dynami slip weakening distane p. 4

6 2. Rupture history from kinemati soure models Earthquake rupture history is often imaged through inverse approahes using a kinemati soure parameterization. Geophysial data (seismograms and geodeti data) are inverted using non-linear algorithms to infer the spatiotemporal distribution of fault slip, slip diretion (rake angle) and slip duration (rise time). The range of numerial approahes in use assume either an analytial soure time funtion (single-window approah) or represent the soure time funtion as the superposition of several triangular funtions (multi-window approah) [see the detailed disussion in Cohee and Beroza, 1994]. The latter method has the advantage of avoiding the seletion of an analytial soure time funtion, but limitations inlude poor resolution and a sparse sampling of the slip veloity time history. On the ontrary, the single window approah has the limitation of the a priori hoie of soure time funtion, but allows higher resolution sampling of the soure time funtion and thus an provide better onstraint on the breakdown proess and the rupture history on the fault plane. Figure 2 displays several examples of slip veloity soure time funtions urrently adopted in the literature. They have different parameterization and for eah model slip veloity reahes its peak in a different time interval. Piatanesi et al (2004) disuss the effet of using different soure time funtions for imaging the distribution of dynami parameters on the fault plane (dynami and breakdown stress drop, strength exess and ritial slip weakening distane). They pointed out that the hoie of the slip veloity funtion affets the inferred dynami parameters; in partiular, as we will disuss in the following, different soure time funtions yield different values of ritial slip weakening distane and a different saling with final slip. Figure 3 shows the final slip distribution for the 1979 Imperial Valley earthquake obtained by Hartzell and Heaton (1993) by inverting strong motion data. The authors used an asymmetri triangular funtion having a rise time of 0.7 seonds and the time to peak slip veloity T a equal to 0.2 seonds. The five panels on the bottom display the slip and slip veloity time histories at seleted positions on the fault plane indiated by letters in the upper panel. This kinemati model is an example in whih the rise time is assumed onstant on the fault (and therefore it is not inverted) and Coo et al. Saling of dynami slip weakening distane p. 5

7 the slip veloity time histories have a onstant shape in all subfaults and vary only in amplitude. As noted by Piatanesi et al. (2004), the use in kinemati modelling of soure time funtions not ompatible with the dynami rupture propagation ould bias the estimate of D and hene the inferred ratio of D to the total slip D tot. Based on these results Tinti et al. (2005b) proposed the use of a new soure time funtion to infer kinemati soure models onsistent with earthquake dynamis, whih they named the regularized Yoffe funtion (shown in Figure 1a, see Tinti et al., 2005b, for the details of its analytial parameterization). Although other andidate soure time funtions are available in the literature (see for instane Nakamura and Miyatake, 2000; Dreger et al., 2007), we emphasize that the regularized Yoffe funtion is onsistent with dynami solutions of the elasto-dynami equation (Nielsen and Madariaga, 2003) and allows a flexible parameterization for our purposes. In order to image the rupture history on the fault plane, robust kinemati inversions have been proposed to improve resolution. A variety of smoothing onstraints have been adopted to ensure stable solutions of the inverse problem (see Hartzell et al., 2007 and referenes therein). It is generally aepted to use positivity and smoothing onstraints (Hartzell and Heaton, 1983; Henry et al., 2000, among many others) for reduing the instability and the omplexity of the inverted models to levels onsistent with resolution of the filtered data. However, depending on the hoie of assumed spatial or temporal onstraints, the results of the inversions (in terms of kinemati parameters) may hange dramatially (see Beresnev, 2003). Moreover, other fators an also strongly affet results suh as signal pre-proessing (Boore, 2005; Boore and Bommer, 2005), model parameterization (Piatanesi et al., 2007) and inversion shemes (see Hartzell et al and referenes therein). Beause of the diffiulties in omputing aurate Green s funtions at high frequenies (f > 2Hz), approahes based on waveform inversion model seismograms in a limited frequeny bandwidth. Applying filters to reorded seismograms helps in imaging the slip distribution but it redues the available resolution of the soure proess at small wavelengths. Spudih and Guatteri (2004) highlighted the effets of the limited frequeny bandwidth of modelled data on the inferred dynami and fritional parameters. Despite these limitations, rupture history an be retrieved only through kinemati soure models, and therefore they represent a unique resoure of information to better understand earthquake dynamis. Coo et al. Saling of dynami slip weakening distane p. 6

8 3. Inferring tration evolution Tinti et al. (2005a) have implemented a 3-D finite differene ode based on the Andrews (1999) approah to alulate the stress time histories on the earthquake fault plane from kinemati rupture models. The fault is represented by a surfae ontaining double nodes and the stress is omputed through the fundamental elastodynami equation (Ide and Takeo, 1997; Day et al., 1998). Eah node belonging to the fault plane is fored to move with a presribed slip veloity time series, whih orresponds to imposing the slip veloity as a boundary ondition on the fault and determining the stress-hange time series everywhere on the fault. This numerial approah does not require speifiation of any onstitutive law relating total dynami tration to frition. The dynami tration evolution is a result of the alulations. The numerial model is onsistent with the analytial model proposed by Fukuyama and Madariaga (1998), where stress hange [ σ ( x, t) ] at a position x of the fault plane is related to slip veloity time history [ v( x, t) ] at time t by means of the following relation: μ t σ ( x, t) = v( x, t) + K( x ξ; t t') v( ξ, t) dt' ds, (1) 2β 0 Σ where β is the shear wave veloity, μ the shear rigidity, K the dynami load assoiated to those points that are already slipping (that is, those within the one of ausality around the rupture front). Piatanesi et al. (2004) used the same approah to infer dynami parameters from kinemati models. The slip veloity time histories at eah point on the fault plane are obtained from the kinemati rupture models inferred by inverting geophysial data. In order to onvert the slip model to a ontinuously differentiable slip-rate funtion, the original kinemati models have to be interpolated and smoothed both in time and spae (see Day et al., 1998; Tinti et al., 2005a, for details). The free surfae is inluded in these omputations and the Earth models are simplified assuming homogeneous half-spaes. As disussed above, the inadequate resolution and the limited frequeny bandwidth, whih haraterize inverted kinemati models, redue the ability to infer the real dynami tration evolution everywhere on the fault plane. Many reent papers have investigated the limitations of using poorly resolved kinemati soure models (Guatteri and Spudih, 2000; Piatanesi et al., 2004; Spudih and Guatteri 2004). We disuss these issues in further detail below. Coo et al. Saling of dynami slip weakening distane p. 7

9 Despite the limitations noted above, the methods proposed by Ide and Takeo (1997), Day et al. (1998), and Tinti et al. (2005a) provide the dynami shear stress time histories on the fault plane. This is an important step in using seismologial observations to understand the breakdown proess during earthquake ruptures. Moreover, soure time funtions ompatible with earthquake dynamis and suitable for waveform inversions are beoming ommonly available (see Piatanesi et al., 2004; Tinti et al., 2005b; Dreger et al., 2007; Cirella et al., 2006). Finally, Tinti et al. (2005a and 2008) have disussed in detail the fidelity of alulations of breakdown work and D and onluded that, in agreement with Guatteri and Spudih (2000), the estimates of W b are quite stable despite the limited available resolution in kinemati soure models, while the D parameter is more diffiult to onstrain. For these reasons, we use the approah of Tinti et al. (2005a) in the present study. Figure 4 shows the tration evolution as a funtion of slip for the 1979 Imperial Valley earthquake. Numbers along the axes represent the relative position along strike and dip on the fault plane of eah target point. The intervals shown for the lower left panel indiate tration (in MPa) and slip (in m) and apply to eah panel of Figure 4. From the inferred slip weakening urves we an measure the breakdown stress drop, D, and the breakdown work at eah grid point on the fault plane. Figure 4 learly shows high values of stress drop in orrespondene to the large slip path. The slip weakening urves also exhibit a slow weakening rate due to the smoothed soure time funtion of the kinemati model. Figure 5 displays shear tration time histories (top panels) and the slip weakening urves (bottom panels) for the five points noted in Figure 3. As expeted, the dynami parameters of strength exess and breakdown stress drop vary over the fault plane (Figure 5). Moreover, D differs for different positions on the fault, varying from ~ 0.7 to 1.5 m for the seleted positions. We also emphasize that the duration of the breakdown phase and the subsequent restrengthening phase vary over the fault plane, even though the rise time (0.7 s) and T a (0.2 s) are assumed to be onstant. The dynami tration evolution urves inferred from kinemati rupture histories display an initial inrease before reahing the upper yield stress value τ y, as learly evident in the examples shown in Figures 1 and 5. This initial slip hardening phase preedes the dynami weakening phase and it is assoiated with relatively small slip amplitudes. We define D a as the slip at the upper yield stress: Figure 1a shows that, at Coo et al. Saling of dynami slip weakening distane p. 8

10 least in this ase, it is muh smaller than both and D. In the framework of lassi dynami models, this initial hardening phase is assoiated with the strength exess and it is an important feature for modeling rupture propagation through spontaneous dynami models. However, it is important to note that lassi slip weakening models impliitly assume D = 0 This implies that the duration of initial slip hardening ( T ) is a negligible and muh shorter than the dynami weakening phase ( T ). se << Ta Tb It is important to note that the initial slip hardening phase has been observed in laboratory experiments (Ohnaka and Yamashita, 1989; Ohnaka, 2003) and modeled using rate and state frition (Bizzarri et al., 2001; Marone et al., 2008). We emphasize that the breakdown work estimates done by Tinti et al. (2005a) and Coo et al. (2006) inlude both the initial slip-hardening and the subsequent slip weakening phases, and onsequently their D estimates inlude a ontribution from. D a se 4. Measuring from peak slip veloity Following the approah proposed by Mikumo et al. (2003) we have measured from kinemati soure model of the 1979 Imperial Valley earthquake, whose slip distribution is shown in Figure 3. Figure 6 displays the distribution of inferred from the slip history imaged by Hartzell and Heaton (1983) by inverting strong motion aelerograms. ranges between 0.5 and 1 m in the entral, high slip path; these values an be onsidered as a lower bound for D. As expeted, the spatial distribution of is strongly orrelated with the slip distribution. This is even more evident in Figure 7a, whih shows the perfet linear saling of with final slip. This saling D tot is simply the result of the initial hypothesis adopted in kinemati inversion of a fixed soure time funtion (an asymmetri triangular funtion) with a onstant rise time and Ta (0.7 and 0.2 s, respetively). Therefore, in this kinemati model, the heterogeneity of inferred slip ompletely ditates the heterogeneity of peak slip veloity and hene. Nowadays, thanks to omputational tools, finite-fault inversions are ommonly performed using non-linear formulations that allow all the kinemati parameters to be inverted (e.g., Ji et al., 2002; Delouis et al., 2002; Piatanesi et al., 2007). To aount for the atual rupture omplexity, slip or peak slip veloity, rupture time, rise time and slip Coo et al. Saling of dynami slip weakening distane p. 9

11 diretion are simultaneously inverted. However, the parameter T a is usually fixed a priori beause of data frequeny bandwidth limitations and, as a onsequene, strongly affets estimates of. Independent inversion for all of the kinemati parameters should produe a natural satter of the orrelation between are still other problems and limitations in measuring and D tot. However, there from seismologial observations. Spudih and Guatteri (2004) pointed out that low-pass filtering of strong motion seismograms an affet the estimates of, beause it biases the inverted rupture models ausing an artifiial orrelation between and D tot. These authors laimed that slip models derived from band-limited ground motion data might not resolve periods shorter than the breakdown time, and therefore the models do not ontain periods shorter than Ta ( Tb ). They defined these inverted models as temporally unresolved. This means that the proess of low pass filtering ground motion data an remove information about D and. Spudih and Guatteri (2004) onluded that filtering ground motion data or the inferred slip models tends to bias upward the D values inferred from the slip weakening urves and to generate artifiial orrelations with final slip. The effet of filtering is to shift the peak slip veloity later in time, whih means that T a is overestimated, and/or that postpeak energy is effetively repositioned to a time before the peak slip veloity (Spudih and Guatteri, 2004). These issues raise the question of whether estimates of from modeling ground motion waveforms are tenable and orroborated by data. Yasuda et al. (2005) performed a partiularly interesting test for the subjet disussed here. They simulated a spontaneous dynami earthquake rupture, by assuming onstant D, and omputed the syntheti waveforms that would be observed at atual reeivers. Therefore they inverted these syntheti seismograms to image the kinemati rupture history and to onstrain the slip rate funtion on eah sub-fault. As expeted, the spontaneous forward dynami model has a spatial and temporal resolution muh higher than the inverted model (nearly 10 times larger). This numerial test allows the omparison of inferred values of both D and with those of the target model. The results of Yasuda et al. (2005) learly show that the values measured from the dynamially generated slip rate funtions range between 0.25 D and D (as expeted sine T T ). Also, for a onstant D model (see Figure 3 in Yasuda et al. (2005)) a b Coo et al. Saling of dynami slip weakening distane p. 10

12 the inferred values of sale with final slip. These results are onsistent with the findings of Tinti et al. (2004), who also demonstrate that the values measured from the inverted slip rate funtions sale with final slip. The values of and D so derived define a roughly linear saling with slope equal to 1/2, although they exhibit larger satter than values measured from the dynamially generated slip rate funtions. Spudih and Guatteri (2004) explain the saling tot = 0.5 as a onsequene of the entral limit theorem: after the filtering operation, a soure time funtion tends to a Gaussian in whih half of the slip ours before peak slip veloity. Yasuda et al. (2005) also onluded that both of their estimates of final slip. The orrelation between D tot exhibit an apparent orrelation with and final slip has also been obtained in other inversions of kinemati models. Figure 7b shows the saling of with D for the 1994 Northridge earthquake. These values derive from the alulations performed by Tinti et al. (2005a) using the kinemati model results of Wald et al. (1996). The latter authors used a multi-windows approah and the soure time funtion onsists of three overlapping isoseles triangles eah with duration of 0.6 s and initiations separated by 0.4 s. This allows a rise time lasting up to 1.4 s. We show here values estimated for the dip-slip omponent of the slip vetor (Figure 7b). D tot tot exhibits a larger satter around a linear saling, whih arises in part beause in this ase the slip rate funtion an ontain multiple peaks at different times (see Figure 2 in Tinti et al., 2005a). Nevertheless, the orrelation of with final slip is still evident (Figure 7b). A onlusion from all of these models is that is an aurate estimate of D only if the inversion method retains information on the breakdown proess. Unfortunately, this ondition is not met, due to inherent limitations on spatial and temporal resolution of kinemati soure models. Moreover, although one might expet to be able to use quite diffiult to onstrain as a proxy for in numerial models of dynami rupture, it is D from the rupture history imaged from kinemati inversions beause in these approahes the slip veloity funtion is hosen a priori (single window) or it is imaged with a poor resolution insuffiient to resolve T Fukuyama et al. (2003) pointed out that also for a smoothly propagating rupture the standard deviation of the measured values an be larger than 30%. We a. Coo et al. Saling of dynami slip weakening distane p. 11

13 therefore onlude that the orrelation of with final slip is often an unavoidable onsequene of a priori assumptions and limited resolution. In the next setions we disuss i) a physial explanation of the deviation of poor resolution of the breakdown proess. from and ii) the problem of D 5. Measuring D from inferred tration evolution urves We have disussed approximations and limitations assoiated with inferring the ritial slip weakening distane from the slip at peak slip rate. Here, we disuss the estimation of D diretly from the slip weakening urves obtained from the inverted kinemati rupture histories. As noted above, we follow the approah proposed by Ide and Takeo (1997), Day (1998) and Tinti et al. (2005a). Figure 8 shows the values inferred from the tration evolution urves displayed in Figure 4 and obtained from the slip history imaged by Hartzell and Heaton (1983) for the 1979 Imperial Valley earthquake. A visual omparison between Figures 3 and 8 suggests that D D is orrelated with final slip. This orrelation is illustrated in Figure 9, whih shows the saling of D with final slip for all the sub-faults of this model (see Tinti et al., 2005a for details). The proportionality between and final slip has been obtained by several other authors (Dalguer et al., 2003; Pulido and Kubo, 2004; Burjanek and Zaharadnik, 2007) and imaged for other earthquakes. Figure 11 of Tinti et al. (2005a) displays a similar saling also for the 1995 Kobe and the 1997 Colfiorito earthquakes. Therefore, we onlude that, similarly to D, the estimates of sale with final slip aording to a nearly D linear relationship. Another interesting features emerges from Figure 9: it shows that most of the subfaults have a D value quite lose to the final slip ( D D tot ). This is even more lear in Figure 10, whih displays the spatial distribution of the ratio D D on the tot fault plane. The ratio is lose to the unity over muh of fault plane (Figure 10). This is due to the soure time funtion seleted for modeling observed ground motions (Piatanesi et al., 2004), and to the limited spatial and temporal resolution employed in numerial alulations (Spudih and Guatteri, 2004; Yasuda et al., 2005). Coo et al. Saling of dynami slip weakening distane p. 12

14 In order to better understand the effets of adopting different soure time funtions in retrieving dynami tration evolution, we show in Figure 11 the dynami tration evolution inferred for two slip rate funtions: f f = f& ( t) = H ( t) 1 tanh (2t T ) 2 R TR TR = f& ( t) = H ( T R 2 t) π T R t + t where H(t) is the Heaveside funtion and T R is the rise time. The first funtion f 1 is a smoothed ramp funtion in slip resulting in a Gaussian slip rate funtion; the seond funtion f 2 is the Yoffe funtion, whih is singular at the rupture time (see Piatanesi et al., 2004). The numerial representation of the latter is obtained by smoothing the funtion in time with a moving triangular window (0.37s) (see Tinti et al., 2005b for details of the analytial regularization of Yoffe funtion). These slip and slip rate funtions are shown in panels a and b of Figure 11. Tration hanges of panels and d of Figure 11 are omputed for the heterogeneous slip model shown in Figure 12a in a point with 2.7 m of final slip. This is the rupture history imaged by Iwata and Sekiguhi (2002) for the 2000 western Tottori, Japan, earthquake and it is haraterized by a nonuniform slip distribution and an extremely variable rupture veloity. Figure 11 allows us to point out the main differenes of slip rates and inferred tration evolution urves retrieved by using these two soure time funtions. To this goal, panels e and f of this figure display the slip, slip rate and shear tration time histories for the seleted target point. It is evident that while T a for the Yoffe funtion is muh shorter than the rise time, the same parameter is half of the rise time for the smoothed ramp. Moreover, funtion f 1 is haraterized by a smooth onset of slip veloity, while in funtion f 2 it is quite sharp. All these features, whih reflet peuliarities of the dynamis of earthquake rupture, yield quite different evolution urves. In partiular, we emphasize that the Yoffe funtion yields: (i) a muh shorter duration of breakdown proess than the smoothed ramp, (ii) smaller values of D and tot T R 1 2 (2) D D than f 1, (iii) smaller values of than those of funtion f1 (nearly half of the latter), and (iv) higher peak slip veloity values. We note that the two rupture histories used to ompute the tration evolution urves illustrated in Figure 11 only differ for the seleted slip rate funtion, while all the other parameters are the same (the rupture model is that shown in Figure 12a). Coo et al. Saling of dynami slip weakening distane p. 13

15 Figure 12 shows the strength exess and the dynami stress drop distributions omputed from the slip weakening urves inferred using the two soure time funtions defined above (f 1 and f 2 ) and two slip models differing only for the rupture veloity. Left panels depit the slip, strength exess and dynami stress drop distributions for the variable rupture veloity model of Iwata and Sekiguhi (2002), while right panels display the same parameters for a uniform rupture veloity model (equal to 2.1 km/s). This figure learly shows that, as expeted, the variability of rupture veloity ontrols the heterogeneity of strength exess and dynami stress drop on the fault plane, but it also shows that the adopted soure time funtion affets the values of these dynami parameters. This is in agreement with Guatteri and Spudih (2000) who onluded that there is a trade-off between strength exess and D in ontrolling the rupture veloity. 6. Saling between D and final slip In previous setions we have disussed estimation methods of both and D as well as the reliability of their saling with final slip. The main limitations arise from: 1) limitations in our ability to model seismi waveforms given the narrow frequeny bandwidths available (i.e., lak of high frequeny omponents, see Spudih and Guatteri, 2004 and referenes therein), 2) adoption of soure time funtions that are not ompatible with dynami rupture propagation (see Piatanesi et al., 2004), and 3) the lak of ausality onstraints on spatial and temporal evolution of slip veloity in seismi inversions (i.e., poor onstraints on spatial gradient of slip veloity, see Tinti et. al, 2008 for details). The duration of the breakdown proess and the peak slip veloity depend on the fritional and onstitutive properties of the fault. Therefore, we expet that differenes between Ta (timing of peak slip veloity) and T b (breakdown duration) and onsequently between and D is ontrolled by the onstitutive properties of the soure. Tinti et al. (2004) have disussed the differene between and D for a 2- D rupture governed by either a slip weakening or a rate- and state-dependent frition law. They demonstrated that suh a differene is ontrolled by the strength parameter S (i.e., the ratio between strength exess and dynami stress drop). These authors have also shown that the rate dependene of the frition law affets slip aeleration and the slip weakening parameter. Coo et al. Saling of dynami slip weakening distane p. 14

16 We have disussed several interpretations of the biases affeting estimates of the slip weakening distane. However, the most important issue onerns the saling of this parameter with the final slip. Although we annot exlude the possibility that suh a saling might be physially tenable for real earthquakes, we present evidene showing that the inferred proportionality between D and D is artifiial. The same is true for the parameter D, whose estimate from kinemati slip models is ontrolled by the adopted soure time funtion and by other a priori assumptions when inverting ground motion data. Tinti et al. (2005b) performed several simulations using a Yoffe funtion to represent slip veloity time history and the tration at split nodes approah to retrieve dynami shear tration hanges aused by oseismi slip. These authors propose that the following relation holds: T tot a D Dtot, (3) TR suggesting that, when the duration of positive slip aeleration T a (i.e., time of peak slip veloity) and the rise time are both onstant or their ratio is onstant, there is a diret proportionality between slip weakening distane and final slip. The values of Ta and TR T R ontrol peak slip veloity, and therefore the same fators, whih bias the estimate of D, explain the diffiulties in onstraining both final slip and slip veloity. These findings are summarized in Figure 13 whih shows the dynami tration hanges (panels, d, e and f) omputed for soure models having a uniform slip of 1 m with a onstant rupture veloity of 2.0 km/s and a slip veloity time history represented by a Yoffe funtion (panels a and b). Left panels display the results of alulations obtained for different T a values and a onstant rise time (1.0 s), while right panels show those omputed for different rise times and a onstant T a equal to s (see Tinti et al., 2005b for further details about these alulations). Panels g and h show the saling of peak slip veloity with D for the two test ases. The dynami tration histories and the slip weakening urves displayed in Figure 13 learly illustrate that D and the weakening rate vary with the parameter Ta. Inverting ground motion data with a limited temporal resolution overestimates the real T a and onsequently produes an overestimate of D. The weakening rate varies in order to maintain the same value of the final slip. As a onsequene, the peak slip veloity dereases for inreasing T a and Coo et al. Saling of dynami slip weakening distane p. 15

17 D, as learly shown in panel g of Figure 13. The effet of using different rise times for the same T a is to affet the D value, but very smoothly affeting the weakening rate. Figure 13 points out that inreasing the rise time auses both D and peak slip veloity to derease. Figure 13 summarizes in a simple way all the diffiulties in imaging dynami tration evolution and measuring the slip weakening parameters. It is also useful to illustrate in a shemati way all the limitations in measuring the slip weakening parameters by modeling observed ground motion data. 7. Disussion and Conluding Remarks We disuss physial models for the harateristi slip weakening distane D and the saling between D and total slip. We show that urrent methods and available seismi data introdue potential biases in estimates of D and its saling with seismi slip due to the limited frequeny bandwidth onsidered during typial kinemati inversions. For studies of dynami slip weakening it is important to analyze time intervals shorter than T a and T b in order to obtain aurate estimates of D and its saling with total slip. Unfortunately, T a is usually fixed a priori in kinemati inversions due to poor data resolution. The same is true for the rise time, whih is poorly onstrained in kinemati inversions and often assumed to be spatially uniform on the fault plane. Therefore, in suh ases, T a (and T a T R ) strongly affet estimates of as well as D, and, as shown in equation (3), this generates artifiial orrelation of D with final slip. In this study, we extend previous works (Guatteri and Spudih, 2000; Tinti et al. 2008) and show that D inferred through seismologial data an be biased unless a proper modeling of high frequeny waves and soure parameterization are adopted. Fukuyama and Mikumo (2007) estimated the slip weakening distane from seismograms reorded at near field stations. These authors laimed that the proposed approah is not signifiantly affeted by spatiotemporal smoothing and resolution limitations. However, this approah also depends on several a priori assumptions onerning the rupture history (rupture veloity, rise time, et ) and soure time funtion. We annot exlude the existene of physial mehanisms ontrolling the saling of D with final slip D tot. Ohnaka and Yamashita (1999) and Ohnaka (2003) suggest that D sales with the roughness of the fault plane in the diretion of slip Coo et al. Saling of dynami slip weakening distane p. 16

18 and that D has a fratal distribution on the fault plane. The idea that D sales with fault maturity was also proposed by Marone and Kilgore (1993) and Aberrombie and Rie (2005). On the other hand, Sholz (1988) argued that rough fault surfaes under lithostati load will develop a harateristi ontat dimension and hene exhibit a onstant value of D, rather than one that sales with roughness (see also Brown and Sholz, 1985; Aviles et al., 1987; Power et al., 1987). Nevertheless, the assumption of a fratal distribution of D and the saling with fault roughness might imply a saling with final slip. Moreover, beause slip is heterogeneously distributed on the fault plane, onstant D models are not physially onsistent (that is, they ould predit D values larger than final slip in loked pathes). A full disussion of the physial mehanisms that ould yield ausal saling between D and final slip is beyond the sope of the present study. We emphasize that the linear saling between D and D tot inferred from kinemati soure models is aused by the poor resolution of the breakdown proess. In this study we have shown that the estimates of breakdown work from kinemati soure models are more reliable. The breakdown work is a more appropriate quantity for assessing the earthquake energy budget than the frature energy as defined in lassi frature mehanis (Coo et al., 2006; Coo and Tinti, 2008). Figure 14 shows the distribution of W b (J/m 2 ) obtained by Tinti et al. (2005a) for a kinemati model of the 1979 Imperial Valley earthquake (Hartzell and Heaton, 1983). The spatial distribution of breakdown work is strongly orrelated with the slip distribution (see Figure 3). The orrelation between the distributions of W b and slip is due primarily to the orrelation of D with slip, but also seondarily to the orrelation of stress drop with total slip. The two fundamental parameters haraterizing dynami fault weakening, breakdown work W b and slip weakening distane D, are intrinsially sale dependent (Ionesu and Campillo, 1999; Ohnaka and Yamashita, 1989; Coo and Tinti, 2008). This means that they annot be assoiated with any other physial proess ontrolling dynami fault weakening at time and length sales smaller than that seleted for the marosopi representation impliit in seismologial observations (see Coo et al., 2006; Coo and Tinti, 2008). Thus, the interpretation of the D estimates inferred from seismologial data are representative of the marosopi sale in whih the fault zone is shrunk to a virtual mathematial plane of zero thikness. Therefore, they annot be easily ompared with estimates retrieved from Coo et al. Saling of dynami slip weakening distane p. 17

19 laboratory experiments on rok frition and frature or from those assoiated with weakening mehanisms ourring at time and length sales smaller than the fault zone thikness. The results of this study point out that the parameter T a and peak slip veloity ontain the same dynami information as breakdown stress drop and D. The inadequate resolution and the limited frequeny bandwidth, whih haraterize inverted kinemati models, redue the ability to infer the real dynami tration evolution everywhere on the fault plane. Future attempts to model high frequeny seismi waves are important and will be aided by high performane omputing failities and high quality seismi waveforms from borehole seismometers. Coo et al. Saling of dynami slip weakening distane p. 18

20 Referenes Aberrombie, R.E., and J.R. Rie, (2005), Can observations of earthquake saling onstrain slip weakening?, Geophys. J. Int., 162, Andrews, D. J., (1976), Rupture propagation with finite stress in antiplane strain, J. Geophys. Res., 81, Andrews, D. J., (1999), Test of two methods for faulting in finite-differene alulations, Bull. Seismol. So. Am., 89(4), Andrews D.J., and Ben-Zion Y., (1997),Wrinkle-like slip pulse on a fault between different materials, J. Geophys. Res.,102 (B1): Aviles, C.A., and C.H. Sholz, (1987), Fratal analysis applied to harateristi segments of the San Andreas fault, J. Geophys. Res.,92 (B1), Beroza G.C., and Mikumo T, (1996), Short slip duration in dynami rupture in the presene of heterogeneous fault properties, J. Geophys. Res.,101 (B10), Beeler N.M., and Tullis T.E., (1996), Self-healing slip pulses in dynami rupture models due to veloity-dependent strength, Bull. Seismol. So. Am., 86 (4): Beresnev, I. A. (2003). Unertainties in finite-fault slip inversions: to what extent to believe? (A ritial review), Bull. Seism. So. Am., 93, Bizzarri A., M. Coo, D.J. Andrews, and E. Boshi, (2001), Solving the dynami rupture problem with different numerial approahes and onstitutive laws, Geophys. J. Int., 144 (3): Boore D.M., (2005), On pads and filters: Proessing strong-motion data, Bull. Seism. So. Am., 76, Boore, DM, and J.J. Bommer, (2005), Proessing of strong-motion aelerograms: needs, options and onsequenes, Soil Dynamis and Earthquake Engineering, 25, Bouhon, M., (1997), The state of stress on some faults of the San Andreas system as inferred from near-field strong motion data, J. Geophys. Res.,102(B6), Brown, S. R. and C. H. Sholz, (1985). Broad bandwidth study of the topography of natural rok surfaes, J. Geophys. Res., 90, 12,575-12,582. Burjanek J., and J. Zaharadnik, (2007), Dynami stress field of a kinemati earthquake soure model with k-squared slip distribution, Geophys. J. Int., 171 (3): Cirella, A., A. Piatanesi, E. Tinti, and M. Coo (2006), Dynamially onsistent soure time funtions to invert kinemati rupture histories, Eos, Trans. AGU, 87(52), Fall Meet. Suppl., Abstrat S41B Coo et al. Saling of dynami slip weakening distane p. 19

21 Coo, M., A. Bizzarri, and E. Tinti (2004). Physial interpretation of the breakdown proess using a rate- and state-dependent frition law, Tetonophysis 378, Coo, M., Spudih P. and E. Tinti, (2006), On the mehanial work absorbed on faults during earthquake ruptures, in Earthquakes: Radiated Energy and the Physis of Faulting, Geophysial Monograph Series 170, Amerian Geophysial Union, /170GM24. Coo M. and E. Tinti, (2008), Sale dependene in the dynamis of earthquake propagation: evidene from seismologial and geologial observations, submitted to Earth Planet. Si. Lett. Cohard A. and R. Madariaga, (1994), Dynami faulting under rate-dependent frition, Pure and Applied Geophysis, 142, (3-4), Cohard A. and R. Madariaga, (1996), Complexity of seismiity due to highly rate-dependent frition, J. Geophys. Res.,101(B11), Cohard A. and J.R. Rie, (2000), Fault rupture between dissimilar materials: Ill-posedness, regularization, and slip-pulse response, J. Geophys. Res.,105 (B11), Cohee, B. P., and G. C. Beroza (1994). A omparison of two methods for earthquake soure inversion using strong motion seismograms, Ann. Geophys. 37, Dalguer L.A., Irikura K., and J. D. Riera JD, (2003) Generation of new raks aompanied by the dynami shear rupture propagation of the 2000 Tottori (Japan) earthquake, Bull. Seismol. So. Am., 93 (5): Day, S. M., G. Yu and D. J. Wald (1998) Dynami stress hanges during earthquake rupture, Bull. Seismol. So. Am., 88, Delouis, B., D. Giardini, P. Lundgren, and J. Salihon (2002), Joint inversion of Insar, GPS, teleseismi and strong-motion data for the spatial and temporal distribution of earthquake slip: appliation to the 1999 Izmit mainshok, Bull. Seismol. So. Am., 92(1), Dreger D., Tinti E and A. Cirella, (2007), Slip Veloity Parameterization for Broadband Ground Motion Simulation, Abstrat for the Annual Meeting of Seismologial Soiety of Ameria. Fukuyama, E. and R. Madariaga (1998) Rupture dynamis of a planar fault in a 3D elasti medium: Rate-and slip-weakening frition, Bull. Seismol. So. Am., 88, Fukuyama, E., T. Mikumo, and K. B. Olsen, (2003), Estimation of the ritial slip-weakening distane: Theoretial bakground, Bull. Seismol. So. Am., 93, Fukuyama, E., and T. Mikumo, (2007), Slip-weakening distane estimated at near-fault stations, Geophys. Res. Lett., 34, doi: /2006gl Guatteri, M. and P. Spudih, (2000), What an strong-motion data tell us about slip-weakening fault-frition laws?, Bull. Seismol. So. Am., 90, Coo et al. Saling of dynami slip weakening distane p. 20

22 Hartzell, S., and T. H. Heaton (1983). Inversion of strong ground motion and teleseismi waveform data for the fault rupture history of the 1979 Imperial Valley, California, earthquake, Bull. Seism. So. Am.,73, Hartzell, S., P. Liu, C. Mendoza, C. Ji, and K. M. Larson, (2007), Stability and unertainty of finite-fault slip inversions: appliation to the 2004 Parkfield, California, earthquake, Bull. Seism. So. Am.,97, Henry C, S. Das, and Woodhouse JH, (2000), The great Marh 25, 1998, Antarti Plate earthquake: Moment tensor and rupture history, J. Geophys. Res., 105 (B7), Ide, S. and M. Takeo, (1997), Determination of onstitutive relations of fault slip based on seismi wave analysis, J. Geophys. Res., 102(B12), Ionesu, I.R., Campillo M., (1999), Influene of the shape of the frition law and fault finiteness on the duration of initiation, J. Geophys. Res., 104 (B2), Iwata, T., and H. Sekiguhi (2002), Soure proess and near-soure ground motion during the 2000 Tottori-ken-Seibu earthquake, Pro. 11th Japan Earthq. Eng. Symp., Ji, C., D. J. Wald, and D. V. Helmberger (2002), Soure desription of the 1999 Hetor Mine, California, earthquake, Part I: Wavelet domain inversion theory and resolution analysis, Bull. Seismol. So. Am., 92(4), Marone C., M. Coo, E. Rihardson, and E Tinti, (2008), Mehanis of the ritial slip distane for seismi and aseismi faulting, Submitted to Speial Issue of Pageoph. Marone C. and B. Kilgore, (1993), Saling of the ritial slip distane for seismi faulting with shear strain in fault zones, Nature, 362, Mikumo, T., K. B. Olsen, E. Fukuyama and Y. Yagi, (2003), Stress-breakdown time and slipweakening distane inferred from slip-veloity funtions on earthquake faults, Bull. Seismol. So. Am., 93(1), Nakamura, H., and T. Miyatake (2000) An approximate expression of slip veloity time funtions for simulation of near-field strong ground motion, Zisin (J. Seism. So. Jpn.), 53, 1-9 (in Japanese with English abstrat). Nielsen, S., and R. Madariaga, (2003), On the self-healing frature mode, Bull. Seismol. So. Am., 93(6), Ohnaka, M. (2003) A onstitutive saling law and a unified omprehension for fritional slip failure, shear frature of intat rok, and earthquake rupture, J. Geophys. Res., 108(B2), 2080, doi: /2000jb Coo et al. Saling of dynami slip weakening distane p. 21

23 Ohnaka, M., Yamashita, T., (1989), A ohesive zone model for dynami shear faulting based on experimentally inferred onstitutive relation and strong motion soure parameters. J. Geophys. Res., 94, Palmer, A. C., J. R. Rie, (1973), The growth of slip surfaes in the progressive failure of overonsolidated lay. Pro. R. So. London Ser. A 332, Perrin, G., Rie, J.R., Zheng, G., (1995), Self-healing slip pulse on a fritional surfae. J. Meh. Phys. Solids, 43, Piatanesi, A., E. Tinti, M. Coo and E. Fukuyama, (2004), The dependene of tration evolution on the earthquake soure time funtion adopted in kinemati rupture models, Geophys. Res. Lett., 31, doi: /2003gl Piatanesi, A., A. Cirella, P. Spudih, and M. Coo (2007), A global searh inversion for earthquake kinemati rupture history: Appliation to the 2000 western Tottori, Japan earthquake, J. Geophys. Res., 112, B07314, doi: /2006jb Power, W.L., T.E. Tullis, S.R. Brown, G.N. Boitnott, and C.H. Sholz, (1987), Roughness of natural fault surfaes, Geophys. Res. Lett., 14, Pulido, N. and T. Kubo (2004), Near-fault strong motion omplexity of the 2000 Tottori earthquake (Japan) from a broadband soure asperity model, Tetonophysis, 390, Rie, J.R., C.G. Sammis and R. Parsons, (2005), Off-fault seondary failure indued by a dynami slip-pulse,, Seismol. So. Am. Bull., 95(1), Rie, J. R., and M. Coo, 2006, Seismi fault rheology and earthquake dynamis, Dahlem Workshop on The Dynamis of Fault Zones, pp , edited by M. R. Handy, MIT Press, Cambridge, Mass. Sholz C.H., (1988), The ritial slip distane for seismi faulting, Nature, 336, Spudih, P., and M. Guatteri, (2004), The effet of bandwidth limitations on the inferene of earthquake slip-weakening distane from seismograms, Bull. Seismol. So. Am., 94, Tinti, E., A. Bizzarri, A. Piatanesi, and M. Coo (2004), Estimates of slip weakening distane for different dynami rupture models, Geophys. Res. Lett., 31, L02611, doi: /2003gl Tinti, E., P. Spudih, and M. Coo, (2005a), Earthquake frature energy inferred from kinemati rupture models on extended faults, J. Geophys. Res., 110, B12303, doi: / 2005JB00364 Tinti, E., E. Fukuyama, A. Piatanesi and M. Coo, (2005b), A kinemati soure time funtion ompatible with earthquake dynamis, Bull. Seismol. So. Am., 95(4), Tinti, E., M. Coo, E. Fukuyama, and A. Piatanesi, (2008), Dependene of slip weakening distane (D) on final slip during dynami rupture of earthquakes, submitted to Geophys. J. Int. Coo et al. Saling of dynami slip weakening distane p. 22

24 Zheng, G., and J. R. Rie (1998). Conditions under whih veloity-weakening frition allows a selfhealing versus a rak-like mode of rupture, Bull. Seism. So. Am., 88, Yasuda T., Y. Yagi, T. Mikumo, T. Miyatake, (2005), A omparison between D -values obtained from a dynami rupture model and waveform inversion, Geophys. Res. Lett., 32, L14316, doi: /2005gl Wald, D.J., T.H. Heaton, and K.W. Hudnut, (1996), The slip history of the 1994 Northridge, California, earthquake determined from strong-motion, teleseismi, GPS, and leveling data, Seismol. So. Am. Bull., 86 (1, Part B Suppl.), Weertman J., (1980), Crak tip advane model in fatigue of dutile material, Journal of Metals, 32, (12), Figure Captions Figure 1: (a) Comparison of slip veloity, slip and tration time histories at a target point on a fault plane using a smoothed Yoffe funtion as a soure time funtion. Blak solid irle indiates the time of peak slip veloity (T a ) and the grey solid irle indiate the end of weakening (T b ); (b) orresponding tration versus slip behavior; the same irles of panel a are indiated in term of slip with D, and D parameters, respetively. a Figure 2: Several analytial soure time funtions proposed in the literature to model the slip veloity evolution on the fault plane: Delta, Box-ar, Triangular, Gaussian, Kostrov and Yoffe funtions. Figure 3: Upper panel: Slip distribution of kinemati model by Hartzell and Heaton (1983) for the 1979 Imperial Valley earthquake. Bottom panel: slip veloity and slip time histories for five subfaults as indiated by the apital letters above. Figure 4: Tration versus slip urves of the 1979 Imperial Valley earthquake for several subfaults. The relative position on the fault plane (dip and strike) is indiated for eah subfault. The Coo et al. Saling of dynami slip weakening distane p. 23

25 intervals shown for the lower left panel ([0 2], [-20 20]) indiate tration (in MPa) and slip (in m) and are the sales for eah panel. Figure 5: Upper panel: Tration hange time histories and tration hange versus slip urves for the Figure 6: same five subfaults of Figure 3. distribution for the 1979 Imperial Valley earthquake inferred from the kinemati model of Hartzell and Heaton (1983). Figure 7a: versus total slip (Dtot) for all subfaults of the 1979 Imperial Valley earthquake using the Hartzell and Heaton (1983) kinemati model. Figure 7b: versus total slip (Dtot) for all subfaults of 1994 Northridge earthquake using the Wald et al. (1996) kinemati model. Values are measured following the interpolation strategy disussed by Tinti et al. (2005a). Figure 8: D distribution for 1979 Imperial Valley earthquake using the Hartzell and Heaton (1983) kinemati model as a boundary ondition to ompute tration history. Figure 9: D versus total slip (D tot ) for all subfaults of the 1979 Imperial Valley earthquake using the Hartzell and Heaton (1983) kinemati model as boundary ondition of the tration hange. Figure 10: D /D tot distribution for the 1979 Imperial Valley earthquake using the Hartzell and Heaton (1983) kinemati model. Figura 11: Top panel: omparison of slip (a), slip veloity (b), tration evolution () and tration versus slip urves (d) at a target point for the two soure time funtions f 1 (in blak) and f 2 (in gray). Bottom panel: normalized time histories of slip, slip veloity and dynami tration alulated with Tanh funtion (e) and Yoffe funtion (f) for the same target point. Figure 12: Distribution of slip and rupture time (a, f); strength exess (b,, g, h) and dynami stress drop (d, e, i, l) on the fault plane retrieved for the two soure time funtion f 1 and f 2 and for heterogeneous (left panels) and onstant (right panels) rupture veloity models. Figure 13: The dynami tration hanges (panels, d, e and f) for soure models having uniform slip of 1 m, onstant rupture veloity (2.0 km/s) and slip veloity time histories represented by Yoffe funtion (panels a and b). Left panels: alulations obtained for different T a values and onstant rise time (1.0 s); right panels: alulations for different rise times and a onstant T a (0.225s). Figure 14: W b distribution (J/m 2 ) of the 1979 Imperial Valley earthquake using the Hartzell and Heaton (1983) kinemati model. Coo et al. Saling of dynami slip weakening distane p. 24

26 Figures: Figure 1 Figure 2 Coo et al. Saling of dynami slip weakening distane p. 25

27 Figure 3 Figure 4 Figure 5 Coo et al. Saling of dynami slip weakening distane p. 26

28 Figure 6 Figure 7 Figure 8 Coo et al. Saling of dynami slip weakening distane p. 27

29 Figure 9 Figure 10 Coo et al. Saling of dynami slip weakening distane p. 28

30 Figure 11 Coo et al. Saling of dynami slip weakening distane p. 29

31 Figure 12 Coo et al. Saling of dynami slip weakening distane p. 30

32 Figure 13 Figure 14 Coo et al. Saling of dynami slip weakening distane p. 31