Transition process from nucleation to high-speed rupture propagation: scaling from stick-slip experiments to natural earthquakes

The process of earthquake generation is governed by a coupled non-linear system consisting of the equation of motion in elastodynamics and a fault constitutive relation. On the basis of the results of stick-slip experiments we constructed a theoretical source model with a slip-dependent constitutive law. Using the theoretical source model, we simulated the transition process numerically from quasi-static nucleation to high-speed rupture propagation and succeeded in quantitatively explaining the three phases observed in stick-slip experiments, that is very slow (1 cm s − 1 ) quasi-static nucleation preceding the onset of dynamic rupture, dynamic but slow (10 m s − 1 ) rupture growth without seismic-wave radiation, and subsequent high-speed (2 km s − 1 ) rupture propagation. Theoretical computation of far-ﬁeld waveforms with this model shows that a slow initial phase preceding the main P phase expected from a classical source model is radiated in the accelerating stage from the slow dynamic rupture growth to the high-speed rupture propagation. On the assumption that the physical law governing rupture processes in natural earthquakes is essentially the same as that in stick-slip events, we scaled the theoretical source model explaining the stick-slip experiments to the case of natural earthquakes so that the scaled source model explains the observed average stress drop, the critical nucleation-zone size, and the duration of the slow initial phase well. The physical parameters prescribing the source model are the weak-zone size L , the critical weakening displacement D 9 c , the breakdown strength drop t : b , and the rigidity m of the surrounding elastic medium. In scaling these parameters, we held a non-dimensional controlling parameter m ∞= ( m D 9 c )/( t : b L ) in numerical simulation constant. From the results of scaling we found the following fundamental relations between the source parameters: (1) the critical weakening displacement D 9 c is in proportion to the weak-zone size L , but (2) the breakdown strength drop t : b is independent of L . initial rupture process of by analysing near-source records and conﬁrmed that seismic phases outward migration of the end zones with high stress concentration. When the stress concentration at the end zones reaches characterized by a low moment-release rate exist for all of the earthquakes ( M 2.6–8.1) which they examined. These obser- a critical level, dynamic rupture starts and propagates outward.

surfaces. The concept of the cohesive force was first introduced 1 INTRODUC TION by Barenblatt (1959) to represent molecular cohesion in brittle tensile fracture and was later extended to the case of shear It is now widely accepted that the source of earthquakes is fracture by Ida (1972) and Palmer & Rice (1973). The essential brittle shear fracture occurring in the Earth's interior. In the property of the cohesive force is that it is a decreasing resistant classical theory based on linear elasticity, the brittle fracture force with the progress of failure. The existence of such a is described as crack extension with sudden stress drop. The resistant force in brittle shear fracture has been confirmed by sudden stress drop necessarily leads to physically unreasonable Okubo & Dieterich (1984) and Ohnaka, Kuwahara & stress singularities at crack tips. The stress singularities can be Yamamoto (1987) through careful stick-slip laboratory experiresolved by introducing a cohesive force acting between crack ments. According to these experimental studies, with the progress of fault slip, shear stress acting on the fault plane first increases rapidly up to a peak value and then decreases essential, as pointed out by Ohnaka & Kuwahara (1990). If the rate effect is weak and negligible, the rate-and state-gradually to a dynamic frictional level. Matsu'ura, Kataoko & Shibazaki (1992) have revealed the physical mechanism of dependent friction law can be reduced to the slip-dependent friction law in form (Matsu'ura et al. 1992). In the present such a slip-strengthening and weakening process by considering microscopic interaction between statistically self-similar study, we use the slip-dependent friction law because our present concern is the transition process from nucleation to fault surfaces, and concluded that diversity in constitutive behaviour of actual faults can be ascribed to differences in the high-speed rupture propagation.
Recently, initial rupture processes of natural earthquakes physical properties, such as roughness and hardness, of fault surfaces.
have been investigated in detail through the analysis of seismicwaveform data. For example, Iio (1992Iio ( , 1995 has carried out The constitutive relation is a physical law of fracture described in terms of elasticity. Therefore, incorporating the high-resolution observations of microearthquakes and confirmed the existence of a slowly rising initial phase preceding constitutive relation into the equation of equilibrium, we can treat the problem of earthquake rupture nucleation as a kind the arrival of the main P phase. The slow initial phase cannot be explained by classical source models, in which rupture is of boundary-value problem in elastodynamics. Following this idea, Matsu'ura et al. (1992) constructed a theoretical model presumed to expand at a constant velocity with a constant stress drop. Umeda (1990Umeda ( , 1992 has also reported the existence and numerically examined the details of the nucleation proceeding in a seismogenic zone with the increase of external of a low-amplitude initial phase preceding the main P phase in large earthquakes. Ellsworth & Beroza (1995) investigated shear stress. These numerical simulations showed that the nucleation process is always accompanied by the growth and the initial rupture process of earthquakes by analysing nearsource records and confirmed that seismic nucleation phases outward migration of the end zones with high stress concentration. When the stress concentration at the end zones reaches characterized by a low moment-release rate exist for all of the earthquakes (M2.6-8.1) which they examined. These obser-a critical level, dynamic rupture starts and propagates outward. Andrews (1976Andrews ( , 1985, Day (1982) and Fujiwara & Irikura vational studies suggest that the existence of slow initial phases is a common feature characterizing the initial rupture process (1991) have studied the details of the dynamic propagation of shear rupture theoretically, assuming a slip-weakening friction of earthquakes.
In the present study, first, we briefly review the basic law. In these studies, however, it was supposed that frictional properties of faults were uniform, and so fracture energy was equations governing the process of earthquake generation. Next, we demonstrate that the transition process observed in constant, at least in nucleation zones. If the fracture energy is constant, the dynamic rupture must start from a pre-existing stick-slip experiments can be quantitatively explained by a theoretical source model with a slip-dependent friction law. crack of a critical size. Fundamental questions are how the crack grows into the critical size, how the dynamic rupture Then, on the assumption that the physical law governing rupture processes in natural earthquakes is essentially the same starts and how the rupture propagation is accelerated to a terminal velocity.
as that in stick-slip events, we scale the theoretical source model explaining the stick-slip experiments to the case of As to the transition process from nucleation to high-speed rupture propagation, several experimental studies of stick-slip natural earthquakes. Finally, on the basis of the results of scaling, we derive the fundamental relations amongst the have been carried out. In recent stick-slip experiments with large rock samples, three different phases of rupture growth physical parameters prescribing the source model. have been observed. Dieterich (1978) has pointed out the existence of two different precursory phases preceding the 2 BASIC EQUATIONS GOVERNING onset of high-speed rupture propagation, namely very slow EARTHQUAKE GENERATION PROCESSES quasi-static nucleation and unstable slow rupture growth. Ohnaka & Kuwahara (1990) performed careful stick-slip We represent frictional interaction between fault surfaces by a slip-dependent constitutive relation and use it as the funda-experiments and confirmed the existence of the second phase. Yamashita & Ohnaka (1991) numerically simulated the quasi-mental law governing the entire process of earthquake rupture. This means that the rupture at a certain point on the fault static nucleation process by using a simplified breakdownzone model. Dieterich (1992) also simulated the nucleation plane must proceed irreversibly along the constitutive relation curve at this point. Then, incorporating the constitutive relation process by using a rate-and state-dependent friction law. Okubo (1989) calculated the entire process of earthquake into the equation of equilibrium (in a quasi-static case) or the equation of motion (in a dynamic case), we can treat the process rupture with the rate-and state-dependent friction law. Shibazaki & Matsu'ura (1992) developed a theoretical model of earthquake generation as a kind of boundary-value problem in elastodynamics. with a slip-dependent friction law and examined the spontaneous process of nucleation, dynamic propagation and stop We take a Cartesian coordinate system (x, y, z) in an infinite elastic medium and consider 2-D, anti-plane shear faulting on in earthquake rupture through numerical simulations.
As experimentally confirmed by Dieterich (1979Dieterich ( , 1981, the x-z plane. The directions of fault slip and fault extension are taken to be parallel to the z-axis and the x-axis, respectively. the frictional behaviour of rocks has a rate dependence. There have been many studies that modelled the earthquake The constitutive relation between shear stress t(x) and fault slip w(x), which is generally a function of position x, is generation process with the rate-and state-dependent friction law (e.g. Tse & Rice 1986;Okubo 1989;Dieterich 1992;Rice expressed as 1993). The results of these studies indicate that the rate effect ( 1 ) is essential in the processes of tectonic loading, nucleation and healing. For the transition process from nucleation to high-When we apply a uniform shear stress t 2 externally to this system, first, the quasi-static nucleation proceeds in the weakest speed rupture propagation, however, the rate effect is not so portion of the fault. In this case the equilibrium condition for 3 INTERPRETATION OF TRANSITION traction on a faulting region, −c≤x≤c, is given by PROCESSES IN STICK-SLIP EV ENTS

experiments
Ohnaka & Kuwahara (1990) carried out stick-slip laboratory where m is the rigidity of the surrounding elastic medium. In experiments with a 28 cm×28 cm×5 cm rock sample of the above equation, which was first obtained by Eshelby, Tsukuba granite, which has a pre-cut fault aligned along the Frank & Nabarro (1951) on the basis of elastic-dislocation diagonal of the sample (Fig. 1). The rigidity and the S-wave theory, the first term represents the stress changes produced velocity of the rock sample are measured as 20 GPa and by the fault slip w(x). Incorporating the fault constitutive 2.9 km s−1, respectively. Fig. 2 is an example of experimental relation (1) into the equilibrium condition (2), we obtain a results, which shows the development of the breakdown zone coupled non-linear system governing the quasi-static nucleation after the onset of dynamic rupture. The hatched portion process. This coupled non-linear system can be numerically indicates the breakdown zone, in which shear stress decreases solved by using the method developed by Matsu'ura et al.
from a peak stress s p to a constant frictional stress s f , as (1992).
shown in Fig. 3. From Fig. 2 it is seen that the rupture front When the external shear stress exceeds a critical level, the propagates very slowly (50 m s−1) from CH5 to CH4 at first. fault system becomes unstable and dynamic rupture starts. In Then, after about 0.6 ms from the start of rupture, the dynamic this case, we incorporate the fault constitutive relation into rupture is rapidly accelerated to a terminal velocity (2 km s−1). the equation of motion and obtain a coupled non-linear system The duration of the accelerating stage from slow dynamic governing the dynamic process of rupture propagation. Just rupture growth to high-speed rupture propagation is about before the initiation of dynamic rupture, the shear stress and 0.06 ms. In Fig. 4(a) we show the changes in the shear stress the fault slip are in critical states, t c (x) and w c (x), which can acting on the fault plane with time after the onset of dynamic easily be found by solving successively the coupled quasi-static rupture. These are the original records from which the diagram equations, (1) and (2), increasing the external shear stress in Fig. 2 was constructed. The dynamic rupture starts at a stepwise. We take these critical states as the reference states to point between CH5 and CH6 and propagates very slowly at measure changes in stress and fault slip, Dt(x, t) and Dw(x, t), first with a gradual stress drop. After about 0.6 ms, the dynamic during the dynamic rupture process. Then, the total stress rupture accelerates to a terminal velocity and propagates t(x, t) and the total fault slip w(x, t) are given by outwards with a rapid stress drop. Fig. 4( b) shows changes in fault slip (thick line) and slip velocity (thin line) with time. At CH6 and CH5, located inside the nucleation zone, the fault slip increases very slowly at first. Here, we call the region in w(x, t)=w c (x)+Dw(x, t) .
(4) which slip weakening proceeds quasi-statically the nucleation zone. After about 0.6 ms, with the onset of the high-speed rupture In the case of a 2-D, anti-plane fault, solving a boundarypropagation, the fault slip increases rapidly. At CH1, where value problem in elastodynamics, Kostrov (1966) obtained a relation between the incremental stress Dt(x, t) and the incremental fault slip Dw(x, t): with S; b2(t−t∞)2−(x−x∞)2≥0 and 0≤t∞≤t , where b indicates the S-wave velocity. We assume that the dynamic rupture process is governed by the same fault constitutive relation as in the case of quasi-static nucleation; that is, if the slip-velocity is positive, but if the slip-velocity is zero, We couple the equations (3)-(5) and (7) or (8) and obtain a non-linear system governing the dynamic process of rupture propagation. This coupled non-linear system can be solved numerically by using the boundary-integral method developed by Andrews (1985). the rupture propagation has accelerated to a terminal velocity, the fault slip increases abruptly with the onset of dynamic rupture, so the slip velocity has a very high and narrow peak. Kuwahara et al. (1986) also examined how the transition from quasi-static nucleation to dynamic rupture propagation proceeds through stick-slip laboratory experiments. Fig. 5 shows the relation between the rupture growth rate normalized by the S-wave velocity of rock samples and the rupture growth distance normalized by the upper corner wavelength l c of the power spectrum of the surface topography. Kuwahara et al. found that the normalized data for various experiments with different l c lie almost in a single curve. This suggests that the upper corner wavelength l c , and thus the roughness of rock surfaces, is a key parameter for scaling the transition process. From Fig. 5 it is seen that the process of rupture growth consists of three different phases: quasi-static nucleation (phase I), dynamic but slow rupture growth (phase II) and high-speed rupture propagation (phase III). It should be noted that the rupture growth rate in phase I depends strongly on the applied strain rate ė.
As shown in Fig. 3, the slip-dependent constitutive behaviour is essentially prescribed by the breakdown strength drop t b , which is defined by the difference between the peak stress s p and the dynamic frictional stress s f , and the critical weakening displacement D c . Ohnaka & Kuwahara (1990) measured the values of these physical parameters directly at each channel. Their results are summarized in Figs 6(a) and (b) for the breakdown strength drop t b and the critical weakening displacement D c , respectively. Here, we take the centre of the nucleation zone to be at 1.5 cm to the left of CH5. These

Theoretical interpretation of the transition process
We now demonstrate that the transition process observed in stick-slip experiments can be quantitatively explained by a theoretical source model with the laboratory-based fault constitutive relation. Taking a dynamic frictional stress s f as the reference to measure shear stress t(x), we may generally express the slip-dependent constitutive relation in the following form as a function of position x (Matsu'ura et al. 1992): On the basis of the experimental results in Fig. 6, we determine the concrete functional forms of t b (x) and D c (x), as shown in Table 1. It should be noted here that the spatial variable x is normalized by the weak-zone size L , and the shear stress t and the fault slip w are normalized, respectively, by the breakdown strength drop t : b and the critical weakening displacement D 9 c in the normal-strength region extending outside the weak zone. In the present case, the weak-zone size L , the breakdown strength drop t : b , and the critical displacement D 9 c are 10 cm, 1.2 MPa, and 3 mm, respectively. In Fig. 7 we schematically represent a 2-D fault with the constitutive relation curves reconstructed from the experimental results. The fault has a broad weak zone, characterized  With the source model reconstructed from experimental stress t 2 , at first the shear stress acting on the fault surface rises uniformly over the whole region. When the shear stress results, we numerically compute the entire process from quasistatic nucleation to high-speed dynamic rupture propagation. reaches a certain level (t=0.72 MPa), quasi-static nucleation starts at the weakest portion and develops outwards with a In the numerical computation, in addition to the normalization of t, w, and x by t : b , D 9 c , and L , the time variable t is also gradual stress drop (phase I). The gradual stress drop in the nucleation zone brings about a weak stress concentration in normalized by a characteristic time L /b of the system. Corresponding to these normalizations of variables, we should the surrounding regions. When the externally applied stress reaches a critical level (t 2 =0.86 MPa), the system becomes replace the S-wave velocity b in the eqs (5) and (6) with unity and the rigidity m in eqs (2) and (5) with unstable and dynamic rupture starts at the end zone. Fig. 8(b) shows the change in shear stress with time after the onset of m∞=( mD 9 c )/(t : b L ) .
(10) dynamic rupture. The stress curve at t=0 in Fig. 8( b) corresponds to that in the final state of the quasi-static nucleation Here it should be noted that the non-dimensional quantity m∞ is the only parameter which essentially controls the quasi-process shown in Fig. 8(a). The dynamic rupture grows very slowly at first (phase II). Then, after about 0.2 ms, the rupture static and dynamic behaviour of the non-linear coupled system in numerical simulations. In the case of the stick-slip experi-growth is suddenly accelerated and high-speed dynamic rupture propagates outwards with a rapid stress drop (phase III). ments performed by Ohnaka & Kuwahara (1990), taking the observed values of m=23 GPa, t : b =1.2 MPa, D 9 c =3 mm, and In phase III, high stress concentration appears on the rupture fronts. L =10 cm, we obtain m∞=0.57. Table 1. The values of the parameters prescribing the slip-dependent fault constitutive relation.   Fig. 8. At CH5 and CH6, located inside the nucleation zone, shear stress and the theoretical results in Fig. 9, we can see that the present theoretical model successfully describes the changes in shear drops gradually with time after the onset of dynamic rupture. After about 0.2 ms, high-speed dynamic rupture starts at the stress and fault slip during the transition process observed in stick-slip experiments. end of the nucleation zone and extends outwards with a rapid stress drop. Inside the nucleation zone, the fault-slip motion is Fig. 10 shows the relations between shear stress and slip velocity during the breakdown process at the five points very slow at first and then gradually accelerates with the onset of high-speed rupture propagation. Therefore, the slip velocity CH2-CH6. At CH5 and CH6, located inside the nucleation zone, the shear stress drops monotonically with the increase at CH5 and CH6 has a low and broad peak. At CH4, located just outside the nucleation zone, the fault slip increases fairly in slip velocity. At CH2 and CH3, where the rupture propagation has accelerated to a terminal velocity, the shear-stress-rapidly with the onset of dynamic rupture, so the corresponding slip velocity has a high peak with a long tail. At CH2 and slip-velocity state changes along an oval trajectory. These theoretical results are also in accord with the experimental CH3, where the rupture propagation has accelerated to a terminal velocity, the fault slip increases abruptly with the results of Ohnaka & Kuwahara (1990). Fig. 11 shows the development of the breakdown zone with time in the transition process from quasi-static nucleation (b) to dynamic rupture propagation (a). Here, the breakdown zone is defined as the region where the process of slip weakening is ongoing. The rupture growth rate in the quasi-static nucleation process (Fig. 11b) is in proportion to the rate of increase in the externally applied stress (strain). In the present case, for comparison with the experimental results, we take the increase rate of externally applied strain to be 10−6 s−1. The quasistatic nucleation begins about 1.4 s before the onset of dynamic rupture. For the quasi-static nucleation (phase I) the average rupture growth rate is about 1.4 cm s−1. When the strain rate is 10−5 s−1, the nucleation begins about 0.14 s before the onset of dynamic rupture. In either case the critical size L c of the nucleation zone is 4.2 cm. The dynamic rupture process shown in phase III expected from the theoretical model are somewhat shorter than those observed in the laboratory experiments. This may be due to the difference in the mode of fracture between the theoretical calculation (mode III) and the laboratory experiments (mode II).
The duration of slow dynamic rupture growth (phase II) depends on the magnitude of a perturbation given for triggering dynamic rupture. If the magnitude of the given perturbation is larger, the duration of phase II becomes shorter. In stickslip experiments it is frequently observed that the main dynamic rupture is triggered by the occurrence of a small AE event in a critical stress state. Therefore, in our numerical simulation, we search the critical stress state by increasing the external shear stress with a very small step (t : b ×10−5), and then start the dynamic analysis by applying a very small perturbation at the edge of the nucleation zone.
From a cohesive zone model it is expected that the breakdown time T b is nearly equal to the width of the slipacceleration pulse, so its inverse gives a rough estimate of the cut-off frequency f s max of the slip-acceleration spectrum (Ohnaka & Yamashita 1989). Fig. 12 shows changes in fault slip, slip velocity, slip acceleration and shear stress with time at a point located in the middle of the normal-strength region, where the dynamic rupture propagates at the terminal velocity. From Fig. 12 the breakdown time and the width of the slipacceleration pulse is found to be about 0.01 ms. From the slip-acceleration spectrum in Fig. 13, on the other hand, the cut-off frequency f s max is found to be about 100 kHz. Therefore, the approximate relation obtained by Ohnaka & Yamashita (1989), is valid for the present theoretical model. Finally, we show a plot of the rupture growth rate versus the half-length of the rupture zone in Fig. 14. Here, the applied  rupture growth rate increases rapidly with the length of the rupture zone. In the last stage of phase II, the rupture growth accelerates to the S-wave velocity. The rapid growth of rupture in the early stage of phase II expected from the theoretical model is in accordance with the experimental results in Fig. 5.

Basic assumptions in scaling
In the previous section we demonstrated that the transition process observed in stick-slip experiments can be completely described by the theoretical model with the laboratory-based fault constitutive relation. As shown in Fig. 5, the experimental data for various cases with different upper-corner wavelengths l c lie almost in a single rupture-velocity versus rupture-length curve, if the rupture length is normalized by l c . On the other hand, as theoretically demonstrated by Matsu'ura et al. (1992), the critical weakening displacement D c has a linear l c dependence. These experimental and theoretical results suggest that the critical weakening displacement D c has a linear scale dependence. This means that the ratio of the critical weakening displacement D 9 c to the weak-zone size L is nearly constant, so the value of m∞=(mD 9 c )/(t : b L ) is independent of the size of stick-slip events, since the weak-zone size L is the only parameter characterizing the system dimension in the numerical simulation, and m and t : b are essentially scale-independent parameters. Therefore, we may conclude that the result of slip velocity during the breakdown process at five representative numerical simulation presented here will have some generality, points, CH2-CH6. The locations of CH2-CH6 correspond to those in Fig. 8. although it is only a special case of m∞=0.57. for shear stress to degrade from a peak stress to a constant frictional stress.
In the present section, on the hypothesis that the physical law governing the rupture process in natural earthquakes is essentially the same as that in stick-slip events, we scale the theoretical source model explaining the stick-slip experiments to the case of natural earthquakes so that the scaled source model explains the observed average stress drop, the critical nucleation-zone size and the duration of a slow initial phase well. Since the rigidity, m, of the surrounding elastic medium can be regarded as nearly constant, key parameters in the scaling are the weak-zone size, L , the critical weakening displacement, D 9 c , and the breakdown strength drop t : b . In scaling these parameters, as a natural extension of the conclusion for stick-slip events, we assume that the nondimensional quantity m∞=(mD 9 c )/(t : b L ) remains constant (0.57) in the case of natural earthquakes.

Theoretical relations between observable quantities and source parameters
From the results of numerical simulation given in Section 3.2, we can derive some theoretical relations between seismologically observable quantities, such as the average stress drop, the critical nucleation-zone size and the duration of a slow initial phase, and the physical parameters prescribing the source model. As shown in Fig. 8, the difference between the The time axis is scaled by L /b. The velocity waveform amplified 10 times is also shown for reference. The first silent period T s and the duration of a slowly rising phase T i correspond, respectively, to the durations of the slow dynamic rupture Figure 14. A plot of the rupture velocity (rupture growth rate) versus growth and the accelerating stage in Fig. 11(a). the half-length of the rupture zone (rupture growth distance) in the transition process from quasi-static nucleation to high-speed dynamic rupture propagation. The rupture growth rate is normalized by the taneous stress drop. In the case of classical source models, the S-wave velocity.
far-field velocity waveform rises linearly with time after the arrival of the initial phase (Sato & Hirasawa 1973). From Fig. 11 the duration of the accelerating stage from phase II to 0.86 MPa during the dynamic rupture, which roughly corresponds to the average stress drop estimated from seismological phase III is found to be 4×10−5 s. This value is about 1.2 times as long as the characteristic time L /b (=3.3×10−5 s) of observations, is about 0.7 times as large as the breakdown strength drop t : b (=1.2 MPa) at the normal-strength region. the system. Therefore, we obtain the following approximate relation between the duration T i of the slow initial phase and Therefore, we obtain the following approximate relation between the average stress drop Dt : and the breakdown strength the characteristic time L /b of the system: drop t : b : The slow initial phases have actually been observed for many natural earthquakes (Umeda 1990;Iio 1992Iio , 1995Ellsworth From Fig. 11, the critical size L c of the nucleation zone is found to be 4.2 cm. This value is about 0.4 times as long as & Beroza 1995). From the relations (13) and (14), eliminating the source the weak-zone size L (=10 cm). Therefore, we obtain the following approximate relation between the critical nucleation-parameter L , we obtain the following relation between two seismologically observable quantities, the duration T i of the zone size L c and the weak-zone size L : slow initial phase and the critical size L c of the nucleation L c =0.4L .
(13) zone: The critical size of nucleation zones for natural earthquakes can be roughly estimated from the extent of immediate foreshock activity in the hypocentral area of the main event Here it should be noted that the above theoretically predicted relation holds, irrespective of the size of events, if our basic (Ohnaka 1993;Shibazaki & Matsu'ura 1995).
Another important observable quantity is the duration of a assumption in scaling, slowly rising initial phase preceding the main P phase. In Fig. 15 we show an example of the far-field waveforms computed from a rupture nucleation model. In the computation, is satisfied, that is we can use the relation (15) in order to check the validity of our assumption in scaling. the 1-D slip-time function Dw(x, t) obtained by the numerical simulation is simply extended to the 2-D slip-time function Dw(r, t) on a circular fault. The factors of radiation pattern 4.3 Scaling to natural earthquakes and geometrical spreading are omitted. From this diagram it is seen that seismic waves are hardly radiated during the process We now scale the theoretical source model explaining the transition process observed in stick-slip experiments to the of slow dynamic rupture growth (phase II). After a silent period, the far-field velocity waveform rises very slowly. This cases of two large earthquakes which occurred in Japan: the 1995 Hyogoken Nanbu earthquake (M7.2) and the 1983 slowly rising initial phase corresponds to the accelerating stage from phase II to phase III ( high-speed rupture propagation) Central Japan Sea earthquake (M7.7). For these earthquakes, both the foreshock activity just before the occurrence of the in Fig. 11. The slowly rising initial phase cannot be explained by classical source models, in which rupture is presumed to main shock and the existence of the slow initial phase have been precisely documented. expand from a point at a constant velocity with an instan-which is very close to the hypocentre of the main shock, by 4.3.1 T he 1995 Hyogoken Nanbu earthquake using the seismogram of the largest foreshock (M3.5) as the empirical Green's function. The deconvolved displacement and From the inversion analysis of far-field waveform data, the average stress drop Dt : of the Hyogoken Nanbu earthquake velocity functions (Figs 18a and b) clearly show the existence of a slowly rising initial phase. From these diagrams the has been estimated as 10 MPa (Kikuchi 1997). Substituting this value into the relation (12), we obtain t : b =14 MPa as the duration T i of the slow initial phase is found to be 0.6 s. The S-wave velocity b of the Earth's crust is 3.5 km s−1 on estimate of the breakdown strength drop, which is about 12 times as large as that in the case of stick-slip experiments.
average, so substitution of L c =0.5 km into the right-hand side of eq. (15) yields T i =0.43 s. This value is nearly equal to the In the case of this earthquake, four foreshocks were observed in the hypocentral area just before the mainshock.
observed value of T i =0.6 s. Thus, our assumption in scaling is valid for the Hyogoken Nanbu earthquake. Then, sub- Fig. 16 shows the hypocentre distributions of these foreshocks, together with the main shock (Nakamura, personal stituting T i =0.6 s into relation (14), we estimate the weakzone size L of this earthquake to be 1.7 km. Furthermore, from communication, 1996). The largest foreshock (M3.5) occurred 12 hours before the main shock. From this diagram it is seen eq. (16), taking the rigidity m of the Earth's crust to be 35 GPa, we find that the critical weakening displacement D 9 c must be that the foreshocks occurred on the fault plane of the main shock, which has a strike C-D. Therefore, we may regard these scaled to 0.4 m in order to hold the value of ( mD 9 c )/(t : b L ) constant (0.57). This value of D 9 c is about 1.3×105 times as foreshocks as the local dynamic rupture of asperities in the weak zone associated with the nucleation of the main rupture large as that in the case of stick-slip experiments. (Shibazaki & Matsu'ura 1995). From the extent of the foreshock activity around the hypocentre of the main shock, we 4.3.2 T he 1983 Central Japan Sea earthquake can roughly estimate the critical nucleation-zone size L c of this earthquake to be 0.5 km.
From the inversion analysis of tsunami data, the average stress drop Dt : of the Central Japan Sea earthquake has been Fig. 17 shows the initial parts of the velocity seismograms of the Hyogoken Nanbu earthquake, recorded by a JMA-87 estimated as 7 MPa (Satake 1985). Substituting this value into relation (12), we obtain t : b =10 MPa as the estimate of the strong-motion seismograph at Okayama station with an epicentral distance of D=103 km. The arrival time of the first breakdown strength drop, which is about eight times as large as that in the case of stick-slip experiments. weak signal is indicated by P 1 on the seismograms. The duration P 1 -P 2 of this initial phase is found to be about 0.6 s. In the case of the Central Japan Sea earthquake, remarkable foreshock activity was observed in the hypocentral area just Shibazaki & Yoshida (1995) have deconvolved the high-gain velocity seismogram of the main shock at Sumoto station, before the main shock (Hasegawa 1983). Fig. 19 shows the  preceding the main P phase. For this earthquake Umeda epicentre distributions of foreshocks (a) and aftershocks (b) of (1990) also reported the existence of a slow initial phase with this earthquake. The star indicates the epicentre of the main a duration of 5-6 s, based on the analysis of JMA-59 seismoshock. The largest foreshock (M4.9) occurred 12 days before grams recorded at various stations with different epicentral the main shock within the 2 km epicentral distance (Umino distances (D=187-294 km). et al. 1985). The extent of subsequent foreshock activity around Substitution of L c =5 km and b=3.5 km s−1 into the rightthe hypocentre of the main shock is about 5 km. On the basis hand side of eq. (15) yields T i =4.3 s. This value is nearly equal of these observations we can roughly estimate the critical size to the observed value of T i =5-6 s. Thus, our assumption in L c of the nucleation zone as 5 km. scaling is valid for the Central Japan Sea earthquake as well Fig. 20 shows the initial parts of the seismograms of the as for the Hyogoken Nanbu earthquake. Then, substituting Central Japan Sea earthquake [(a) medium-period displace-L c =5 km into relation (13), we can estimate the weak-zone ment and ( b) long-period displacement] recorded at Dodaira size L of this earthquake as 12 km. Furthermore, from eq. (16) station with an epicentral distance D=480 km (Tsujiura 1988).
we find that the critical weakening displacement D 9 c must be The arrival time of the first weak signal, confirmed by a shortscaled to 1.9 m in order to hold the value of (mD 9 c )/(t : b L ) period seismogram at the same station, is indicated by the constant (0.57). This value of D 9 c is about 6×105 times as large arrows P and P 1 on the seismograms. The duration P 1 -P 2 of as that in the case of stick-slip experiments. the weak signal preceding the main P phase is 5-6 s. In Fig. 21 we show the initial parts of JMA strong-motion seismograms recorded at two stations, Akita (D=113 km) and Morioka 4.4 Fundamental scaling relations (D=193 km), near the hypocentre of the main shock. The arrows indicate the arrival time of the first weak signal, In Section 4.2 we theoretically derived relation (15), confirmed by high-sensitivity seismograms at the same stations. These records support the existence of a slow initial phase between the duration T i of the slow initial phase and the scaling relation critical size L c of the nucleation zone on the basic assumption Given the physical parameters prescribing the source model (Table 2a), we can theoretically compute the seismologically observable quantities, the average stress drop Dt : , the critical in scaling. Fig. 22 is a plot of observed T i versus observed L c nucleation-zone size L c , the duration of the initial phase T i for two natural earthquakes, the Hyogoken Nanbu earthquake and the cut-off frequency of the slip-acceleration spectrum (M7.2) and the Central Japan Sea earthquake (M7.7), and f s max for each case. All of the results are summarized in one stick-slip event. From this diagram, it is seen that the Table 2( b), together with the observed values. Here it should theoretically predicted relation between T i and L c , which is be noted that the cut-off frequency of the slip-acceleration indicated by the thick line, is roughly realized, irrespective of spectrum f s max is nearly in proportion to the inverse of the sizes of the events. This means that our estimation of the breakdown time T b in the stage of high-speed rupture source parameters based on the basic assumption in scaling is propagation, as demonstrated in Section 3.2. The breakdown valid for the natural earthquakes.
time T b is scaled by the characteristic time L /b of the system: In Table 2 we summarize the values of the physical parameters prescribing the theoretical source model (a) and the seismologically observable quantities ( b) for the Hyogoken so, substituting this equation into eq. (11), we obtain the Nanbu earthquake and the Central Japan Sea earthquake, approximate relation together with those for the stick-slip event. Since the rigidity m and the S-wave velocity b are scale-independent in any sense, f s max =3b/L . (21) the essential parameters prescribing the source model are Furthermore, from relations (13) and (21), eliminating the the breakdown strength drop t : b , the critical weakening source parameter L , we obtain the following relation between displacement D 9 c and the weak-zone size L . From Table 2(a) the cut-off frequency f s max and the critical nucleation-zone size we find the following fundamental relations between the source L c : parameters: For the Hyogoken Nanbu earthquake, the cut-off frequency t : b =c (c$10 MPa) .
(18) of 4-5 Hz has been obtained from the spectral analysis of the acceleration seismogram recorded at Kobe University, which That is, the critical weakening displacement D 9 c increases linearly with the weak-zone size L , but the breakdown strength is very close to the earthquake fault (Kamae, personal communication, 1996). This value is in accord with the theoretical drop t : b is nearly constant, irrespective of the size of events. From these fundamental scaling relations, defining the fracture estimate (6 Hz). In the case of the Central Japan Sea earthquake, from the analysis of a strong-motion seismogram surface energy by G 9 c =t : b D 9 c /2, we can obtain the secondary  recorded at Furofushi station with an epicentral distance D=73 km, we can evaluate the cut-off frequency of the

DISCUSSION OF RESULTS
the onset of dynamic rupture, dynamic but slow rupture growth without seismic-wave radiation and subsequent high-Using a fault model with the slip-dependent constitutive law reconstructed from the results of stick-slip experiments, we speed rupture propagation. As experimentally confirmed by Dieterich (1979Dieterich ( , 1981, the frictional stress of rocks depends numerically simulated the transition process from quasi-static nucleation to high-speed rupture propagation and succeeded not only on fault slip but also on slip rate. The success in quantitatively explaining the experimental results with the slip-in quantitatively explaining the three phases observed in stickslip experiments: very slow quasi-static nucleation preceding dependent constitutive law indicates that the rate effect has only secondary significance as far as the transition process and the duration of the slow initial phase well. The essential parameters prescribing the source model are the weak-zone from nucleation to dynamic rupture propagation is concerned. In fact, Okubo & Dieterich (1986) have suggested that the size L , the critical weakening displacement D 9 c and the breakdown strength drop t : b . In scaling these parameters, we assumed rate effect has a high-speed cut-off. In the present numerical simulation, the slip velocity during slow dynamic rupture that the non-dimensional quantity m∞=( mD 9 c )/(t : b L ) is kept constant (0.57) irrespective of the size of the events. The growth is of the order of 10−4-10−2 m s−1, which is above the average high-speed cut-off velocity (2×10−4 m s−1) estimated validity of this basic assumption in scaling was confirmed by checking the theoretically predicted relation (15) between two by them.
Computation of far-field waveforms with the theoretical seismologically observable quantities, the duration T i of the slow initial phase and the critical size L c of the nucleation source model shows that a slowly rising initial phase preceding the main P phase is radiated in the accelerating stage from slow zone. From the results of scaling we found the following fundamental relations between the source parameters: the dynamic rupture growth to high-speed rupture propagation (Fig. 15). The actually observed slow initial phases (e.g. Fig. 18) critical weakening displacement D 9 c increases linearly with the weak-zone size L , but the breakdown strength drop t : b is nearly are, however, not as simple as theoretically predicted, but contain several small pre-events. These small pre-events can constant irrespective of the size of events. These fundamental relations lead directly to the secondary scaling relation: the be considered as the local brittle rupture of small asperities distributed in the nucleation zone of the main rupture. fracture surface energy, G 9 c =t : b D 9 c /2, increases linearly with the weak-zone size L . Recently, Shibazaki & Matsu'ura (1995) have succeeded in explaining the occurrence of foreshocks and pre-events associ-As a consequence of the fundamental scaling relations, it can theoretically be expected that the duration T i of the slow ated with the nucleation of large earthquakes by considering the distribution of locally strong parts (asperities) in the broad initial phase is in proportion to the weak-zone size L ; that is, taking the value of b to be 3 km in eq. (14), weak zone on a fault plane. In the present study, we considered a simple weak-zone model for scaling, because the overall T i =0.4L (T i in s and L in km) .
(23) features of nucleation are essentially controlled by the frictional properties of the weak zone with the largest scale. In this sense From the analysis of far-field seismograms for large and we may regard the weak-zone size L as the largest wavelength intermediate earthquakes, on the other hand, Umeda (1992) characterizing strength variation along the fault.
has obtained an empirical relation between the duration T i of On the hypothesis that the physical law governing the slow initial phases and the magnitude M of earthquakes: rupture process in natural earthquakes is essentially the same as that in stick-slip events, we scaled the theoretical source log T i =0.5M−3.4 (T i in s) . (24) model explaining the stick-slip experiments to the cases of the Hyogoken Nanbu earthquake and the Central Japan Sea Recently, from the analysis of near-field seismograms, Ellsworth & Beroza (1995) have also confirmed the existence earthquake, so that the scaled source model explains the observed average stress drop, the critical nucleation-zone size of a similar relation over a wide range of magnitude