Effects of Partial Relaxation of the
Effective Stress upon Seismic Radiation
著者
Sato Tamao, Hirasawa Tomowo
雑誌名
Science reports of the Tohoku University. Ser.
5, Geophysics
巻
22
号
3-4
ページ
153-165
発行年
1975-04
URL
http://hdl.handle.net/10097/44726
Sci. Rep. TOhoku Univ., Ser. 5, Geophysics, Vol. 22, Nos. 3-4, pp. 153-165, 1975.
Effects of Partial Relaxation of the Effective Stress
upon Seismic Radiation
TAMAO SATO and Tomowo HIRASAWA Geophysical Institute, Faculty of Science
TOhoku University, Sendai, Japan
(Received January 11, 1975)
Abstract: To study the effects of a change in sliding friction during slip, an earthquake source model is derived by assuming that an increase in sliding
tion with time causes a partial relaxation of the effective stress. Our model is similar
to Brune's in the sense that an initial rupture is followed and superimposed by a
secondary one accompanying a reverse slippage with a proper time lag. For each
individual rupture we specify the dislocation function for the crack model, which
approximately satisfies the condition that the shear stress on the slip plane is free or
constant during the growth of rupture. Together with an assumption of similarity
condition, a constant ratio (E-0.1) of final stress drop to the effective stress and a
constant delayed-time (td=0.16 sec) irrespective of earthquake size give a scaling law
of seismic spectra similar to that of the Aki's revised model B which well explains
various observational results. From this, the slip process is inferred as follows: The
sliding friction starts to increase after a short time interval of 0.16 sec, causing a
sudden decrease in slip velocity, and thereafter the slip continues with a relatively
small slip velocity until the rupture front stops its propagation.
1. Introduction
In recent studies on source mechanisms, attempts have been made to eliminate the arbitrariness in specifying the relative slip as a function of time and space on the fault plane from physical viewpoints. Burridge (1969) described an idealized dynamical model for which plausible relative motions might be obtained by solving boundary-value problems. A constant drop of shear stress across the fault plane was assumed to be the boundary condition, which is valid when the fault plane is a free surface or when the relative motion is resisted by the sliding friction invariant with time and space. While he treated the problem of expanding cracks, whose area varies with time, Ida and Aki (1972) considered the case of uniformly propagating cracks.
In addition to these mathematically rigorous approaches, Brune (1970) introduced an intuitive approach to the boundary-value problem to clarify the basic nature of the relative motion on the fault plane. His model corresponds to the case where a constant drop of shear stress simultaneously takes place across the finite fault plane in the prestressed medium. The intuitive method adopted by Brune seems to be oversimplified, and yet its simple result has the advantage of practical use; we can roughly estimate from seismic observations the effective stress defined as the difference between the initial tectonic stress and sliding friction at the initial stage of slip (Brune, 1970; Trifunac, 1972). The idea of the effective stress is different from that of the
154 T. SATO and T. HIRASAWA
stress drop. If the sliding friction changes with time during slippage , it may sometimes occur that the final stress drop is not equal to the effective stress.
More recently, another attempt was made by Sato and Hirasawa (1973) to eliminate the arbitrariness in specifying the relative slip. They specified a dislocation function on the fault by making use of the static solution of cracks so that the resultant relative slip might represent a first approximation to the dynamical process of stress relaxation. In the sequal, we refer to the source model as 'crack model', because it was derived by use of the static solution of cracks. In the crack model the effective stress and rupture velocity are the basic parameters that characterize the slip process. They considered only the case where the final stress drop was equal to the effective stress.
For most large earthquakes, the effective stress is of the order of 100 bars when estimated from seismic observations at near-field by using the Brune's model (Brune, 1970; Trifunac, 1972; Hanks, 1974). Similar estimates are given by use of the crack model. However, these estimated values of the effective stress are very large as compared with the stress drop of the order of 10 bars observed for the same earthquakes. Only a small fraction of the effective stress appears to be relaxed and this indicates the complexity in releasing the tectonic strain energy.
The purpose of the present study is to investigate the effects of partial relaxation of the effective stress upon seismic radiation. Our method is similar to that taken by Brune (1970), who modeled the effect by superimposing a reverse slippage at a time shortly after the start of slippage governed by the initial rupture. Our approach is different from the Brune's in that the time lag between the initial and secondary ruptures is not apriori assumed to be proportional to the length of fault but determined with reference to the Aki's (1972) revised scaling law of seismic spectra which is based on various observational results.
2. Crack Model
In this section, a brief review is made of the crack model proposed by Sato and Hirasawa (1973), since the source model is frequently referred to in the subsequent
sections.
With such problems as considered by Burridge (1969) in mind, they offered an earthquake source-model in which the rupture originates at a point and expands circularly at a constant speed until it finally stops, accompanying a dislocation motion specified by making use of the static solution of cracks. If a2h62,--__3 is assumed, the dislocation function D (p, t) of time and space was given by
D(p,t) = K f(v02.—p2)112 plv) fl-11(p—L)); vt <L , (1) D(p,t) = K fL2—p2)112 (I —H(p---L)); vt > L . where, K-= (24/7n-) (4oly), du; stress drop,
PARTIAL RELAXATION OF THE EFFECTIVE STRESS 155
tt; shear modulus,
L; radius of circular fault,
p; distance of a point on the fault from its center, v; rupture velocity,
H(t); the Heaviside unit step function.
Fig. 1 shows the slip process expressed by Eq. (1), indicating that the center of the fault slips for a longer time than the edge and that the final relative displacement near the center is larger than that near the edge. The particle velocity at the center of the fault is given by /4= (1/2)Kv—(1/2)(24/77c) • (da fp,)v (L,_, (1 /2) (40./y)v.
The solution of the circular self-similar crack has the same functional form as the dislocation function for vt<L in Eq. (1). According to Dahlen (1974), the constant K is proportional to (doly) through a relation of the form K--A(v1,8)(dal,u), where A(v113) is a function of vh3 and ,8 the shear wave velocity. In the limit as vh8 tends to zero, the dislocation function with K= (24177) (401,a) is actually the exact solution of the circular self-similar crack. In the other limiting case v=f3, the numerical evaluation shows K=--(8/9)(4al,u).
It should be noted that for the crack model the termination of the rupture front is sudden and abrupt. This assumption would not always be reasonable. The
Fig. _1 -N Q 1.0 0.8 0.6 0.4 0.2 0.0 ^ 0 .0 1.0
(V/ L) t
1. Dimensionless displacement time functions (solid curves) at different points (p/L= 0.0, 0.2, 0.4, 0.6, 0.8) on the fault plane. The scale of each time-function is given at the right-hand side. The dashed line indicates the arrival time of rupture front with respect
to piL for which the ordinate is given at the left-hand side.
I/I /7 , /,, , / / I/ 1 /e ,--," ^• ... _... .__I -.. .^ Y .--..." 1,
156 T. SATO and T. HIRASAWA
termination may sometimes be smooth. It would seem that whether the termination is abrupt or smooth depends on the tectonic structure and the distribution of the in situ stress in the source region.
Factors for the geometrical spreading and radiation patterns being omitted, the far-field spectrum of body waves radiated from the crack model is given by
B
c(co)
L2k(1
k2)[k
cos
(co
Lk)
cosCOL
+ sin
(co
Lk)
sinCOL
•
c 1
k2)
((1+k2)
sin
(coLk)
cos
coL
—2k
cos
(coLk)
sincoL)1
(2)
+2 co
L2k(1
—
k2)[sin
(co
Lk)
cos
CO
L
—
k cos
(cu
Lk)
sin
to
L
•
to L(1— 1
k2)(2k—(1+0)
sin
(co
Lk)
sinCOL—2k
cos
(co
Lk)
cos
coLli
,
co = (1,1v) co, k = (v I c) sin 0 ,
where co is the angular frequency, 0 is the angle measured from the normal of the fault plane, and c implies either a or (3 which is the seismic velocity of P or S wave.
103 to 11 1-] 2 10° cr _ io' a_ cr) icf 0 102 30° V/13 = 0 .5 Fig. 2. WAVE WAVE 10° 01a, D. H a_ <io° H Cl_ (f) 0 L1-110° O 10-2 30° V/i3 = 0 .8 50° 90° WAVE 1.0 wL/V10.0 1.0cuL/V10.0 Typical examples of far-field spectra for the case vh3-0.5, 0.8.
PARTIAL RFLAXATION OF THE EFFECTIVE STRESS 157
The amplitude spectrum IB,(w)I is flat at low frequencies with the height proportional to the seismic moment and decreases proportionally to w-2 beyond the corner frequency as shown in Fig. 2. It is important to note that the corner frequency of P waves is higher than that of S waves at every observation point; the ratio of P to S wave corner frequency is about 1.3 on the average. This result is attributed entirely to the slip characteristics on the fault, that is, the center of the fault slips for a longer time than the edge and the amount of final relative slip near the edge is much smaller than that near the center.
The far-field source spectrum decays as co--2 beyond corner frequency for the crack model, whereas co-3-decay should be expected at very high frequencies for a model of equidimensional rupture propagation with a ramp time function. We ascribe this co-2-decay to the existence of the singularity of D(p, t) at the rupture front. From physical viewpoint the singularity of D(p, t) should be eliminated. However, the spectral decay of co-2 would not be affected if the singularity is eliminated by modifying D(p, t) within a sufficiently small region compared with wave-lengths under consideration.
3. Partial Relaxation of the Effective Stress
From a number of previous studies, it seems that most earthquakes do little to relieve the accumulation of the tectonic strain which gives rise to them. For example, Healey et al. (1968) estimated a lower limit for the in situ stress in the source region of Denver earthquakes from the rate of fluid injection and the fluid pressure. They obtained a value of 203 bars at the time of the initiation of faulting. Wyss and Molnar (1972) showed that the Denver earthqueks relieved only a small fraction of the available tectonic strain. A similar effect is observed also in laboratory experiment and called `stick slip' (B
race and Byerlee, 1966).
Although we have the knowledge that most earthquakes appear to relieve a small fraction of the available tectonic strain, it is difficult to picture the slip processes on faults without an additional information about the sliding friction operative across the fault plane. In order to put the problem into mathematical setting, Burridge (1969) assumed an idealized dynamical process in which the sliding friction is independent of the slip velocity and takes a constant value smaller than the static friction. The crack model in Section 2 was derived from a similar idealization.
Since the time when Brune (1970) introduced a simple model that enabled us to estimate the effective stress defined as the stress difference between the initial tectonic stress and the sliding friction at the initial stage of slip, it has been clarified that the final stress drop differs from the effective stress and the time history of sliding friction has been studied in more detail. In view of the Brune's model, the particle velocity at the initial stage of slip is given by the equation it where a is the effective stress, kt the shear modulus, and ,8 the shear wave velocity in the source region. The crack model gives a similar relation it,=(1/2)(24/7n) (cf/p)v, where /.4 is the particle velocity at the center of fault. Putting ,u=3 x 1011 dyne/cm2, and 13 r,2(24/77r)z, km/sec, we find approximate relations u (cm/sec)---cr (bars) for the Brune's model
158 T. SATO and T. HIRASAWA
and is (cm/sec) = (1/2) a (bars) for the crack model, respectively.
From seismic observations at near-field Brune (1970) considered that 100 cm/sec is a reasonable value for the upper limit of the particle velocity at the initial stage of slip process. If it is admissible to infer that the initial particle velocity for most large earthquakes ranges from 50 to 100 cm/sec, the effective stress for these earthquakes is roughly estimated to be of the order of 100 bars through the relations mentioned above. This estimate of the effective stress should be compared with the stress drop of the order of 10 bars observed for the same earthquakes. Only a small fraction of the effective stress seems to be relaxed and this suggests a likely change in sliding friction with time or even more complicated effects. Brune (1970) modeled the effect by applying a reverse stress 0- (1—e) at a time td shortly after the effective stress a is applied. Here e is the ratio of final stress drop to the effective stress. As a result the long-period spectra and seismic moment are reduced to 6 times the values for the complete drop of effective stress. However, the very high frequency spectra are much less affected. The time history of sliding friction considered by Brune is shown in Fig. 3. td -4- A b u.) €(0O- oi CC
I,
1 k td qE(0- 0-0
T
cr,i
cr f TIME, tFig. 3. Approximate time history of average stress on the fault plane, reproduced from Trifunac (1972). 0-0 and 0-1 are the stresses before and after the earthquake, respectively. o- is the sliding friction at the initial stage of slip.
To take into account the complexity in releasing the tectonic strain, we model the effect of partial relaxation of the effective stress in a way similar to Brune's. First, an initial rupture nucleates at a point and starts to propagate accompanying a dislocation motion specified by Eq. (1). At a time td shortly after the initial rupture started, a reverse slippage of, relative magnitude (1-E) occurs and propagates following the initial rupture until it finally stops at the same time as the initial rupture. If the dislocation motion Dr (p, t) of the reverse slippage is assumed to have the same func-tional form as that of the initial rupture, we have
D f(p,t) — (1 --E)K {v2(t02_ p2)112 H(t—t d ___ pie H ) (L(1 —Td) p), vtd < vt < L (3)
PARTIAL RFLAXATION OF THE EFFECTIVE STRESS 159
Dr(p, t) — (1 ___E)K (L2(1 _ Td)2 —p2, 1/2 H (L (1 —T d) — p) ; vt > L ,
where Td=td•v1L. The total slip motion 40, can be obtained by superimposing the initial and secondary ruptures, i.e.,
D toe(p, t) D(p,t)+Dr(p,t) • (4) Since the secondary rupture originates at a time td shortly after the initial one and stops at the same time as the initial rupture, the rupture front of the reverse slippage reaches to a radius of L(1 — T d) which is always smaller than the fault radius L of the initial rupture.
The modification of Eq. (2) gives the far-field spectrum of body waves radiated from the seismic source described by Eq. (4) as
1 3c' (w) = [U (w) —(1-0(1— T 3 (cos (CO LT d)U-' (W) + sin (coLTd) V '
(5) + i [V(w) —(l (1—Ta)3 [cos (COL Td) (t0) — sin (wLT d)U' ,
where U(w) and V (co) are the real and imaginary parts of B (w), respectively, and U' (w) and V' (w) are derived from U(w) and V (w) by replacing wL in U(w) and V (w) with w L(1-T a). We investigate interference patterns for the amplitude spectra 1B c' (co) by drawing spectral curves for several values of e and Td. In the following, we give v 3
=0.7 and fix observation point 0 at 60° where the spectrum shows a fault radius versus corner frequency relation nearly equal to that averaged over ( 3. In the limiting case where w tends to zero, Eq. (5) becomes
B c' (w) — (1 — (1 —e) (1 —T d)3) (U (w) iV (w)) .
Hence the long period spectra and seismic moment are reduced to 1--(1-e)(1-T 03 times the values for 100% drop of the initial effective stress. In case Td tends to zero, seismic moment is nearly e times the value for the complete relaxation of the effective stress as is the case considered by Brune.
We find three distinct types of amplitude decay depending on the value of Td in the frequency range higher than corner frequency f0. For example, we show interference patterns of I Bfi'((0)1 for the case 6=0.1. (i) For Td<0.04 the spectrum falls off as (w-2, co—', (0-2), that is, the spectral amplitude decreases first as w-2, second as co-4, and finally as co-2, with increasing frequency. (ii) For 0.04<Td<0.9 it decreases as (w-i, w-2). For Td>0.9 the spectrum demonstrates a simple co-2-decay. The spectral pattern for each case is shown in Fig. 4. The above ranges of Td shift accord-ing as the value of e changes. As 6 increases, however, these three types of spectral patterns become obscured and the spectrum decays simply as co-2 independently of the value of Td. The relation between the fault radius L and the corner frequency is much less affected by the partial relaxation of the effective stress as shown in Fig. 4. The ratio of P to S wave corner frequency is about 1.3 on the average.
In order to see how E and Td characterize the slip motion on faults, Dio,(t) at the center of fault is illustrated in Fig. 5. The smaller the value e becomes, the greater the change in slip velocity. The parameter Td gives the ratio of the time
160 T. SATO and o° 10° 01-6 D 1— 0_ 10° < _I < CC 10°1--0 LLI Cl_ U) (:) I-1-1 1 01NI =J < 2 Er 0 102 z T• HIRASAWA
Al°
r
( € = T O. d=00 €.0.1 Td =0.0E Td=0.95 10010 I w L/vFig. 4. Interference patterns in amplitude spectra caused by partial relaxation of
the effective stress.
interval of slip governed only by the initial ru motion. It is noteworthy that the characteri: obtained for a very small value of Td. This si larger earthquakes in the Aki's (1972) revised
4. Scaling Law of Seismic Spectra
Recently, Aki (1972) re-examined the val newly available data. He found that the ne co-square model for earthquakes with surface longer than 10 sec. For smaller earthquakes showed that the new observations undoubtedly model. In particular, the co-square model relation (Basham, 1969; Liebermann and Pome which follows the linear extrapolation of the
x ntb —3.97).
Since the co-square model explains most ( longer than 10 sec, Aki left it unchanged for period range shorter than 10 sec so as to sati
1 J Y •-^
g Td
De. Tv (i-Td) Td (WU tplitude Fig. 5. Dislocation motion at the tion of center of circular fault.
initial rupture to the total duration time of slip characteristic spectral decay of (o)-2, co 1, co-2) is This spectral shape is quite similar to that for 2) revised. model B.
tra
:d the validity of the co-square model by using at the new evidences, in general, supported the -1 surface wave magnitude Ms>6, or for periods .rthquakes or for shorter periods, however, Aki doubtedly required some revision of the co-square e model fails to explain the obsevred Ms-m1 and Pomeroy, 1969, 1970; Evernden et al., 1971) Dn of the Gutenberg-Richter formula (Ms=1.59
ins most observational results for a period range tnged for this period range and modified it for a as to satisfy the observed Ms—mb relation. He
PARTIAL RELAXATION OF THE EFFECTIVE STRESS 161
considered two extreme cases. In one revision (model A), he kept the spectral decay of co-2, but rejected the similarity assumption by changing the relation between M
, and f0. In the other (model B), he abandoned co-2-decay and adopted co-1-decay in the period range from 0.01 to 10 sec without modifying the M 5-f, relation for the co-square model.
Both models are capable of explaining most observational results: (1) fault length, (2) stress drop , (3) seismic moment, (4) spectral ratios for periods longer than 10 sec, and (5) M ,-mb relation. Nevertheless , there exists a clear difference between the two models with regard to the M 5-f, relation. The M 5-f, relation for the model B is expressed by the same curve as that for the co-square model . On the other hand, the corner period for the model A deviates from the curve for smaller earthquakes with M,<6 (3/4); the corner period has a nearly constant value of 6 sec for 4 (1/2) >M 5>3 and bends sharply down to the period range from 0.1 to 0.01 sec for M,<3. This bend in the magnitude-corner period relation is demonstrated in the data reported by Terashima (1968). However, it is still open to question whether the apparent bend in the magnitude-corner period curve is due to the source effect or to some other effects such as a gap in response of recording instrument and a difference in attenuation and scattering between short- and long-periods. More recently, Thacher and Hanks (1973) reported the observed relation between f, and seismic moment M0 for moderate earthquakes with 2<ML<7. The observed MS, relation does not show a sharp bend expected for the revised model A but agrees with the predicted curve for the co-square model or the revised model B, though the data are largely scattered.
Although available data are not sufficient for a unique revision of the co-square model, the revised model B is taken as a working model in this paper and its implica-tions for the slip process on faults will be discussed.
To study the effect of partial relaxation of the effective stress upon seismic radia-tion, we offered an source model in Section 3 and derived the far-field displacement spectrum Bc' (co) of Eq. (5) radiated from the seismic source expressed by Eq. (4). The spectral shape of 1B' c(co)I depends on such parameters as e, Td, and k(=v sin 0/c). Brune (1970) assumed that the delayed-time td was proportional to the fault radius, which was equivalent to the assumption that Td is constant irrespectively of earth-quake size. Our approach is different from Brune's, because we take the standpoint that td should be so determined as to explain the spectral behavior for the Aki's revised model B which is based on various observational results .
The amplitude spectrum for the model B decreases as (6)-2, 60--1, (.4)-T) for M,>6 and as (co-1, co-7) for M,<6 in a frequency range higher than A . The decay ratio at very high frequency is not clear, so the power is denoted by y. At any rate , y must be larger than 1.5 in order for the seismic energies to be finite. In case the amplitude spectrum I B pi(co)1 expressed by Eq. (5) demonstrates a spectral decay of
0)-2) for Td<0.04 and transforms into a (co-', co-2)-decay for Td>0.04 in agreement with the spectral patterns for the revised model B. This resemblance of spectral behavior induces us to interpret the scaling effect of the revised model B in
162 T. SATO and T. HIRASAWA
terms of the present source-model which takes the effect of partial relaxartion of the effective stress into account.
We try to relate the S wave spectrum 11313/(c0)1 with earthquake magnitude in such a way that 111,-M0 and MS- relations are the same as for the revised model B and that the spectral patterns of the S waves coincide with those for the revised model B at frequencies higher than f0. The difference between P and S wave spectra being ignored, the S wave spectra are made representative of seismic spectra of both waves . Together with the assumption of similarity, we assume 6=0 .1 and v/fi=0.7
independ-ently of earthquake size in order to scale earthquakes with fewer parameters . Observation point 0 is fixed at 60°.
First, we derive the relation between fault radius L and corner frequency fo from 1,6 p'(w) I as
L 1.79p/27f, . (6) If we assume )3=3.5 km/sec, Eq. (6) becomes
L(km) — 1.0T, , (7) where T, is the corner period in sec. According to the 111,-f, relation for the revised model B, earthquakes with have a corner period of 10 sec. Eq. (7) gives a fault radius L of 10 km for these earthquakes. In order that Td for earthquakes with Ms= 6 may take a value of 0.04, the delayed-time td must be
td LTdiv = 10X0.04/2.45 = 0.16 (sec) .
To obtain a family of spectral curves which are similar to those for the revised model B, we have to set Td < 0.04 for Als> 6 and Td > 0.04 for Ms< 6. This can be accomplished by assuming td to be constant irrespectively of earthquake size, because earthquakes with larger magnitude have larger source dimensions. The resultant family of spectral curves generally agree with those for the revised model B as shown in Fig. 6, though there is some difference in the spectral decay for smaller earthquakes. For M,<2, the present model predicts a spectral decay o f co-2, whereas the revised model B a spectral decay of co-1. Several workers (Terashima, 1968; Douglas and Ryall, 1972) have reported observed spectral decays for very small earth-quakes, however, the source spectra appear to be still uncertain for lack of the knowledge about the propagation effects for shorter periods. In view of the coherency expected for microearthquakes, it is important to observe and make clear the spectral patterns for them.
5. Discussions
In this section we intend to study how the family of spectral curves shown in Fig. 6 are associated with the dislocation motion on faults. From Eq. (4), the disloca-tion modisloca-tion at the center of the fault is
Kvt ; 0 < t < td D(t) =(8)
PARTIAL RFLAXATION OF THE EFFECTIVE STRESS 163 a _I w I- CC LA_ I-11 1:r 0 I— U .< LL 1.1.1 U Cr D 0 (r) >- 1— (r) z w a _J •ct CC H C.) LJJ Cl_ (f) H Z L.LJ 2 w 0 c = i _J Cl_ (f) a 2 (J 1 11.1 Z >- 0 Z 24 li J 0 2 23 c) 2 u.) EI-5 10 100 1000 PERIODS I N SEC
Fig. 6. Amplitude spectra of far-field displacement from earthquakes with different Ms.
The slip motion yields a final relative displacement Do=KL(Td+e(1—Td)). from Eq. (8) the slip velocity at the center of the fault as
{Kv
; 0
<t
<td,
,b(t)
eKv ; td < t < Liu .
Taking the Fourier transform of b(t), we have
Mar) = KG(co) ,
We obtain
(9)
164 T. SATO and T. HIRASAWA
6
F- ‹ 0,-, 011-1 I I I i _1--I Ms= 8 (n< —0 0 0) ou-Lut000-- > >---7 1--1--(75:1
6
Wzwioo- r. , cr 5 ..._ ,--- _ _J>— '1I---2 H[0—,- D 10 j„,- 00 W ._]Pf5.
iI;I
1 1 10 100 1000 PERIODS IN SECFig. 7. Amplitude spectra of slip velocity at the center of fault for different M.
where
G(co) = [1(1 --e) Cos w LTd-1 +E cos coL) 2
COL
(1 —6) sin COLTd-FE sin coL}9112
Putting e=0.1 and td-0.16 into Eq. (10), we obtain a spectral curve of slip velocity for each magnitude M, as illustrated in Fig. 7. It is assumed here from the assumption of similarity that K (—(24177r)(altt)) is constant independently of earthquake size.
It is found that the spectral amplitude of slip velocity at periods shorter than 5 sec does not depend on the magnitude for earthquakes with /1/5>5. This scaling law of dislocation motion is quite similar to that obtained by Aki (1972), who applied a procedure of inversion to the far-field spectra for the revised model B by assuming such a simple source-model as considered by Haskell (1964). In terms of the present source-model, the scaling law of slip velocity spectra shown in Fig. 7 results from the effect of partial relaxation of the effective stress and also from the assumption that the delayed-time td is constant irrespectively of earthquake size.
The slip process inferred above can grossly be described as follows: At the initial stage of slip, when the initial effective stress of the order of 100 bars is operative across the faults, the slip velocity is about 100 cm/sec. The time interval of this stage is 0.16 sec independently of earthquake size, and then the effective stress suddenly starts to decrease or the sliding friction starts to build up, causing a rapid decrease in slip velocity. Thereafter, the slip continues with a relatively small slip velocity until the rupture front stops its propagation. The total duration time of slip increases with increasing magnitude.
PARTIAL RELAXATION OF THE EFFECTIVE STRESS 165 mental studies to see if the present interpretation of the revised model B is reasonable or not. Especially, the near-field observation of seismic waves is significant to find whether there is a sudden change in slip velocity and to check the validity of the assumption that a delayed-time td is constant irrespectively of earthquake magnitude. Acknowledgments: We wish to thank Prof. Z. Suzuki and Dr. K. Yamamoto for their valuable suggestions. This study was supported by Grant in Aid for Scientific
Research, the Ministry of Education of Japan.
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