Determination of constitutive relations of fault slip based on seismic wave analysis

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 102, NO. B12, PAGES 27,379-27,391, DECEMBER 10, 1997 Determination of constitutive relations of fault slip based on seismic wave analysis Satoshi Ide and Minoru Takeo Earthquake Research Institute, University of Tokyo, Tokyo Abstract. Constitutive laws define the boundary conditions on fault plane and govern many aspects of earthquake failure. Although several constitutive laws have been formulated based on laboratory rock experiments and applied to theoretical studies in various fields, no actual relation during a natural earthquake has been determined. The 1995 Kobe earthquake is suitable for detailed kinematic analysis, and this enables the first evaluation of constitutive relations for a natural earthquake. In this study, we determine spatiotemporal slip distribution on an assumed fault plane of the 1995 Kobe earthquake by waveform inversion and then solve elastodynamic equations using a finite difference method to determine the stress distribution and constitutive relations on the fault plane. An inversion method based on Bayes theorem is employed to obtain a spatiotemporal slip distribution, and enables us to ensure the objective uniqueness of the solution with numerous parameters and smoothing constraints. This slip distribution then used as part of the boundary condition in the finite difference calculation. The time histories of slip and shear stress obtained then provide a constitutive relation at each point on the fault plane. They show slip weakening relations almost everywhere on the fault plane, while slip rate dependency is not clear. The slip weakening behavior has a clear depth dependency indicating that the slip weakening rate (dr./du) is smaller in the shallow crusthan that in the deep crust. This may be associated with the paucity of shallow seismicity observed in the source region of this earthquake as reported for many mature fault systems. 1. Introduction determine whole fault behavior. Up until now, many studies have presented various constitutive relations based on laboratory Detailed analyses of seismic sources have revealed their experiments of rock friction in which frictional sliding is regarded complexity and diversity. Generally, an earthquake consists of as an analog to fault slip in natural earthquakes, and various many small-scale ruptures which are often called subevents, and constitutive laws are suggested from these experiments, or just its rupture propagation is irregular. Moreover, one earthquake theoretically. The most simple constitutive law is the classical may behave differently from another one, even if both have friction law of static and dynamic friction levels, which results in similar magnitude; one example is the so-called slow earthquake. a stress singularity at the crack tip when it is introduced as the Is it possible to interpret such complexity and diversity of boundary condition of the crack surface. To mathematically earthquakes within a unified physical model? eliminate this singularity, slip weakening friction law was To interpret earthquake dynamics, we regard the source as introduced by Ida [1972] using the notion of a crack tip cohesive distribution of discontinuities of displacement, namely, slip, on a zone introduced by Barenblatt [1959] for tensile cracks. As fault plane. Although actual faults have many kinks or jogs and shown in Figure 1, the slip weakening friction law is form a fault zone having a certain thickness, they are successfully characterized by four parameters: an initial stress to, a peak stress approximated as a fault plane if we perform an analysis of the rp, a residual, or dynamic, stress rr, and a critical displacement Dc. seismic waves whose wavelength are longer than the fault zone From experimental observations of shear stress and frictional thickness. The surrounding body is successfully approximated sliding amount during stick-slip instability along a precut fault in as an elastic material as long as a phenomenon of the earthquake rock samples, Okubo and Dieterich [1984] and Ohnaka et al. rupture concerns. Thus an earthquake source is represented by [1987] found the slip weakening relations. Further, Matsu'ura an elastic body and its boundary on which a constitutive relation et al. [1992] interpreted these relations by considering the controlling earthquake complexity and diversity is given as a microscopic interaction between statistically self-similar fault boundary condition. surfaces and presented a slip-dependent constitutive law. Using A constitutive relation is dependence among fault slip, slip rate, such a slip-dependent constitutive law, Shibazaki and Matsu'ura and stress; moreover, it is controlled by pressure, te nperature, [1992] simulated the nucleation process of earthquakes. some chemical effects, and other factors. In this paper, we use Dynamic rupture propagation based on a slip weakening law was "constitutive relation" as any observable relation between numerically studied by Andrews [1976, 1985] for twophysical quantities and distinguish it from "constitutive laws" or dimensional (2-D) shear crack and by Day [1982] for 3-D crack. "friction law," which are thought to be physical constraints and Another familiar constitutive law is the rate- and state- Copyright 1997 by the American Geophysical Union. Paper number 97JB /97/97 JB $ ,379 dependent friction law first presented by Dieterich [1979] and mathematically refined by Ruina [1983]. The advantage of this law over the slip-dependent constitutive law is that the rate- and state-dependent law allows a retrieval of fault strength following a

2 27,380 IDE AND TAKEO: DETERMINATION OF CONSTITUTIVE RELATIONS modeling methods for this purpose. We overcome these difficulties by determining stress distribution directly from a kinematic model which is objectively unique and more detailed than the previous results for the 1995 Kobe earthquake. The procedure is divided into two successive calculations. First, we invert seismic waves to obtain the spatiotemporal slip distribution and then solve the elastodynamic equations to obtain the spatiotemporal stress distribution. In laboratory experiments (a) 36' 35 ø Figure 1. Idealized slip weakening relation showing shear stress as a function of fault displacement. 34' dynamic rupture. Therefore it is applied in numerical simulations of long-term duration, e.g., the seismic nucleation process [Dieterich, 1992], the seismic cycle [Tse and Rice, 1986] and afterslip [Marone et al., 1991 ]. As another constitutive law, the slip rate-dependent friction law was applied in the statistical study of seismicity [Carlson and Langer, 1989]. Various rate dependence of friction laws are used in simulation of dynamic rupture propagation [Cochard and Madariaga, 1994; Pertin et al., 1995; Beelet and Tullis, 1996] to explain a slip pulse-like propagation of fault rupture. In spite of its significance in the many theoretical studies referred to above, the constitutive relation has not been estimated for the faulting process of natural earthquakes. The scaling relation between experimentally derived constitutive relations and those of natural earthquakes is also unknown. In the present paper, we try to obtain constitutive relations between slips and shear stresses on a fault plane, namely, as slip-dependent relations. We also compare slip rate with stress to verify the possibility of slip rate-dependent relations. Previous studies of the earthquake source processes have been mainly kinematic ones, in which the spatiotemporal distributions of slip are determined on assumed fault planes. To determine the constitutive relations, the spatiotemporal distribution of stress is also necessary. The dynamic models presented by Quin [1990] and Miyatake [1992] determined the stress distribution during earthquake rupture based on the linear elasticity theory with crack propagation and the classical friction theory of static and dynamic friction levels. In these dynamic models the kinematic parameters, slip amount and rupture time, are converted to dynamic parameters, static and dynamic stress levels on the crack plane. However, this approach has some defects, i.e., (1) stress and slip rate singularities at the crack tip due to the assumption of the classical friction law, (2) nonuniqueness arising from subjectivity due to the insufficient information from the kinematic model which they used, and (3) the difficulty in explaining detailed kinematic models because a part of the procedure is resolved by trial-and-error method. The third 33' 133' 134' 135' 136' 137' (b) 35.0ø1,,,,,,,.,,,.,, 1' o ø,½oo o / i g " oo o o o ø oo o øo o o - i 'ø i i o o 34.5 ø 35.0 ø 35.5 ø (C)surfac o o o I. o '.o 50 km o, o o ø N52øE, Figure 2. (a) Location map of strong-motion stations used in our analysis. The mainshock hypocenter and the assumed fault plane are shown by a star and a solid line, respectively. (b) Locations of aftershocks determined by Japanese University Group for Urgent Joint Observation of the 1995 Hyogo-ken defect results in difficulties in constructing such a dynamic model Nanbu Earthquake [Hirata et al., 1996]. The mainshock hypocenter is also represented by a star. Hatched rectangle that can explain the observed seismic waves, though Fukuyama represents the assumed fault plane. (c) Same as Figure 2b and Mikumo [1993] and lde and Takeo [1996] proposed iterative projected onto the assumed fault plane. o

3 IDE AND TAKEO: DETERMINATION OF CONSTITUTIVE RELATIONS 27,381 of stick-slip instability [Okubo and Dieterich, 1984; Ohnaka et al., 1987], slip and shear stress are individually measured by strain meters located along a precut fault in the rock sample. By [Akaike, 1980]. ABIC is based on the entropy maximization principle and has been successfully used in various fields including geophysics, e.g., Yabuki and Matsu'ura [1992], monitoring these meters during stick-slip instability, a local slip- Yoshida [1989], and Yoshida and Koketsu [1990]. In the stress relation is obtained at each point on the fault plane. On the other hand, in the analysis of natural earthquakes the only quantity obtained by waveform inversion is the distribution of slip. Nevertheless, the stress distribution can be obtained by solving elastodynamic equations with the boundary conditions of slip distribution. We employ a finite difference method (FDM) for the calculation of the elastodynamic equations. By comparing the slip and the stress on the fault plane at each time step, a local slip-stress relation is obtained at each point. The target earthquake is the 1995 Kobe (Hyogo-ken Nanbu), Japan, earthquake (Ms 6.8); a shallow intraplate strike slip earthquake which was recorded at numerous near-field stations. previous study of the Kobe earthquake, we adopted ABIC to the linear inversion of Takeo [1992] and Yoshida [1992] and determined the rupture history uniquely from a statistical standpoint [Ide et al., 1996]. Although a subfault discretization system is convenient for the representation of spatial distribution of slip, it is not continuous at the subfault boundaries and hence unsuitable for the following FDM calculation. A more general representation of slip distribution is expansion by 2-D spatial and temporal basis functions, with the expansion coefficients being unknown parameters. The spatiotemporal distribution of slip rate / i(xl, x2, t) is expanded as Figure 2 shows the hypocentral location determined by Japan Meteorological Agency (JMA) together with the station l i(x,t)= _ ailmn(p/(xl)(pm2 (X2)lffn(t), (1) distribution used in the analysis (Figure 2a) and the aftershock where aitmn are the expansion coefficients and O/(x ), Ore(X2), 2 and distribution (Figures 2b and 2c) [Hiram et al., 1996]. Some kinematic source models of this earthquake have been presented [Ide et al., 1996; Horikawa et al., 1996; Sekiguchi et al., 1996; Wald, 1996; Yoshida et al., 1996]. Although these models are determined on different conditions with different data sets, all the gt,,(t) are the basis functions in strike direction, dip direction, and time, respectively. If one selects a boxcar function as spatial basis function, (1) is identical with the conventional subfault discretization system. In this study, each basis function is an isosceles triangle determined by three knots, and hence the slip seismic moments are almost 2.0 x 10 9 N m. The spatial patterns distribution is continuous everywhere spatially and temporally. of slip are similar in the point that the areas of largest slip are the shallow region southwest of the hypocenter and the region near the hypocenter. Northeast of the hypocenter, all models show relatively smaller slip on the deeper part of the fault plane than in the southwest of the hypocenter. Slip or slip rate at a point in this 3-D source volume is easily determined by use of (1) with obtained model parameters a i m. Although a itmn can be determined by just fitting kinematic model with subfault system, this must yield another problem, namely ambiguity of smoothness. Hence, in the present study we determine them by inverting seismic data as follows. The relation between the synthetic displacement at a station 2. Determination of Slip Distribution and the slip rate distribution ti (x, t) on the fault plane is represented as 2.1. Method uj(x,t) = Ilg/j (x,t; :,'r)/ i ( e, ')d ed ', (2) First, we determine the spatiotemporal distribution of slip on a fault plane by inversion of near-field strong-motion seismograms. where gij(x, t;,,) is a Green's function representing the jth Many studies have revealed the source process of various component of the synthetic displacement when an impulsive slip earthquakes as spatiotemporal distributions of slip on fault planes rate in the ith direction is applied at x=, t= ' [e.g., Aki and using near-field seismic waves. Various inversion methods have Richards, 1980]. Substituting (1) into (2), an equation with been developed to do that. Fault planes are conventionally observedisplacement u '(x,t) is expressed as divided into many subfaults to represent spatial complexity, and the total rupture processes are composed of their individual slip histories. Hartzell and Heaton [1983] allowed subfaults to slip repeatedly in the manner of circular time window propagation u (x,t) = aitmn II gij (x,t; e, ')0] ( )0m 2 ( 2)gt,, ( ')d +ej(x,t), (3) from the hypocenter with fixed velocities. Because the starting where e (x,t) is the error between the observed and synthetic time of rupture at each subfault is fixed, this method enables a linear inversion that ensures the uniqueness of the solution. Takeo [1992] and Yoshida [1992] developed slightly different displacement. This equation is rewritten in vector form as d = Grn + e, (4) approaches of linear inversion by expanding the source time where d, rn, and e are the data vector composed of sampled data, function of each subfault with basis functions. These methods u (x,t), the parameter vector of a i m, and the error vector of e j generally require numerous parameters and various constraints (x,t), respectively. G is an N, the number of data, x M, the adopted to perform stable inversion, namely, with small model number of parameters, matrix obtained after calculating covariance. Without objectivestimation of the weights of these integration in (3) and sampling corresponding to d. It should be constraints, an objectively unique solution cannot be obtained. noted that the error vector e contains measurement errors and However, the significance of this objectivity has been almost modeling errors, e.g., inaccuracy of (1) the employed Green's neglected in most of the previous studies. A useful solution to maintain objectivity under linear functions and (2) assumption of fault setting. Although it cannot be assured, for simplicity, error e is assumed to be Gaussian with constraints on model parameters is by constructing a Bayesian zero mean and covariance 0'2I, where I is an identity matrix. model, which consists of a family of parametric models having We introduce two types of smoothing constraints as prior hyperparameters that correspond to the weights of constraints. information of Bayesian modeling: the temporal constraint and To select a specific model from the family of parametric models, the spatial constraint. For the temporal constraint the we employ Akaike's Bayesian information criterion (ABIC) smoothness of the source time function is assumed as

4 27,382 IDE AND TAKEO: DETERMINATION OF CONSTITUTIVE RELATIONS Dlm=e 1, (5) where D1 is a N1 x M matrix which is the finite difference operator of first-order partial derivative of slip rate by time, and el is error vector. For the spatial constrainthe smoothness of the spatial distribution of total slip is introduced by a Laplacian finite difference operator; i.e., D2m=e 2, (6) m* = (GtG+oq2DltD1 +ot2d 2 2 t D 2 )-1 G td. (8) Superscript t indicates the transposition of the matrix. The appropriate value of al and a2 based on entropy maximization principle are objectively obtained by minimizing Akaike's Bayesian information criterion (ABIC) [Akaike, 1980]. The derivation of the representation of ABIC is described by Yabuki and Matsu'ura [1992] and Ide et al. [1996], and it is written in the present case as To determine a spatiotemporal distribution of fault slip, we initially arrange an assumed fault plane in the source region and determine a spatiotemporal slip distribution on the plane. The geometry of the fault plane is assumed based on the observed aftershock distribution (Figures 2b and 2c) [Hirata et al., 1996]. The assumed fault plane strikes along N232øE and has a dip angle of 85 ø in a northwest direction, being inclined only 5 ø away from vertical. The length and the width of the fault are 50 km and 20 km, which covers the whole source extent as shown by the analysis of Ide et al. [ 1996]. To expand the spatiotemporal distribution on the assumed fault using isosceles triangle functions, the knots of the function are arranged in the 3-D source volume, a 2-D fault plane and a time axis (Figure 3). The front view (Figure 3d) shows a spatial distribution of knots whose interval is set at 2.5 km, and the top (Figure 3b) and the side (Figure 3e) views show the locations of where D2 and e2 are N2 x M matrix and error vector, respectively. It should be noted that this smooths the total slip distribution and knots in the time direction with an interval of 0.6 s. To save does not affect the instantaneou smoothness of the spatial slip computational costs, knots are activated for each point on the distribution. As with e, error vector el and e2 are assumed as fault plane after an imaginary rupture front with a propagation Gaussian with zero mean and covariance cry21 and or5 2 I, respectively. velocity of 3.0 km/s reaches there. This rupture velocity limits Best estimates of model parameters in (4) with temporal (5) only the initiation time of rupture and the local rupture velocity and spatial (6) smoothing constraints are given as the solution of a between neighboring knots, and a proper choice of the expansion damped least squares problem written as, coefficients enables a faster or a slowe rupture than this velocity. The numbers of the basis functions in strike, dip, and time G d+e direction are 19, 8, and 11, respectively; the total number oqd 1 = e 1, (7) necessary to represent slip of a specific direction is then o 2D 2 e 2 Since the overall mechanism of this earthquake is almost pure right-lateral strike slip, two directions of slip on this fault plane where al = cr/crl and a2 = cr/or2 are hyperparameters which are assumed: the rake angles of 135 ø (right-lateral strike slip with represent the relative weights of the corresponding constraints. thrust slip component) and 225 ø (right-lateral strike slip with As such, the best estimates of model parameter m* are given for any combination of al and a2, using the method described by Jackson and Matsu'ura [ 1985], thereby obtaining normal slip component), and slip at each point is represented by the composition of these slip vectors. Since slip distribution in one direction is expanded by 1672 basis functions and they are used for each of two direction, the total number of basis functions and expansion coefficients is By constraining expansion coefficients to be positive in the following inversion, slip direction at any point on the fault plane is limited between 135 ø and 225 ø. The original data are strong-motion seismograms recorded by acceleration seismometers operated at 18 JMA stations located less than 150 km away from the hypocenter (Figure 2a). Eight stations provided seismograms from A-type units that have a flat frequency response from 0.01 to 10 Hz [Katsumata, 1989]. ABIC(oq2,0t22) = (N + N '+N, '-M) log s(m*) Each station furnished three components of acceleration data, - Nl'logoq 2 - N2'logot22 namely, up-down (U-D), north-south (N-S), and east-west (E-W) for a total of 24 components. The U-D component of Kobe + logllgtg + ZD,tD, (9) station (KOB) located I km northwest of the assumed fault plane was eliminated from the data set because this component suffers where N ' and N2' are the rank of DltD and D2tD2, respectively, the effect of fault mislocation and is almost independent of the and whole rupture process, while the horizontal components are well s(m) = lid-gmll = + 711DI = +illd=l =. C0) associated with the source process near the hypocenter as well as the fault area near KOB station. The other 10 stations provided In the calculation, the best estimate of model parameter m* is seismograms, each with the same acceleration component, from first obtained using particular values of c and a2 and (8). In new B-type units that have a flat frequency response from 0 to 8.2 parallel, equation (9) gives ABIC by numerical computation using Hz [Yoshida and Yokota, 1994]. Eliminating the seismograms resultant m*, a and a2. The values of a and a2 are that indicate some recording trouble, we used 22 of 30 subsequently varied, and this procedure is repeated until the components of acceleration. Therefore 45 seismograms were minimum ABIC value is found. After determining the set of oq processed into displacement data by integrating twice and then and or2 minimizing the ABIC, m* previously obtained with these band-pass filtered from to 0.5 Hz to remove low-frequency values tums into the optimum model parameter m* that constructs noise amplified by the integration. Use of a 0.2 s sampling the spatiotemporal distribution of fault slip. interval and a 50 s period extracted from each of the Preparation for Inversion displacement records provided 11,295 data points. Each data point was equally weighted in the inversion analysis so as to fit the larger data values better. The Green's function in (2) is calculated using a point source of impulsive slip rate for each slip direction at each calculation point arranged at I km interval on the assumed fault plane by the discrete wavenumber method of Bouchon [1981] in which the reflection-transmission matrices of Kennet and Kerry [ 1979] are

5 ß IDE AND TAKEO: DETERMINATION OF CONSTITUTIVE RELATIONS 27,383.' (a)... ',)'j,:.;..,_;;,,,:,,.[?.,.-;..-: _.. _.. ß..../'.-'L--'"' "'' '. '', 2', ',- '",.., ''' "-'"'.. '' ' - ß '-" ',, ß ',"_;,'5,';;";../] ß x :-.'.'" '' ' _ "" '" "'" ""-""' ''-* ' ','--. "" -. ' ',.,- _' "-?;,, '.;,',,;.,"....- I % '-'-''' ',, '"",.. ' ".. '. ', 1 ' '.,".., ';.,:..%/ (b) TOP [] n [] (c) Linear B-Spline o (d) E 50 km 7.2 s Figure 3. (a) The knot distribution of basis function in the present study in 3-D source volume spanned by the assumed fault plane and the time axis. (b) The distribution the plane shown in Figure 3d by a horizontal gray line. (c) The shape of basis functions with knots. (d) The front view of Figure 3a. Two horizontal and vertical lines show the projected plane of Figures 3b and 3e, respectively. (e) The distribution of knots on the plane shown in Figure 3d by a vertical gray line. used. The effect of anelasticity is also introduced by the use of complex velocities [Takeo, 1985]. The interval of 1 km is fine enough to be assumed a point source because the rupture propagation at this distance about s and comparable to the duration of the discretized version of a delta function at a sampling interval of 0.2 s. For the structure of seismic velocity in the analysis region of western Japan, a horizontal crustal structure is assumed based on the P wave velocity structure Table 1, Parameters of the Velocity Structure Used in the Inversion Analysis Layer Vt, Vs p h Qt, km/s km/s g/cm 3 km Q$ deduced from an explosion survey performed along a line passing right through Kobe and Awaji Island [Aoki and Muramatsu, 1973]. The S wave velocity structure is estimated from P wave velocities assuming a Poisson's ratio of 0.25, and p, Qp, and Qs are tailored to be appropriate for the velocity and depth of each layer (Table 1). This structure is the same as the one of Ide et al. [1996]. Even a small error in timing may generate a large error in the waveform analysis; therefore the time alignments are applied to all combinations of Green's functions and data. The P wave arrival times are aligned for the U-D component, while S wave arrival times were similarly aligned for the horizontal E-W and N-S component. After the same band-pass filtering as data and convolution with temporal basis function, the surface integral in (3) is performed numericalb at 1 km sampling interval Inversion Results To solve the damped least squares problem (7) with a positivity constraint the model parameters, we employed the nonnegative least squares (NNLS) algorithm of Lawson and

6 27,384 IDE AND TAKEO: DETERMINATION OF CONSTITUTIVE RELATIONS a b c A 0.0,,, 0.0,, 0.0,/,2., e f 0.0 cl 0... i'' i'i 0.01',, 0.5,., ' ' ,, O.Ot'... T"=., ,/"., 0.0,,,/',, Time {s) Figure 4. Total slip distribution of the Kobe earthquake and its source time functions at 10 selected points shown by a-j in the slip distribution. The source time functions show change of slip rate with 7.2 s time window after the arrival of an imaginary rupture front. Star represents the mainshock hypocenter. i j Hanson [1974]. After determining the appropriate value of hyperparameters a] and a2 by ABIC, the best estimates of model parameters m* are obtained. Figure 4 shows the obtained spatial distribution of total slip and the corresponding source time (slip rate) functions at 10 selected points on the fault plane. This solution has the same following characteristics as the previous kinematic models. The areas of large slip occur near the hypocenter and the shallow southwest part of the fault plane, where the maximum slip is 2 m. This large shallow slip corresponds to the surface offsets of about 1 m along the Nojima fault [Nakata and Yomogida, 1995; Awata et al., 1995]. The total seismic moment is calculated as 1.9 x 10 ]9 N m, comparable with the previous results. The shapes of the source time functions differ from place to place on the fault plane (Figure 4). The duration of each source time function, or risetime, at shallow depth (a-d) is about 5 s and its shape resembles a boxcar function, whereas the risetime is less than 3 s (f, i), and the shape resembles a triangle (e, h) at deeper depth. The long duration of the source time function at shallow depth has been reported by Ide et al.'[1996], Sekiguchi et al. [1996], and WaM [1996], though they could not separate the risetime at each point of the fault and the effect of rupture propagation within each subfault from the source time functions. In this study, these two effects can be considered separately using the expansion of slip distribution by basis functions if the data supply enough information2 However, the information does not seem to be sufficient to successfully separate them due to the limited frequency band, up to 0.5 Hz, of the data. The highest frequency of 0.5 Hz suggests an upper spatial resolution of 5-6 km, assuming the characteristic rupture velocity of km/s. As a consequence, the obtained slip distribution and source time I I I I I I I I I '1 0.0 Slip Velocity (m/s) 0.9 Figure 5. The history of rupture propagation of the Kobearthquake. The amplitude of slip rate at 1 s time intervals is shown. The last rectangle shows the locations of three distinct rupture phases discussed in the text.

7 IDE AND TAKEO' DETERMINATION OF CONSTITUTIVE RELATIONS 27,385 AIDA-U HIk-n AIDA-E HIK-E MINB-U OKA-E TOT-U AWAJ-U.: KOYA-U... MONO-E...,..O :.. TSU-U AWAJ-E KOYA-N MZH-U SAKA-U TSU-N HEGU-U KOYA-E MZH-N SAKA-E TSU-E HEGU-E KYOS-U MZH-E SHJ-U WACH-U HIK-U KYOS-N OKA-U SHJ-N WACH-E... \.,, ,,.... Observed...C..a..!..c...u!.a...t..e..d.. I14 cm I 50 s I Figure 6. Observed (solid lines) and calculatedisplacement (dotted lines) for the spatiotemporal distribution of slip. Station code and component are indicated on the top of each set of trace. functions are quite similar to the previous result obtained using 5 km x 5 km subfaults [Ide et al., 1996]. Figure 5 depicts the determined rupture history of this earthquake as slip rate distribution at each 1 s time interval. The rupture begins from the hypocenter and propagates mainly toward the surface in the northeast direction from 0 to 3 s (phase A) and has a relatively large thrust component. The rupture propagation in this direction soon weakens (5 s), whereas its southwesterly propagation begins at 3 s and continues progressing upward with an increasing slip velocity (phase B). Slip rate reaches a maximum (0.8 m/s) at the surface corresponding to the location of the Nojima fault (8 s). After northeastern rupture propagation of phase A weakens, the rupture to the northeast continues until 11 s after the origin of the mainshock (phase C). The rupture in phase C is characterized by highly irregular propagation with small total slip. The rupture in phase C is deeper than that in phase B, consistent with the fact that there is no surface fault rupture in the Kobe region. As summarized by Heaton [1990], many recent studies of earthquakes found the risetimes to be much smaller than total rupture duration. This is unexpected in the conventional dynamic crack model [Madariaga, 1976] in which slip continues at the center of the crack until stopping information reaches there from the crack tip. The view that emerges from these studies is that an earthquake consists of a slip pulse, a propagating band of slip, whose duration is not controlled by fault dimensions [Heaton, 1990]. Beroza and Mikumo [1996] explained slip pulse as a consequence of spatial heterogeneity of rupture. Figure 5 also shows such slip pulse propagation, prominent at 4 s and 9 s, which seems not to be controlled by overall fault dimension. Although the assumed propagating knots distribution of the basis function (Figure 3) inherently produces a slip pulse distribution, the source time functions determined by inversion indicate that the whole rupture is well constrained within the assumed time window, and the width of the obtained slip pulse (about 10 km) is smaller than those of the inherentime window (21.6 km, Figure 5). Therefore these slip pulses are not inherent and are well determined by the waveform inversion. Figure 6 compares the observed and calculated displacement records, in which the correlations between the two traces are quite good for nearly all components. The lack of agreement in the records of the OSA (Osaka) station is probably because the station is located on thick alluvial layers: a local geological structure that is not accounted for in the employed Green's functions. Nevertheless, the long-period components of these records are well explained by the model.

8 27,386 IDE AND TAKEO: DETERMINATION OF CONSTITUTIVE RELATIONS 3. Determination of Stress Distribution and Constitutive Relations 3.1. Method U x To determine the stress distribution on the fault plane from the spatiotemporal distribution obtained by inversion, we employed a finite difference method (FDM) [e.g., Mikumo et al., 1987] with the slip distribution as a boundary condition. The equations of motion, and Hooke's law, P// = r0,j (11) ij = uk,k$ij + #(Ui,j + Uj,i ) (12) are solved in 3-D elastic body surrounded by a fault plane, a free surface, and the four other rigid boundaries whose effects are negligible. The parameters ui and r0 represent the i component of the displacement vector and the ij component of the stress tensor, respectively, and j means partial derivative in j direction; p is density, and # are elasti coefficients, and ts/ is a Kronecker's delta. Since the initial values of the stress components are unknown, and only the difference from the initial state is important to solve (11) and (12), ui and rij are regarded as the differences from the initial values before the rupture initiation (t = 0). The x, y, and z axes are in strike direction, perpendicular to the fault plane, and vertical, respectively. For simplicity, a vertical fault plane is adopted in FDM calculation, although the slip distribution determined on a slightly dipping (dip angle 85 ø in northwest) fault plane. Combining (11) and (12) yields the following set of wave equations: ii x _ + la 32u + la V2u + 2Uy + +# 2,+# p 3x 2 p p 3xo3, p +_ _ A a2u, A + # &V 2 A + # a2u / Y-- /93XO'qY -I 2UY q- Uy q- z //z-- A+#32u, q- A+#3 2UY + A+#32uz +--V2u (13) p x& p 3x& p &2 p The spatial grid spacing in the FDM calculation is taken as 1.0 km in every direction, and the time increment is taken as 0.05 s. Although we tried calculation with half grid interval, the results did not change. The crustal structure is the same as the one used in the Green's function calculation (Table 1). Then, if every boundary condition is given by displacements, equations (13) are solved for each time step. Assuming symmetry across the fault plane (y - 0), the displacements in two directions, u, and uz, on the fault plane are given as half of the slip of the kinematic model. The faultnormal displacement uy is calculated by the continuities of faultnormal traction yy. In this case, the symmetry reduces this condition as 'vy =0, (14) which determines the value of uy on the fault plane. The confinuities of xy and y are easily confirmed from equation (12) using the fact that u, and uz are odd functions and uy is an even function. The boundary condition on the free surface is a traction free condition, i.e., z = y = z =0 z =0. (15) I:-: -.:.'--.-i : ':....:: i: :::i:::::i:i:i ':?i- i!!!...!!!.' -'"' i _i!iiii:.::i:!iiif: ::::::::::::::::::::::::::::!!i! :::.::i!i!i i Os 2:::....'-'-... ::::::::!:::.. ::!!!!sii!:!:!!i!i!!!iii-:-- ::':. =========================':i:!:e::.. ::!:i:i:i:!:i:i:i i:3::' ::: Figure 7. The histories of the slip, u,, and shear stress, %, in the strike direction. The history of u, is determined by the inversion analysis, and that of is calculated by FDM. In the FDM calculation, we used a vertical fault plane instead of the slightly inclined fault plane. We maintain (15) numerically by introducing the concept of fictitious displacements [Alterman and Karal, 1968; Mikumo et al., 1987] at the free surface. Without boundary condition on the other four boundaries, the calculation is carded out until the reflected waves from these boundaries arrive at the assumed fault plane. Once every displacement in the elastic medium is determined, the stress components other than 'ryy are calculated by equations (12) Stress History and Constitutive Relations Figure 7 shows the result of the FDM calculation described in section 3.1. Although three components of the slip vector and six components of the stress tensor are determined everywhere on the fault plane at every 1 s time interval, we show only shear stress, and slip u, since the rupture of this earthquake is dominated by right-lateral strike slip. The maximum stress drop in rxy is about 5 MPa near the hypocenter, not at the region of maximum slip. A small stress increase of less than 2 MPa appears outside of the ruptured area, much smaller than stress increase at the theoretical crack tip with the classical friction law

9 IDE AND TAKEO' DETERMINATION OF CONSTITUTIVE RELATIONS 27,387 1 i i i i i i i d i i I i i i i i i i i i a Slip (m) b '! 2 corresponds to the differences of the durations of source time functions, or risetime: longer risetime with small slip weakening rate and short risetime with large slip weakening rate. The slip weakening critical displacement Dc in the shallow part of the crust is about 1 m and tends to be larger than the one deeper in the crust, which is about 50 cm. One possible factor that may explain such difference is the assumed horizontally layered structure in which the rigidity of the shallowest layer (0-3 km) is lower than those of deeper layers. However, the difference in slip weakening rate is too large to be explained by low rigidity. As discussed in section 3.3, these values of Dc are limited by resolution of data. Spatial resolution is similar order to the slip pulse width, e.g., 10 km, which is comparable with the breakdown zone size of slip weakening crack with Dc -- 1 m, since Dr(breakdown zone size) is roughly equal to (breakdown stress drop)/(rigidity) [ Ohnaka, 1996]. The relation between slip rate and stress is sometimes used as a constitutive law in simulation of earthquakes [Carlson and i i i i i i i i I i i i i i i i i i illinmini lllnw! Slip (m) Figure 8. The constitutive relation between slip and stress on the fault plane. Each trace is the function calculated at the corresponding location in the middle figure of slip distribution Slip Rate (m/s) of static and dynamic friction levels. This may be because of the insufficient resolution due to the basis function expansion, the smoothing constraint in the inversion, and frequency band-limited data smeared out the fine structures of the stress buildup process. To estimate the maximum stress before the rupture, namely, strength at each point, finer knot spacing and more precise knowledge about crustal structure are necessary. The distribution of slip and stress can be used to give the constitutive relation at each grid point in space. Figure 8 shows the relations plotted at each time step up to 30 s after the rupture initiation at the points whose source time functions are shown in Figure 4. At almost every point, stress decreases with increasing slip (slip weakening) as seen in frictional experiments of rocks. Evidently, the slip weakening rate (du /d y) in the shallow parts (a-d) are significantly smaller than those in the deeper parts (e-j). At the shallow part, stress decreases 1-2 MPa during the slippage of about m, while stress decreases 2-4 MPa during the slippage of m at the deeper part. This difference 2 o.o i 'x e h o J5 Slip Rate (m/s) Figure 9. The relation between slip rate and stress. Each trace is the function calculated at the corresponding location in the middle figure of slip distribution.

10 27,388 IDE AND TAKEO: DETERMINATION OF CONSTITUTIVE RELATIONS Langer, 1989; Cochard and Madariaga, 1994]. Heaton [ 1990] suggested that the velocity weakening constitutive law is important in the slip decelerating process to explain the slip pulse-like propagation of rupture that is common in many earthquakes. This is demonstrated by the simulation of dynamic rupture in a 2-D crack with velocity weakening constitutive law [Cochard and Madariaga, 1994] or modified rate- and statedependent constitutive laws [Perrin et al., 1995; Beeler and Tullis, 1996]. The slip pulse is observed in Fig. 5, and then we show how stress depends on slip rate in the same manner as the slip-stress relations (Figure 9). Although the velocity weakening in the slip accelerating process is visible this is trivial and easily predicted by slip weakening behavior. On the other hand, velocity weakening behavior is unclear in the slip decelerating process. However, it is difficult to determine the slip at low rate by waveform inversion. This result alone cannot prove that stress is independent of slip rate. It is difficult to explain slip pulse shown in Figure 5 with rate-dependent constitutive relation, and the explanation with spatial heterogeneity of rupture [Beroza and Mikumo, 1996] is valid in this case Resolution Analysis Since our analysis adopts expansion by basis functions, bandpass filtering of the data and Green's functions, and smoothing constraints in the inversion, the result is affected by these smoothing factors. To examine the effect of these factors, we perform a numerical simulation whose procedure is illustrated by Figure 10. First, a dynamic rupture of a 3-D crack is calculated by the method of Mikumo et al. [1987], with a classical friction law of static and dynamic friction, namely instantaneou stress drop (Figure 10b). The shape of the crack area is assumed to be similar to the ruptured area of the Kobe earthquake, and the rupture velocity and the stress drop are 2.5 km/s and 5 MPa, everywhere in the crack area (Figure 10a). Seismic waves excited by this source are calculated using Green' s functions used d c Slip (m) a b c d e (a) "20 km : Slip (m) Figure 11. The constitutive relation between slip and stress on the fault plane as the result of the simulation. Each trace is the function calculated at the corresponding location in the middle figure of slip distribution.! 50 km Initial Stress- u3_ I' Dynamic Friction T SLIP MPa Figure 10. Schematic illustrating (a) the dynamic model of crack propagation with (b) a friction law as a constitutive relation. in the actual inversion, and we constructarget synthetic seismograms by using the band-pass filter between the frequency range from to 0.5 Hz, the same as the actual analysis. These target seismograms are then inverted as data, and the spatiotemporal distributions of slip are obtained. The basis functions are the same as the previous inversion, and the proper weights of smoothing constraints are estimated by ABIC. FDM calculation followed using this kinematic model to obtain the constitutive relations. Figure 11 represents the constitutive laws introduced in the crack model and the calculated ones at five points on the crack surface. Although the original constitutive law should impose prompt reduction of stress as slip reaches zero, the FDM calculation produces artificial slip weakening relations due to finiteness of the time increment and spatial spacing. The calculated constitutive relations have a smaller slip weakening rate. This is mainly due to the bandlimited data, up to 0.5 Hz, and the smoothing constraints, and the

11 IDE AND TAKEO: DETERMINATION OF CONSTITUTIVE RELATIONS 27,389 effect of expansion by basis function is smaller because its width is sufficiently fine in this frequency range. Unlike the actual data analysis (Figure 8), these relations show little depth dependency and the slip weakening rates are almost the same at every depth. The slip weakening rate is about 5 MPa/m which represents the resolution limit inherent in our analysis and is similar to those in the deeper part of the previous results (Figure 8, e-j). Therefore it is quite questionable whether the constitutive relations on the deeper part of the fault plane are realistic or not. The estimated values of Dc of about 50 cm from these relations has no significance other than an upper limit of Dc. On the other hand, the slip weakening rate in the shallow crust is smaller than this limit, and it is plausible that the relations on the shallower part is well determined by our method. The slip weakening rate may be small or the critical displacement Dc is large in the shallow crust. 4. Discussion 4.1. Constitutive Relations in Shallow Crust The constitutive relations in the shallow parts are different from those in the deeper parts, which must be associated with some characteristic phenomena in the shallower part of the crust. A well known such phenomenon is paucity of very shallow seismicity. The aftershock seismicity of the Kobe earthquake in the shallowest (0-3 km) region was quite low along the Nojima fault (Figure 2c), where the total slip amount is the largest. It is generally known that seismicity is usually low in the shallow part of faults with well-developed gouge [e.g., Marone and Scholz, 1988]. Some mechanisms are proposed to explain the shallow paucity of seismicity. Based on laboratory experimental friction studies indicating that thick layers of simulated gouge exhibit a positive slip rate dependence of frictional resistance (velocity strengthening) at slow slip velocities of about grn/s [e.g., Marone et al., 1990], Marone and Scholz [1988] attributed the paucity of shallow seismicity to velocity strengthening behavior of unconsolidated materials between fault planes. Velocity strengthening behavior may stabilize fault slip and prevent earthquake instability, and moreover this mechanism may play key roles for afterslip occurrence and creep phenomenas pointed out by Marone et al. [1991]. On the other hand, Scholz [1988] suggested that a large critical distance Dc stabilizes slip in the shallow region of the fault. He related Dc to the minimum size of the contact junctions of geometrically unmated fractal surfaces under a normal stress, and showed only large-scale junctions are built by relatively low normal stress in the shallow region. Therefore Dc becomes large enough to prevent the seismic nucleation process in the shallow crust. Marone and Kilgore [1993] presented another interpretation of large Dc, in which Dc is controlled by the thickness of gouge zones that may be thick in the shallow region. In the present study, no velocity-strength constitutive relation appears, while Dc is large (about 1 m) in the shallow region. This supports the possibility of large Dc as a mechanism related to the paucity of shallow seismicity rather than the velocitystrengthening behavior. Our model supplies no information about the physical interpretation of large Dc, and there are other mechanisms accountable for paucity of shallow seismicity; combined brittle-creep process is suggested by Ben-Zion [ 1996] and the effects of fluid must be considered. Whether Dc is generally responsible for seismicity should be studied for many other fault systems Other Earthquakes The analysis of the Kobe earthquake shows that small slip weakening rate in the shallow crust corresponds to a long risetime. Since a small slip weakening rate is generally associated with a long risetime, the depth dependency of constitutive relations can be roughly estimated by the depth dependency of risetimes determined by waveform inversions for other earthquakes. An earthquake that has large surface offset and was analyzed in detail by near-field strong-motion records is quite rare and the only example other than the Kobe earthquake is the 1992 Landers, California, earthquake [Wald and Heaton, 1994; Cohee and Beroza, 1994; Cotton and Campillo, 1995]. For this earthquake, the risetime in the shallow crust is reported to be longer than in the deeper parts [Wald and Heaton, 1994]; hence it is probable that the Landers earthquake was characterized by the same dependency of the stress decrease ate as in the Kobe earthquake. However, as Cotton and Campillo [1995] questioned this long risetime in the shallow crust because of the lack of resolution, more study is necessary to assure this conclusion. The 1979 Imperial Valley earthquake is similar to the Kobe earthquake and the Landers earthquake in that it is a strike slip event in the shallow crust. Unlike the Kobe and the Landers earthquakes, this earthquake has small slip in the shallow part of the assumed fault plane, and the depth dependency of the risetime is not clear [Archuleta, 1984]. Quin [1990] obtained a negative stress drop, namely, stress increase, during slip in the shallow crust by constructing a dynamic crack model with a classical friction law based on the kinematic model of Archuleta [1984]. Since Quin did not determine constitutive relations but assumed a constitutive law of instantaneou stress drop, it should be discussed if this negative stress drop is necessary even in slip weakening model. In any case, this implies that the constitutive relations in the shallow crust are different from those in the deeper crust for the Imperial Valley earthquake. Existence of unconsolidated material along the fault and the low normal stress in the shallow crust also support the possibility that the constitutive relations are different in the shallow crust. Although the difference of constitutive relations at depth may be found in other earthquakes that produced surface fault offsets, further studies on many other actual earthquakes are necessary to conclude how it differs. 5. Conclusion The main purpose of this study is to determine constitutive relations of an actual earthquake, the 1995 Kobe earthquake. This is the first attempt in which shear stress on the fault plane is calculated by solving elastodynamic equations with boundary conditions of fault slip distribution based on inversion of nearfield seismic waves. Bayesian modeling with ABIC ensures objectivity of the solution of the inversion with temporal and spatial constraints. The resultant spatiotemporal distribution has many similarities with previously presented kinematic model of the Kobe earthquake [Ide et al., 1996; Sekiguchi et al., 1996; Horikawa et al., 1996; Yoshida et al., 1996; Wald, 1996]. Computation by finite difference method (FDM) yields all components of the displacement vector and the stress tensor everywhere in the elastic medium surrounding the fault plane, and on the fault plane, too. Constitutive relations are then obtained as the relation between the shear stress and the slip in the strike direction on the fault plane, by connecting them at each time step of FDM calculation. At almost all points having large slip, the relations show slip weakening behavior, whose rates are smaller

12 27,390 IDE AND TAKEO: DETERMINATION OF CONSTITUTIVE RELATIONS in the shallow region than those deeper. A simulation with a Andrews, D. J., Rupture propagation with finite stress in antiplane strain, crack model to verify the resolution of our method concludes that J. Geophys. Res., 8I, , Andrews, D. J., Dynamic plane-strain shearupture with a slip weakening the resolution is not high enough in the deep region of the fault friction law calculated by a boundary integral method, Bull. SeismoI. plane and the obtained slip weakening rates should be regarded as Soc. Am., 75, 1-21, lower limits. The slip weakening characteristic distance Dc is Aoki, H., and I. Muramatsu, Crustal structure in the profile across Kinki estimated to be about 1 m or more in the shallow region, while the and Shikoku, Japan, as derived from Miboro and the Toyama explosions (in Japanese), Zishin 2, 27, , upper limit of Dc is 50 cm and can be much smaller in the deeper Archuleta, R. J., A faulting model for the 1979 Imperial Valley part. earthquake, J. Geophys. Res., 89, , The most serious problem in the study of an actual earthquake using observed seismic waves is that the resolution is insufficient Awata, Y., K. Mizuno, Y. Sugiyama, K. Shimokawa, R. Imura, and K. Kimura, Surface faults associated with the Hyogoken-nanbu to reveal fine structures of the earthquake source. The lack of earthquake of 1995 (in Japanese), Chishitsu News, 486, 16-20, Barenblatt, G.I., The formation of equilibrium cracks during brittle knowledge about crustal structure is significant in this problem. fracture: general ideas and hypotheses, J. AppI. Math. Mech., 23, 622- However, even when accurate crustal structure is known and 636, Green's functions are calculated precisely for every source to Barka, A. A., and K. Kadinsky-Cade, Strike slip fault geometry in Turkey receiver combination, another difficulty arises from the basic and its influence on earthquake activity, Tectonics, 7, , Beeler, N.M., and T. E. Tullis, Self-healing slip pulses in dynamic assumption that the constitutive relations are defined on a fault rupture models due to velocity-dependent strength, Bull. SeismoI. Soc. plane without thickness. As Barka and Kadinsky-Cade [1988] Am., 86, , reported for earthquakes in Turkey, the ruptures of earthquakes do Ben-Zion, Y., Stress, slip, and earthquakes in models of complex singlenot propagate past stepovers that are wider than 5 km, which fault systems incorporating brittle and creep deformations, J. Geophys. Res., IOI, , roughly limits the wavelength that can egard a fault zone as a Beroza, G. C., and T. Mikumo, short slip duration in dynamic rupture in plane without thickness in the analysis of large earthquakes. The the presence of heterogeneous fault properties, J. Geophys. Res., 101, resolution of my analysis is of this order, and more detailed , analysis by the method of this study may be impossible with the Bouchon, M., A simple method to calculate Green's functions for elastic exception of smaller earthquakes and rare large earthquakes layered media, Bull. SeismoI. Soc. Am., 7I, , Carlson, J. M., and J. S. Langer, Mechanical model of an earthquake fault, whose fault system can be successfully represented by a plane. Phys. Rev. A, 40, , Nevertheless, it is still important to estimate constitutive Cochard, A., and R. Madariaga, Dynamic friction under rate-dependent relations even with poor resolution, as long as the difference of friction, Pure Appl. Geophys., 142, , the relations can be detected as in this study. As example, small Cohee, B. P., and G. C. Beroza, Slip distribution of the 1992 Landers slip weakening rate may exist in many earthquakes in the shallow earthquake and its implications for earthquake source mechanics, Bull. SeisrnoI. Soc. Am., 84, , region of the fault plane, since earthquake damage is sometimes Cotton, F., and M. Campillo, Frequency domain inversion of strong anomalously low along the surface ruptured fault trace when large motions: Application to the 1992 Landers earthquake, J. Geophys. earthquake occurs, which cannot be explained if shallow slip obeys a classical friction law of static and dynamic friction. This Res., lo0, , Day, S. M., Three-dimensional finite difference simulation of fault implies that the constitutive relations in the shallow crust have dynamics: Rectangular faults with fixed rupture velocity, Bull. SeisrnoI. Soc. Am., 72, , great significance to strong-motion displacement excited near Dieterich, J. H., Modeling of rock friction, 1, Experimental results and fault planes in practical use. As another-example, small slip constitutivequations, J. Geophys. Res., 84, , weakening rate may be a candidate to explain slipping behavior of Dieterich, J. H., Earthquake nucleation on faults with rate- and stateso-called slow earthquake having a long risetime, expected from dependent strength, Tectonophysics, 2 I I, , Fukuyama, E., and T. Mikumo, Dynamic rupture analysis: inversion for the negative correlation between risetime and slip weakening rate. the source process of the 1990 Izu-Oshima, Japan, earthquake (M - If analysis with high resolution is possible, more discussion is 6.5), J. Geophys. Res., 98, , possible about constitutive relations in many aspects of the Hartzell, S. H., and T. H. Heaton, Inversion of strong ground motion and earthquake source. By accumulation of study about constitutive teleseismic waveform data for the fault rupture history of the 1979 Imperial Valley, California, earthquake, Bull. Seisrnol. Soc. Am., 73, relations of actual earthquakes, the complexity and diversity of , the earthquake source will be understood with the various Heaton, T. H., Evidence for and implications of self-healing pulses of slip constitutive relations and the physics behind them. in earthquake rupture, Phys. Earth Planet. Inter., 64, 1-29, Hirata, N., et al., Urgent joint observation of aftershocks of the 1995 Hyogo-ken nanbu earthquake, J. Phys. Earth, 44, , Acknowledgments. We thank M. Matsu'ura and T. Miyatake for Horikawa, H., K. Hirahara, Y. Umeda, M. Hashimoto, and F. Kusano, their useful suggestions on the fundamentals of this work. Discussion Simultaneous inversion of geodetic and strong motion data for the with T. Mikumo and Y. Ben-Zion were helpful to us. We also would like to thank R. Madariaga, A. Cochard, and G. Beroza for their detailed source process of the Hyogo-ken Nanbu, Japan, earthquake, J. Phys. Earth, 44, l, reviews. The English in the text was improved by the comments of T. Ida, Y., Cohesive force across the tip of a longitudinal-shear crack and Tada, J. M. Johnson, and A. Odedra. 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