Bulletin of the Seismological Society of America, Vol. 81, No. 4, pp. 1115-1128, August 1991

H I G H - F R E Q U E N C Y SEISMIC WAVE RADIATION FROMANTIPLANE COHESIVE ZONE MODEL AND f~a~ AS SOURCEEFFECT
BY HIROYUKI FUJIWARA AND KOJIRO ~RIKURAABSTRACT

While a high-frequency cutoff, fmax, is widely observed in strongmotion seismic data, there is no consensus on whether it is due tosource processes or to attenuation and scattering of high-frequency
radiation in the crust or near the surface. To investigate the ability ofsource processes to control fmax, we use a standard antiplane crack
propagation formalism to numerically model the effect of rupture nucleation and arrest on high-frequency radiation based on a simple physicalhypothesis, a slip-weakening model. We model rupture arrest due tothree types of inhomogeneity: (1) a strong portion of rupture medium(barrier); (2) a drop in the pre-existing stress distribution of rupture
medium; and (3) a finite length of unruptured medium (asperity) lying
between previous ruptures. For cases (2) and (3) high frequencies fall off
more steeply than ~ - 2 , and fmax cannot be properly defined. For case(1), we find that fmax = VfLflLj 2, where Vf is the final crack velocity, Lf
is the final rupture length, and L i is the initial crack size. We extrapolate
this result to the rupture for a three-dimensional model and try to
explain observed fm~x" If we assume that an earthquake is a single
crack, Li is large for a large earthquake. However, if we assume that an
earthquake is made up of set of cracks and asperities, fmax will be
determined by the interaction of small cracks and barriers. If the distribution of these cracks and asperities is independent of source size, thenfmax will be nearly constant for all earthquakes.
INTRODUCTION

Using observations of strong ground motions near seismic faults, H a n k s(1982), Gusev (1983), Papageorgiou and Aki (1983), Faccioli (1986), andPapageorgiou (1988) observed an upper limit on the frequency of acceleration
spectra, f~ax" It is controversial whether f~ax is due to source processes(Papageorgiou and Aki, 1983) or to attenuation and scattering in the crust ornear the surface (Hanks, 1982). Observed seismic waves are affected by attenuation and scattering and have fmaxpath due to such effects (e.g., Hanks, 1982).
However, Fujiwara et al. (1989), Faccioli (1986), U m e d a et al. (1984), and Aki
and Papageorgiou (1988) reported that fmaxs°"rce due to source processes exists
by eliminating the local-site and path effects from observed data. According totheir studies, fmaxS°urce is nearly constant or slightly depends on earthquake
size. In this article, f~ax refers to f~a~ S°urce.
In order to explain fmax, it is assumed that near the crack tip there is acohesive zone that weakens stress concentration at the crack tip. The cohesivezone is introduced by considering a slip-weakening model. Based on the experiments of rock rupture, Ohnaka et al. (1986) and Okubo and Dieterich (1984)
reported that the critical displacement De, which is one of the parameters in theslip-weakening model and relates to the size of the cohesive zone, depends on
1115

1116

H. FUJIWARA AND K. IRIKURA

the geometrical shape and the roughness of the fault plane. The greater theroughness is, the larger D c is. Using a rupture velocity V and a cohesive zone
size L, Papageorgiou and Aki (1983) and Aki (1985) showed that V, L, and fmax
have the following relation:
fm,x = V/L.
They obtained this relation, however, under static conditions and did notconsider the dynamic properties of V and L. In this study, fmax is simulated
under dynamic conditions.
Another important point is that seismic waves from earthquakes have been
observed to follow the scaling law predicted by the ~0-2 model. It is necessary toinvestigate whether or not the seismic radiation from each simulation follows
the scaling law of the w-2 model. As the fmax discussed in this article means
the cut-off frequency in the o~-2-model, if the radiated seismic waves do notfollow the o~-2-model, we can not discuss the fmax for such models.
The rupture propagation is simulated, using an antiplane shear crack. Numerical simulations of rupture propagation have been carried out by many
researchers. In 2-D crack studies, Das and Aki (1977a) used Hamano's criterion,
and Yoshida (1985) used Irwin's criterion. In 3-D crack studies, Day (1982)
simulated the rupture process with a slip-weakening criterion, while Miyatake(1980a, b). Virieux and Madariaga (1982), and Das (1981) used a maximum
stress criterion. Furthermore, Das and Aki (1977b) simulated rupture propagation for a 2-D inplane barrier model and Das and Kostrov (1983) for a 3-D single
asperity model. The method of simulation used in this article is equivalent tothat of Andrews (1985). Andrews (1976a, b, 1985) emphasized rupture propagation, neglecting the effects of rupture arrest on seismic radiation. Here we
account for both the growth and the arrest of the rupture.
Inhomogeneities of fault planes affect the radiation of high-frequency seismic
waves. In this article, crack propagation and arrest are simulated for threecases of inhomogeneities: case 1: a strong portion of rupture medium (barrier);case 2: a drop in the pre-existing stress distribution of rupture medium; andcase 3: a finite length of unruptured medium (asperity) lying between previous
ruptures. The properties of high-frequency seismic waves radiated from each
inhomogeneity model are studied from the simulated results.
The main purpose of this article is to demonstrate a source model that
explains both the o~-2 scaling law and the cutoff frequency fmax"
SLIP-WEAKENING MODEL AND QUASI-STATIC CRACK

A crack in a perfectly elastic body creates a 1 / v / x stress singularity with x
being the distance from the crack tip. Barenblatt (1959) and Ida (1972) showedthat the singularity can be eliminated by assuming the stress to be a function of
the displacement in the region of the crack tip. Ohnaka and Yamashita (1988)
gave the following relation, based on laboratory experiments:
7 = (% - Tf)[1 + a log(1 + / ~ A u ) ] e x p ( - ~ A u ) + rf,(1)

where % is initial stress, ~f is dynamic frictional stress, A u is displacement,
and a, /3, and y are constants. This equation eliminates not only the stresssingularity in the vicinity of a crack tip but also the singularity in acceleration.

H I G H - F R E Q U E N C Y SEISMIC WAVE RADIATION1117

We have approximated (1) with the following equation, which is equivalent tothe slip-weakening model of Andrews (1976a, b) (Fig. 1).
T =TO - - T f ,AU= Tu - ( T . - T f ) " A u / D c , O=----- 0

< a u <_ De,(2)

AU > D e

Tf,

where Tu is yield stress, To is initial stress, 7f is frictional stress, D e is criticaldisplacement, and A u is displacement. Ohnaka et al. (1987a, b) confirmed fromanalysis of experimental data that ru, T[ and Dc can be regarded constants
during rupture propagation.
If the slip-weakening model is assumed, a crack whose size is less than 2 L~ isstable. L c is determined from the parameters in the slip-weakening model. The
quantity G o = De( % - T f ) / 2 is defined from equation (2) and provides a rupture
criterion. G O is the m a x i m u m value of the energy stored per unit area of thecohesive zone. If the crack is small and the energy stored per unit area of thecohesive zone is less than Go, rupture does not nucleate. In this article, such acrack is called quasi-static. An upper bound on the quasi-static crack length 2 L~
exists and L c is regarded as the m a x i m u m length of the static cohesive zone
size.
Solving the following equation, we can obtain L~ for the antiplane shear
crack.
O2AU

02Au

- - +

Oy 2

OX 2

-0

OAU
#

= %-

Oy

To -

(T u -TW)'AulD

c,

Ixl

<_L

Ix I > L

A u = O,

(3)

OAu I

lim # - e-~o

-~

Oy

T u

--

T O

L+e

METHOD OF COMPUTATION
We consider a two-dimensional antiplane shear crack (Fig. 2). The fault plane
is taken on the x - z
plane. The crack is infinite along the z axis andsymmetric about the z axis.

r~ Tu
.kJ
r.~

To

r~
!

Dc

Slip

Fro. 1. Relations between stress and slip in our calculation. 7u, To, and Tf are yield stress, initialstress, and frictional stress, respectively. D c is critical displacement.

H. FUJIWARA AND K. IRIKURA

1118

Y

FIG. 2. Antiplane shear crack with cohesive zones. The shaded areas are cohesive zones. The
crack is infinite along the z axis and symmetric about the z axis.

T h e d i s p l a c e m e n t /x u satisfies
1

02Au

02Au
--

132

cgt 2

-

-

02Au
-If"

c3x 2

-

(4)

-

Oy2

"

T h e b o u n d a r y conditions for (4) a r e
OAu
A r = tL O y= rf-

A r = ru - ro -%,inside t h e c r a c k( % - r f ) A U / Dc , inside t h e cohesive zone

A u = 0,(5)

outside t h e c r a c k

a n d t h e i n i t i a l conditions a r e

Au(x,o,o)= aUo(X)=(6)

w h e r e A u is d i s p l a c e m e n t , Ar is s t r e s s drop, ~ is s h e a r - w a v e velocity, Tf is
frictional stress, r u is y i e l d stress, r o is i n i t i a l stress, a n d AUo(X ) a n d A r o ( X )a r e t h e d i s p l a c e m e n t a n d t h e s t r e s s drop of t h e i n i t i a l crack, r e s p e c t i v e l y .
T h e b o u n d a r y conditions (5) a n d i n i t i a l conditions (6) a r e not e n o u g h to solved i f f e r e n t i a l e q u a t i o n (4), as t h e y do not include i n f o r m a t i o n a b o u t t h e locationof t h e c r a c k tip. A r u p t u r e c r i t e r i o n g i v e s t h i s i n f o r m a t i o n . T h e c r i t e r i o n we use
is g i v e n b y G > Go, b a s e d on t h e s l i p - w e a k e n i n g model. T h i s c r i t e r i o n is b a s e d
on a n e n e r g y b a l a n c e a n d is e q u i v a l e n t to t h e G r i f f i t h criterion.
F r o m t h e d i f f e r e n t i a l e q u a t i o n (4), we o b t a i nu(x,y,t)2 f/sg(x,y,t;x,,O,t,)Ar(x,,O,t,)dx,
#

dt, '

(7)

HIGH-FREQUENCYSEISMIC WAVE RADIATION

1119

where

H{(t-t')g(x,y,t;x',O,t')[(x-x') 2 + y211/9"/~}=2 7 r V / ( t - t')2 - { ( x -x')2 + y2}/~ 2(8)and S is the region such t h a t t - t' < I x - x' ]/ ~.
Discretizing (7) we obtain

Alt( XM, O, tN)
J. L

=

E

7r~ i = 1

N

E AT(xi,O, tj)g(xM, tN; xi, tj),

j=l

(9)

where

&x

At

= fxi+-~- f t j + ~J~x ]. At g(XM'O' tN; x',O, t' ) dx dt'.
xi- --~ tj(10)

When we take A x and A t such t h a t A x = ~A t, thena U(XM, 0,
1

=--

L

N

E

E AT(xi,O, tj)g(xM, tN;xi, ty)

7r/t i = 1 j = l

1+

Ar(XM,O, t N ) g ( x M , tN;XM, tN).

(11)

If we know ~ u and 47 for all time t < tN_l, the first term of the right side of(11) is known, and (11) gives the relation between AU(XM, 0, ty) and AT(XM, 0,tg). Adding this equation to equation (5), which is obtained from the slipweakening model, we can solve for A u and A7 at time t N. Then we have todetermine whether the point (x M, 0, tN) is inside the crack, in the cohesivezone, or out of the crack. The slip-weakening criterion gives the information forthis judgment.
Theoretically, we should take the quasi-static crack whose size is 2 L c as theinitial conditions. However, if we use this critical crack, too much time isconsumed before the rupture starts. We therefore use a crack size slightly
larger t h a n L c in the initial conditions.
MODELS FOR COMPUTATION

Strength and initial stress are inhomogeneous on fault planes. High-frequency
seismic waves are radiated when the rupture velocity changes rapidly due tothese inhomogeneities (Madariaga, 1977; Yamashita, 1983). In this article,
these inhomogeneities are classified into three cases. For each case, the rupture
propagation and arrest of a crack with a cohesive zone is simulated and the

1120

H. F U J I W A R A AND K. I R I K U R Achange of the rupture velocity, the size of cohesive zone, and the seismic wave
radiation are investigated. We can regard each case as a single earthquake or
as components of an earthquake.
In case 1, there are regions where the strength is very high and where thestrength is relatively low. The regions of high strength are called barriers.
When a crack nucleate in a region of low strength and the crack tip reaches abarrier, the rupture propagation temporarily or permanently stops. We assume
here that the strength of the barrier is infinite.
In case 2, the strength is uniform, however the prestress is inhomogeneous.
We assume there is a region of high initial stress surrounded by regions inwhich the initial stress % is equal to the frictional stress rf. The rupture starts
at the high prestress region and the rupture propagates into the low prestressregion.
In case 3, a region where strength and prestress are high exists surroundedby regions where strength and prestress are very small. This means that an
unbroken region is left within a broken region. This unbroken region is anasperity. In our calculation, r o = T f and strength G = 0 are assumed out of theasperity. This model is proposed by Rudnicki and Kanamori (1981).
RESULTS

Initiation o f R u p t u r e

The rupture starts spontaneously in all cases, the rupture process is shown in
Figures 3, 4, and 5. The rupture velocity is slow at first and gradually

CASE 1

AT

2o

6~

x

4o

o

2'0

8~

FIG. 3. Plots of AT(X, t) and Au(x, t) for the barrier model (case 1). The region for X > 65 is abarrier.

HIGH-FREQUENCYCASE 2&~1121

SEISMIC WAVE RADIATION

Au

t

i

i

20

45

X

20

45

X

FIG. 4. Plots of At(x, t) a n d Au(x, t) for t h e p r e s t r e s s d i s c o n t i n u i t y model (case 2). r o = ~rf in t h e
r e g i o n for X > 45.

CASE 3
AT

~U
t

--BO

-40

0

40

80

X

-

0

-

0

0

40

80

X

FIa. 5. Plots of At(x, t) a n d Au(x, t) for t h e a s p e r i t y model (case 3). The region for - 2 0 < x < 20
is asperity. T h e r e g i o n s for x < - 4 0 or x > 40 are u n b r o k e n regions.

1122

H. F U J I W A R A

A N D K. I R I K U R A

accelerates to the shear-wave velocity. The cohesive zone size is largest at the
beginning of rupture propagation. As the crack grows, the cohesive zone size issmaller in inverse proportion to the crack size (Fig. 6).

Stopping of Rupture
Case 1 (Barrier Model). When the crack tip reaches a barrier, the rupture
stops. In the calculated example used here, the rupture propagates at nearly theshear-wave velocity just before the crack tip reaches the barrier and suddenly
stops, as shown in Figure 3. Therefore, the rupture velocity has a discontinuity
at this point. The size of the cohesive zone becomes one third of the initialcohesive zone size just before the crack tip reaches the barrier. From thiscalculation we find that the size of cohesive zone is inversely proportional to thecrack size.
Case 2 (Prestress Discontinuity Model). The results for this case are shown in
Figure 4. The rupture velocity is nearly the shear-wave velocity just before thecrack tip reaches the region where ro = r f . After the crack tip crosses into that
region, the rupture velocity in the calculated example decelerates to one third of
the shear-wave velocity. We did not continue the calculation for the completearrest of rupture to save computational time; however, by calculating rupturestability for the static model, we can estimate that the crack will be stable when
the crack tip is at x = 85 and the cohesive zone size is 7 for this example. We
guess that the rupture stops when the crack tip reaches around the point
x = 85. The cohesive zone spreads after the crack tip crosses into the region
where TO Tf. Inside the region of r o = Tf, the size of the cohesive zone remains
constant as the crack grows.
Case 3 (Asperity Model). As shown in Figure 5, the asperity is the region
between two cracks. The strength in the region I x l > 80 is infinite. In thisexample, the rupture starts at both sides of the asperity. When the rupture
fronts from both sides reach to the center of the asperity, the asperity iscompletely broken. However, the slip on the fault plane continues until theregion made up of two initial cracks and the asperity is completely broken.
=

Displacement and Slip Velocity on the Fault Plane
If a cohesive zone does not exist, the displacement on the fault plane is
proportional to H ( t ) v ~ at the rupture front, and the slip velocity has a

CASEI

CASE 2

CASE 3

t

p

o

so

f

x

0

50

0

FIG. 6. Locations of cohesive zones.

5O

HIGH-FREQUENCY SEISMIC WAVE RADIATION

1123

singularity of H(t)/~f[. If a cohesive zone does exist, such a singularity is
weakened in proportion to the cohesive zone size. For case 1 (Fig. 7), the slip
velocity has a discontinuity of H(t) form due to the stopping phase if cohesivezone does not exist. Such a discontinuity is weakened by the cohesive zone
effect. The cohesive zone effect used here means the effect of the cohesive zone
on seismic waves radiated as a so-called stopping phase. For case 2 (Fig. 8),
since the rupture velocity changes continuously, there do not appear to bephases corresponding to the stopping phase in case 1. The slip velocity in case 2changes more smoothly than in case 1. For case 3 (Fig. 9), the slip velocity has
peaks due to phases radiated when the asperity is completely broken. However,the slip velocity does not include a step-function form.

Near-Field Displacement and Velocity
Madariaga (1983) obtained an analytic solution for the semi-infinite antiplane shear crack without a cohesive zone. He also discussed the highfrequency radiation due to discontinuities in strength and that due to variations
in stress intensity. He considered a semi-infinite crack, while our calculations
are for a finite crack. We cannot, therefore, compare all seismic phases but only
individual phases radiated by the discontinuities in rupture velocity and dynamic stress drop.
According to the analytic solution obtained by Madariaga (1983), the velocity
waveform for case 1 has a discontinuity proportional to H(t) due to the stopping
phase if a cohesive zone does not exist. Because of this discontinuity, thespectral shape of the seismic wave follows the ~-2-model. As the model we usedhere has a cohesive zone, this discontinuity is weakened (Fig. 10). The size of

CASE IDISPLACEMENTS SLIPVELOCITIES/(10,0)(20,0)/(30,0)(40,0)/(50,0)(60,0)

2222t

loo

o

lOO

t

FIG. 7. Displacements and slip velocities on the fault plane for case 1. The slip velocities are
obtained from finite differences of the displacements. Small negative-spikes seen in the velocity
waveforms are caused n u m e r i c a l noise.

1124

H. F U J I W A R A AND K. I R I K U R A
CASE 2DISPLACEMENTS S L I P

VELOCITIES(0,0)(10,0)(20,0)(30,0)i(40,0)(50,0)
i

,

,i

,

,

(60,0)
/,

,

10o

t

o

lOO

t

FIG. 8. Displacements and slip velocities on the fault plane for case 2.

CASE 3IDISPLACEMENTS
ISLIPIVELOCITIES

I(20,0),

,/ ,

,

,

(6o,o)
o

lOO

t

lOO

t

FIG. 9. Displacements and slip velocities on the fault plane for case 3.

the cohesive zone becomes s m a l l e r t h a n L c w h e n t h e r u p t u r e arrests. Therefore,
the effect of t h e cohesive zone is r e s t r i c t e d to a h i g h e r f r e q u e n c y t h a n t h a t
expected f r o m t h e static condition. F o r cases 2 a n d 3 (Figs. 11 a n d 12), thevelocity w a v e f o r m s h a v e no step-function discontinuities. This m e a n s t h a t t h eseismic w a v e s r a d i a t e d in cases 2 a n d 3 do not follow t h e scaling law of theHIGH-FREQUENCY SEISMIC WAVE RADIATION

1125

CASE 1DISPLACEMENTS

VELOCITIES( O, 50)

0

100

'

~t

t

(30,50)

0

100

t

0

100

~ t

100

t

100

t

(5o.5o)100

t(80,50)
I
i

100

t

FIG. 10, Near-field displacements and velocities for case 1. The velocity wave has discontinuities
due to the stopping phase. However, these discontinuities are weakened by the existence of cohesive
zones.

CASE 2
VELOCITIESDISPLACEMENTS( O, 50)

100

t

0

100

t

0

100

t

100

t

100

t

(30,50)0100

t(50,50)100

t(80,50)

L

0

100

t

0

FIG. 11. Near-field displacements and velocities for case 2.

~o-e-model. Therefore, in order to discuss fmax under the condition t h a t theseismic radiation should follow the o~-2-model, we h a v e to consider case 1. The
seismic w a v e s radiated in cases 2 and 3 are less important t h a n those in case 1,
if we restrict our interest to fmax" However, w h e n a lower frequency t h a n fm~x
is studied, it is important to consider cases 2 and 3.

H. FUJIWARA AND K. IRIKURA

1126

CASE 3DISPLACEMENTSI

VELOCITI ES(0,50)

,

'

o

16o

~

~oo

t

100

't

100

t

100

t

(30,50)100t( so, 5o)100

t(80,50)100~

0FIG. 12. Near-fielddisplacementsand velocitiesfor case 3.

DISCUSSION AND CONCLUSIONSIn our simulation, we showed t h a t the seismic waves t h a t have fmax andfollow the scaling law of the ~0-2-model are only radiated in case 1. Following
Papageorgiou and Aki (1983) and Aki (1985), the relation between rupture
velocity V(t), cohesive zone size D(t), and fmax is obtained for case 1:

fmax---- V(to)/D(to),(i2)

where t o is the time t h a t the stopping phase is radiated.
According to our results, D(t) is inversely proportional to the crack size.
Therefore

D(t) = Lc2/L(t).

(13)

L c is the m a x i m u m cohesive zone size and L(t) is the crack size at time t. From(12) and (13), we obtain&ax = V(to)L(to)/Lc 2.

(14)This relation is obtained from a restricted model for a two-dimensions
antiplane. However, it is expected t h a t a similar relation holds for threedimensional models. Using (14), w e a t t e m p t to explain the observed fm~x"
At first, we assume t h a t both large earthquakes and small earthquakes
consist of a single crack. If the initial crack sizes are the same, then the fma~ ofthe large earthquake will be high. However, observed fma~ only slightly depends on earthquake size (e.g., Faccioli, 1986; Fujiwara et al., 1989). The fmax'Sof large earthquakes are a little lower t h a n those of small earthquakes. There-HIGH-FREQUENCY SEISMIC WAVE RADIATION

1127

fore, in order to explain observed fm~ by the single crack model, it must be
assumed that Lc's are different between large earthquakes and small earthquakes. We need to consider that large earthquakes have large initial cracks
and small earthquakes have small initial cracks.
Another model is obtained if we assume that an earthquake consists of a setof inhomogeneities. These inhomogeneities include cases 1, 2, and 3. In therupture process, cases 1, 2, and 3 locally appear many times. In this model, thehigh-frequency seismic waves related to fm~ are locally radiated when small
cracks temporally stop at barriers. Lower-frequency contents than fma~ areradiated when asperities are broken. The asperities behave as barriers until
they are eventually broken. In this model, fma~ only relates to cohesive zone
sizes of small cracks. Therefore, if the distribution of these cracks and asperities
is independent of source size, then f~.~ will be nearly constant for all
earthquakes.
ACKNOWLEDGMENTS
We thank Drs. K. Aki, Y. Ben-Zion, P. Leavy, and W. S. Phillips for helpful suggestions andcriticisms.
Partial support for this work was provided by a grant from the Science Research Fund of the
Ministry of Education of Japan.
REFERENCES
Aki, K. (1985). Origin of fmax, Paper presented at the 5th Maurice Ewing Symposium on Earthquake Source Mechanics, AGU, Harriman, New York, 20-23 May.
Aki, K. and A. A. Papageorgiou (1988). Separation of source and site effects in acceleration power
spectrum of major California earthquakes," in Proc. Ninth World Conf. Earthq. Eng., Vol. 8,
163-167.
Andrews, D. J. (1976a). Rupture propagation with finite stress in antiplane strain, J. Geophys. Res.
81, 3575-3582.
Andrews, D. J. (1976b). Rupture velocity of plane strain shear cracks, J. Geophys. Res. 81,
5679-5687.
Andrews, D. J. (1985). Dynamic plan-strain shear rupture with a slip-weakening friction law
calculated by a boundary integral method, Bull. Seism. Soc. Am. 75, 1-21.
Barenblatt, G. I. (1959). The formation of equilibrium cracks during brittle fracture, general ideas
and hypotheses, axially symmetric cracks, Prikl. Mat. Mek. J. Appl. Math. Mech. 23, 622-636.
Chen, Y. T. and L. Knopoff (1986). Static shear crack with a zone of slip-weakening, Geophys. J. R.
Astr. Soc. 87, 1005-1024.
Das, S. (1981). Three-dimensional spontaneous rupture propagation and implications for the earthquake source mechanism, Geophys. J. R. Astr. Soc. 67, 375-393.
Das, S. and K. Aki (1977a). A numerical study of two-dimensional spontaneous rupture propagation, Geophys. J. R. Astr. Soc. 50, 643-668.
Das, S. and K. Aki (1977b). Fault plane with barriers: a versatile earthquake model, J. Geophys.
Res. 82, 5658-5670.
Das, S. and B. V. Kostrov (1983). Breaking of a single asperity: rupture process and seismic
radiation. J. Geophys. Res. 88, 4277-4288.
Day, S. M. (1982). Three-dimensional simulation of spontaneous rupture: the effect of nonuniform
prestress. Bull. Seism. Soc. Am. 72, 1881-1902.
Faccioli, E. (1986). A study of strong motions from Italy and Yugoslavia in terms of gross source
properties, in Earthquake Source Mechanics, S. Das, J. Boatwright, and C. H. Scholz (Editors),
American Geophysical Union Monograph 37, 297-309.
Fujiwara, H., T. Mikoshiba, and S. Kinoshita (1989). Source parameter of earthquake swarms in
the east region of Izu Peninsula, Prog. Abstr. Seism. Soc. Japan 2, 128.
Gusev, A. A. (1983). Descriptive statistical model of earthquake source radiation and its application
to an estimation of short-period strong motion, Geophys. J. R. Astr. Soc. 74, 787-808.
Hanks, T. C. (1982). fm~, Bull. Seism. Soc. Am. 72, 1867-1880.

1128

H. F U J I W A R A AND K. IRIKURA

Ida, Y. (1972). Cohesive force across the tip of a longitudinal-shear crack and Griffith's specificsurface energy, J. Geophys. Res. 77, 3796-3805.
Madariaga, R. (1977). High-frequency radiation from crack (stress drop) models of earthquake
faulting, Geophys. J. R. Astr. Soc. 51,625-651.
Madariaga, R. (1983). High frequency radiation from dynamic earthquake fault models, Ann.
Geophys. 1, 17-23.
Miyatake, T. (1980a). Numerical simulations of earthquake source process by a three-dimensional
crack model. Part I. Rupture process, J. Phys. Earth 28, 565-598.
Miyatake, T. (1980b). Numerical simulations of earthquake source process by a three-dimensional
crack model. Part II. Seismic waves and spectrum, J. Phys. Earth 28, 599-616.
Ohnaka, M., Y. Kuwahara, and K. Yamamoto (1987a). Constitutive relations between dynamical
physical parameters near a tip of the propagating slip zone during stick-slip shear failure,
Tectonophysics 144, 109-125.
Ohnaka, M., Y. Kuwahara, and K. Yamamoto (1987b). Nucleation and propagation processes ofstick-slip failure and normal stress dependence of the physical parameters of dynamic slip
failure, J. Nat. Disaster Sci. 9, 1-21.
Ohnaka, M., Y. Kuwahara, K. Yamamoto, and T. Hirasaw a (1986). Dynamic breakdown processes
and the generating mechanism for high-frequency elastic radiation during stick-slip instabilities, in Earthquake Source Mechanics S. Das, J. Boatwright, and C. H. Scholz (Editors),
American Geophysical Union Monograph 37, 13-24.
Ohnaka, M. and T. Yamashita (1989). A cohesive zone model for dynamic shear faulting based on
experimentally inferred constitutive relation and strong motion source parameters, J. Geophys.
Res. 94, 4089-4104.
Okubo, P. G. and J. H. Dieterich (1984). Effects of physical fault properties on frictional instabilities produced on simulated faults, J. Geophys. Res. 89, 5817-5827.
Papageorgiou, A. S. (1988). On two characteristic frequencies of acceleration spectra: patch cornerfrequency and fmax Bull. Seism. Soc. Am. 78, 509-529.
Papageorgiou, A. S. and K. Aki (1983). A specific barrier model for the quantitative description of
inhomogeneous faulting and the prediction of strong ground motion. Part I. Description of themodel, Bull. Seism. Soc. Am. 73, 693-722.
Rudnicki, J. W. and H. Kanamori (1981). Effects of fault interaction on moment, stress drop, andstrain energy release, J. Geophys. Res. 86, 1785-1793.
Umeda, Y., Y. Iio, A. Kuroiso, K. Ito, and H. Murakami (1984). Scaling of observed seismic spectra,
J. Seism. Soc. Japan (Zisin) 37, 559-568 (in Japanese).
Virieux, J. and R. Madariaga (1982). Dynamic faulting studied by a finite difference method, Bull.
Seism. Soc. Am. 72, 345-369.
Yamashita, T. (1983). High-frequency acceleration radiated by unsteadily propagating cracks andits near-source geometrical attenuation, J. Phys. Earth 31, 1-32.
Yoshida, S. (1985). Two-dimensional rupture propagation controlled by Irwin's criterion, J. Phys.
Earth 33, 1-20.
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