Bulletin of the Seismological Society of America, 90, 2, pp. 387–400, April 2000

A Theoretical Omega-Square Model Considering the Spatial Variation
in Slip and Rupture Velocity
by Yoshiaki Hisada

Abstract A theoretical model for constructing the x-squared model is proposedby modifying the k-squared model of Bernard et al. (1996). The k-squared model
provides a theoretical basis for the empirical x-squared model under the assumptions
that (1) the spatial wavenumber spectrum of the slip distribution falls off as theinverse of the wavenumber squared (k-squared), (2) the Fourier amplitudes of theslip velocity are independent of x at high frequencies, and (3) the rupture velocity
is constant. In this study, a more realistic model is proposed by modifying the last
two assumptions. First, a Kostrov-type slip velocity model is proposed by superposing equilateral triangles, in which a source-controlled fmax is imposed by the minimum duration among the triangles. The Fourier amplitude of our slip velocity model
falls off as the inverse of x at high frequencies less than fmax. Next, in order tomodel variable rupture velocities, the incoherent rupture time (Dtr), namely, the difference between the actual rupture time and the coherent (average) rupture time, isintroduced. After checking various models for Dtr distributions, the k-squared modelfor Dtr, similar to that for the slip distributions of the k-squared model, is found tobe the most plausible. Finally, it is confirmed that the proposed source model (wecall it as the x-inverse-squared model), which consists of the combination of the slip
velocity proposed here and the k-squared distributions for both slip and Dtr, not onlyis consistent with the empirical x-squared model, but also provides the theoreticalbasis for constructing realistic source models at broadband frequencies.
Introduction
al. (1996) proposed the k-squared model as a theoretical xsquared model. The main assumption of this model is thatthe spatial wavenumber spectrum of the final slip distribution falls off as the inverse of the wavenumber squared (ksquared). This model yields a stress drop independent ofseismic moment, which is consistent with the stochastic
model of Andrews (1981) and the fractal model of Frankel(1991) with the fractal dimension of 2. In addition, Somerville et al. (1999) found that slip distributions derived fromsource inversion studies using strong motion records are
consistent with the k-squared model up to about 1 Hz. The
k-squared model is also consistent with the x-squared model
under the assumptions of constant rupture velocity and scaledependent rise time (Bernard et al., 1996). However, theactual rupture velocity probably fluctuates significantly inrealistic situations (Das and Aki, 1977; Mikumo and Miyatake, 1978; Archuleta, 1982). Since the fluctuation of the
rupture front excites strong high-frequency waves (e.g., Madariaga, 1977), the k-squared model should be modified inorder to construct a more realistic x-squared model.
The purpose of this study is to propose a theoretical xsquared model. For this, we will modify the k-squared model

Since the pioneering work of Aki (1967) to construct
the scaling law of seismic source spectra, the x-squaredmodel has been the most widely used empirical tool for predicting strong ground motions (e.g., Aki, 1972; Hanks,
1979; Hanks and McGuire, 1981; Boore, 1983; Joyner,
1984; Irikura, 1986). Originally, its spectrum was rather simple, with one or two corner frequencies corresponding to thefault dimension and/or the rise time using smooth and homogeneous faulting models (Haskell, 1964; Kostrov, 1964;
Brune, 1970; Madariaga, 1976). Later, a cut-off frequency,
fmax, was introduced to reproduce the steep falloff of theobservational spectra at higher frequencies (Hanks, 1979,
1982). On the other hand, observational and numerical results have shown that earthquake source processes are highly
heterogeneous. Thus, barrier/asperity models were proposed
to include the added complexities (Das and Aki, 1977; Kanamori and Stewart, 1978; Mikumo and Miyatake, 1978;
Boatwright, 1982). Subsequently, the x-squared model has
been modified by including the so-called patch corner frequencies, which correspond to the barrier/asperity size (e.g.,
Gusev, 1983; Papageorgiou and Aki, 1983; Koyama, 1985).
Recently, Herrero and Bernard (1994) and Bernard et
387

388

Y. Hisadaof Bernard et al. (1996) by considering realistic spatial variations in both slip and rupture velocities. This article consists of five sections. First, I show the formulation of the farfield displacement using a unilateral source model. Second,
I briefly review the formulation of the k-squared model.
Third, I propose a realistic and practical slip velocity modelby considering the results from dynamic source modelings
and laboratory experiments. Fourth, I introduce the effectsof the variable rupture velocities in the model using the incoherent rupture time. Finally, I propose a “omega-inversesquared model” for constructing the x-squared model, anddiscuss its characteristics in comparison with the k-squaredmodel and the empirical x-squared model.

Source Spectra for Far-Field Displacements
Following Bernard et al. (1996), I consider a 1-D fault
model with unilateral rupture propagation, embedded in a
homogeneous full-space (Fig. 1). The far-field displacements, Ui , at a station Y from this source model can be
written as
Ui(Y; x) 4

Ri
M (x),
4pqrc3 o(1)where x is the circular frequency, the subscript i denotes theith component in the spherical coordinate system, and Ri is
the radiation pattern for the ith component. The variable cis the medium velocity (VP for the radial, and VS for thetransverse and vertical directions), q is the density, and r is
the distance from the rupture front, x, to Y. Mo in equation(1) is the source spectrum (e.g., Dan and Sato, 1999), and isexpressed as
Mo(x) 4 lW

Figure 1.

Geometry of a unilateral source and an
observation station in the far-field (top), and modelingof variable rupture time using the incoherent rupture
time Dt (bottom).

L# V(x; x) exp{ix(t ` t )}dx, (2)0crwhere, l is the rigidity, V is the Fourier transform of the slip
velocity at a specific point x on the fault plane, and W and
L are the width and length of the fault, respectively. In equation (2), tc and tr are the arrival time of the seismic waves
and the rupture time, respectively, and are given as follows:rr 1 x • cos htc 4 ' 0, (x K r0)ccture time tr into the coherent (average) rupture time and theincoherent rupture time Dtr . I introduce Dtr to include theeffects of fluctuations at the rupture front (see Figure 1). For
convenience, I express the slip velocity function V by the
product of the slip (dislocation) and the slip velocity functionof the unit dislocation.
V(x; x) 4 D(x)FV (x; x)

(5)

(3)
Substituting equations (3), (4), and (5) into equation (2),I obtain the following source spectra,

and,
x
tr 4
` Dtr(x).
V̄r(4)

where, r0 is the distance from the origin of the rupture front
to Y, h is the angle between the directions of the rupture
propagation and the seismic waves, and ¢Vr is the averagerupture velocity. As shown in equation (4), I divide the rup-L

# D(x)F (x; x)

Mo(x) 4 lW exp(ixro /c)

5 1

0

V

(6)

26

x
exp ix` Dtr dx,
V̄rCd
where, Cd is the directivity coefficient (Ben-Menahem,1961),

389

A Theoretical Omega-Square Model Considering the Spatial Variation in Slip and Rupture Velocity

Cd 4

1
.
V̄r
1 1 c • cos h(7)

Substituting equations (8), (9), and (10) into equation(6), and changing the bounds of the integration, the source
spectrum can be expressed as:Mo(x)K-Squared Model with Scale-Dependent Rise Time
Here I briefly explain the k-squared model (Herrero andBernard, 1994; Bernard et al., 1996) and discuss some physical problems that suggest the need to modify this model.
The main assumptions for the k-squared model are as
follows:
a. The Fourier amplitudes of spatial wavenumber (k) spectraof the slip distribution D decay as k12.
b. The slip velocity function FV is the rectangular boxcar
whose rise time is inversely proportional to wavenumber(or frequency) at high frequencies. This assures frequency-independent Fourier amplitudes of FV, as will beshown below.
c. The rupture velocity Vr is constant, namely Dtr 4 0 in
equation (6).
Assumption (a) is based on the self-similarity of thespatial distribution of the slip, and this explains a stress drop
independent of seismic moment and fault size. Assumptions(b) and (c) were introduced to produce the x-squared model.
Regarding assumption (b), Herrero and Bernard (1994) originally adopted the delta function (instantaneous slip), butlater Bernard et al. (1996) used the rectangular boxcar withthe scale-dependent (or, frequency-dependent) rise time. The
slip-velocity function FV of the boxcar can be expressed asfollows,
FV (x; x) 4

sin vs ivs
e ,
vs

(8)

xs(x; x)2(9)where, vs 4

Using assumption (b), the rise time s is inversely proportional to x (or wavenumber) at high frequencies, as
follows,5

smax ,(x # 2paCd /smax)s(x; x) 4 2paC
(10)
d
,
(x
$
2paC
/s
)
d
max
x
¢r
with smax 4 L0 /V
where smax is the total slip duration at x, and L0 is somecharacteristic dimension representing the barrier/asperity
size. A nondimensional coefficient, a, is the ratio of the local
rise time over the propagation time of the average rupture
front along a wavelength (Bernard et al., 1996); this coefficient was introduced to incorporate the observed short slip
duration (Heaton, 1990).

#

4 lWFV exp(ixro /c)with k 4

(11)

``

1`

D(x) exp(2pikx)dx,xf4,
2pCdV¯ r
CdV¯ r(12)5sin(pfsmax),
pfsmax( f # f0)
and FV ( f ) 4,(13)
sin (paCd), ( f $ f0)
paCd
with f0 4 aCd /smax
The spectrum amplitudes of slip velocity function (equation[13]) are independent of frequency for frequencies greaterthan f 0. It should be noted that equation (13) can lead to
unrealistic results for aCd 4 1; FV is zero for f $ f 0 .
The integration in equation (11) is the Fourier transformof the slip distribution with respect to the wavenumber. Under assumption (a), Herrero and Bernard (1994) proposed
the following integrand:
``

#

1`

¯ Fph(k)D(x) exp(2pikx)dx 4 DL
2

5

1,
1
,
(kL)2

(k # 1/L)(k $ 1/L)(14)

where D̄ is the average slip over the fault plane, and Fph isa phase function with unit amplitude.
Finally, the source spectrum of the k-squared model isobtained by substituting equations (12) and (14) into (11).
Using frequency f instead of x, the source spectrum can be
expressed asMo( f ) 4 M0 FV exp(2pfiro /c)Fph ( f )1,(f # fC)
V̄ C
2, with fC 4 r d
2fCL, ( f $ fC)
f

51 2

(15)

where M0 is the seismic moment (4 lD̄LW; D̄ is the averageslip). Combining equations (13) and (15), the k-squaredmodel yields the x-squared model with two corner frequencies ( f 0 and fC : usually f 0 . fC).
On the other hand, the k-squared model is based on two
physically unrealistic assumptions: (1) the boxcar model forthe slip velocity, and (2) a constant rupture velocity. The
Fourier amplitude of a boxcar falls off as the inverse of x,
and generates infinite slip accelerations at the starting and390

Y. Hisada

stopping phases. To produce finite accelerations, smootherslip velocities have to be used, such as a combination oftriangles as used for source inversion studies. Since the Fourier amplitudes of triangles fall off as the inverse of xsquared, it is difficult to model the amplitudes of slip velocity as being independent of x. Consequently, the variablerupture velocity should be incorporated to excite highfrequency waves. Next, I investigate the effects of a more
realistic slip velocity, together with a variable rupture
velocity.

Modification of K-Squared Model Considering
Realistic Slip Velocity and Variable Rupture
Velocity
Fourier Amplitude Spectra of a Realistic Slip
Velocity Function
An analytic solution for the slip velocity was derived
by Kostrov (1964) using a self-similar circular crack. Its
amplitude in time history is infinite at the crack tip, and falls
off as the inverse of a square root of time. More realistically,
dynamic source modeling and laboratory experiments onrock indicate that the slip velocity exhibits a sharp rise witha finite amplitude and is followed by a relatively smoothdecay (e.g., Madariaga, 1976, 1977; Day, 1982; Ohnaka andKuwahara, 1990; Miyatake, 1998). The finiteness of the velocity amplitude at the crack tip is probably caused by nonlinear breakdown processes, such as local yielding (e.g.,
Madariaga, 1977). Accordingly, it may cause the sourcecontrolled f max (Papageorgiou and Aki, 1983).
In order to construct the realistic slip velocity functions,
I adopt a simple and practical model. As shown in Figure 2,
I construct the slip velocity function fv by superimposing
equilateral triangles with different durations. In the time domain, it can be written as follows;NVfV(x; t) 4

o

NV

o

1
Arj11 ˙fj(x; t) with A 4
Arj11,
j41
A j41

(16)

where NV is the total number of the triangles, Ar is the ratioof the area of the jth triangle with respect to the area of thej1lth triangle.
In equation (16), ḟj is the equilateral triangle of the jthelement,5

Aj t,ḟj(x; t) 4 Aj(sj 1 t),
0,with Aj 4 2Vj /sj ,(t # sj /2)(sj /2 # t # sj)(sj # l)and Vj 4 2/sj .

(17)

(18)

Aj and Vj are the maximum acceleration and velocity of thejth triangle, respectively. sj is the duration of the jth triangle,
and I adopt here the following values,

sj 4

2 j11
,
fmax

with smin [ s1 4

1
,
fmax

and smax [ sNm , 4

(19)
Nm11

2
fmax

where I use a coefficient of the j-1th power of 2 to avoid theartificial peaks and troughs not only in the time domain (seeFigure 2), but also in the frequency domain (Fig. 3). Note
that, the reciprocal of the minimum duration (s1) gives asource-controlled f max as shown subsequently. smax is thetotal duration of the slip, which would be controlled by thecharacteristic size of barrier/asperity and/or the healing front(Heaton, 1990).
The slip displacement and acceleration functions areeasily obtained by replacing ḟj in equation (16) byfj(x; t)52(t/sj)2,(t # sj /2)
4 0.5 {1 ` (2t1sj)(3sj12t) /sj2} (sj /2 # t # sj),1,
(sj # t)(20)

for displacement, and5

Aj ,(t # sj /2)f̈j (x; t) 4 1Aj , (sj /2 # t # sj)0,(sj # t)(21)

for acceleration.
Figure 2 shows examples for slip velocities and displacements in the following four cases, for fmax 4 10 Hz(s1 4 0.1 sec),
Case (A) NV 4 3 and Ar 4 1.0,
Case (B) NV 4 3 and Ar 4 !2
Case (C) NV 4 5 and Ar 4 1.0,
Case (D) NV 4 5 and Ar 4 !2.
The total slip durations (smax) for NV 4 3 and 5 are 0.4
and 1.6 seconds, respectively. The velocity for the smaller
NV and Ar shows the sharper and larger peak velocity. The
velocities for Ar 4 !2 decay as 1/!t, which is consistent
with Kostrov’s slip velocity. Although it is not shown, theslip velocity for Ar 4 2 is a scalene triangle, which seems
unrealistic as compared with the Kostrov-type slip velocity.
On the other hand, results for Ar , 1.0 show unrealisticallylarge slip velocities and accelerations for f max 4 10 Hz.
Thus, Ar probably should take a value between 1 and 2.
The Fourier transform of equation (16) can be obtained
analytically:N2m
sin vj1
Fm(x; x) 4 o Aj11
exp(ivj) ,
r
A j41
vj

5

6

xsj
and vj 4
4

(22)

391

A Theoretical Omega-Square Model Considering the Spatial Variation in Slip and Rupture Velocity

Figure 2.

The slip velocities (thick black lines) and displacements (thick gray lines)
for four cases. The slip velocities consist of equilateral triangles with different durations(thin lines). The reciprocal of the minimum duration among triangles generates thesource-controlled fmax. The maximum duration smax among triangles equals to the total
slip duration at a point on a fault plane, and the reciprocal of the double of smax isnearly equal to the first corner frequency f1 (see Fig. 3).

Note that the Fourier amplitude of each triangle has an x12
decay.
Figure 3 shows the Fourier amplitudes of the slip velocities and accelerations corresponding to those in Figure
2. The amplitudes of all the slip velocities have two cornerfrequencies; f 1 ' 1/2smax and f max. The spectra are constantfor frequencies lower than f 1, fall off as the inverse of x
between f 1 and f max and then fall off as the inverse of xsquared at frequencies greater than f max. It is easy to control
the spectrum shape and amplitude by changing these frequencies. In addition, the spectra do not exhibit artificial

sharp peaks and troughs at frequencies lower than f max,which would be advantageous for predicting strong motions.
Source Spectra Considering Variable Rupture Velocity
To investigate the effect of the variable rupture velocity
on the source spectra, I evaluate the source spectra for Dtr? 0, by assuming temporarily the instantaneous slip (FV (x,x) 4 1) in equation (6). The source spectra can beS(x) 4# D(x) exp5ix1V¯ C ` Dt 26dx.
L

x

0

r

r

d

(23)

392

Y. HisadaFigure 3.

The Fourier amplitude spectra of the slip velocities for the four cases
shown in Figure 2. The Fourier spectra of the slip accelerations are also shown in graylines. Note that the velocity spectra fall off as the inverse of x between f1 and fmax.

Here I use the simplest k-squared slip distribution,
namely, the equilateral triangle,
D(x) 42D x/L,(0 # x # L/2)
(24)
52D
(L 1 x)/L, (L/2 # x # L)mm

decay at high frequencies; this would be the simplest ksquared model.
In order to evaluate equation (23) for Dtr ? 0, I subdivide the fault length L into small segments DLj (see Figure1), and carry out the analytic integration on each subfault,
NL

where Dm is the maximum slip at the center. For the constantrupture velocity (Dtr 4 0), the analytical solution can beobtained by substituting equation (24) into equation (23),LDm sin vs ivs 2
S(x) 4
e
2
vs

5

6

xL
with vs 4 ¯
4VrCd

S(x) 4 o

j41

xj

#

xj11

5 1

26

¯ j exp ix x ` Dtrj dx,
DV¯ rCdNL¯ jDLj sin vj exp ix x̄j ` Dtrj ,
4 o D
j41
vj
V¯ rCd

5 1

26

(26)

xj11 ` xj
.
2

(27)

(25)

Note that the Fourier amplitude of equation (25) has the x

12with vj 4

xDLj,
2V¯ rCdand x̄j 4

393

A Theoretical Omega-Square Model Considering the Spatial Variation in Slip and Rupture Velocity

where NL is the total number of subfaults. Note that thesource spectrum S is the superposition of the functions withthe x11 decay.
As an example, I will use the following parameters tocompute the source spectra up to 20 Hz: L 4 10 (km), andVr 4 2.5 (km/s).
Dm 4 1 (m), NL 4 2000, Cd 4 1, and ¢First, I check the very basic effects of Dtr using thefollowing two simple cases:
1. As shown in the left of Figure 4a, the rupture stops
abruptly at the center of the fault for Dt second, thencontinues to rupture. Thus, Dtr is expressed as,
Dtr 4

50,Dt, (0(L/2##x x##L/2)L)(28a)

2. As shown in the right of Figure 4a, the rupture stops andstarts two times. Thus, Dtr is expressed as,
Dtr 4

50,Dt, (0(L/4##x x##L/4,L/2,andandL/23L/4# x##x 3L/4)
# L)
(28b)

Figure 4b and 4c show the source spectra and theirwaveforms (i.e., the source time functions), respectively, forcases 1 and 2. The amplitudes of the source spectra are multiplied by x, and thus the constant amplitude represents thex11 decay. The source time functions in Figure 4c correspond to the displacement waveforms in the far-field. For
reference, I plot the source spectra and the source time functions of the k-squared model using equation (25) by graylines (Case 0; Dt 4 0) in the figures. For Case 1, the abrupt
stop at the center of the fault for Dt second excites large
high-frequency waves, and its spectrum shows the x11 decay with frequency. The source time function shows the extremely strong stopping phases, which seem unrealistic.
Similarly, the spectrum for case 2 shows the x11 decay athigh frequencies, but note that the maximum amplitudes aretwice as large as those of case 1. The waveform shows theunrealistically strong stopping and starting phases at threelocations. The abrupt stops and starts along the whole rupture front cause mathematical singularities and the results
are physically unacceptable.
Next, I investigate the following more realistic threemodels considering smooth Dtr distributions: the k-inversed,
k-squared, and k-cubed models.
The k-inversed model for Dtr . The distribution of Dtr isassumed to be inversely proportional to the spatial wavenumber k. I construct this k-inversed model by superposing
the cos functions, as follows.
M12 (28c)Dtxcos 2pm ` hm
2L
m41 !1 ` m

Dtr(x) 4 o

where hm is a random phase between 0 and 2p (radians), m
is the wavenumber normalized by the fault length, and M is¢r /L gives the limit of resolution
the maximum value of m. MV
in frequency.
The k-squared model for Dtr . The Dtr distribution is assumed to be inversely proportional to k-squared. Similar to
the k-inversed model, I construct the model by superposing
the cos functions, as follows.
M

Dtr(x) 4 o

m41

Dt

!1 ` m4

1

cos 2pm

2 (28d)x` hm
L

The k-cubed model for Dtr . The Dtr distribution is assumed
to be inversely proportional to k-cubed. Similarly, I construct
the model as follows.
M

Dtr(x) 4 o

m41

Dt

!1 ` m6

1

cos 2pm

2 (28e)x` hm
LFigure 5 shows the amplitudes of the k-inversed, k-squared,
and k-cubed models by normalizing by Dt.
Figure 6a shows examples for Dtr distributions using
the k-inversed, k-squared, and k-cubed models. I use Dt 4
¢r 4 4 sec0.1(sec), which is 1/40 of the total duration (L/V
ond), and M 4 500 in equations (28c) to (28e), together
with the same source parameters in cases 1 and 2. Figure 6bshows the source spectra corresponding to the three models
in Figure 6a. Similarly to Figure 4b, the amplitudes of thesource spectra are multiplied by x. The high-frequency amplitudes are nearly constant for the k-inversed model, fall off
as the inverse of x for the k-squared model, and fall off as
the inverse of the x-squared for the k-cubed model. Figure
6c shows the source time functions for the three models. For
reference, I plot the functions for Dt 4 0 using equation(27) by gray lines in the figure. The waveform of the kinverse model shows unrealistically strong and incoherent
high-frequency waves; it is hard to identify the coherent
waveform of the gray line. In contrast, the source time function for the k-squared model shows the high-frequency fluctuations around the coherent waveform, which seems realistic. The functions of the k-cubed model does not show
incoherent high-frequency fluctuations; the smooth fluctuation corresponds to the sin-type Dt seen in Figure 6a. Basedon general expectations, the result of the k-squared modelseems the most plausible.
Finally, I investigate the effect of Dtr on the k-squaredmodel by changing the amplitude Dt in equation (28d). Figure 7a shows the source spectra for Dt 4 0.1, 0.2, 0.4, and
0.8 seconds, which correspond to 1/40, 1/20, 1/10, and 1/5of the total duration, respectively. The Fourier spectra increase with increasing Dt at high frequencies, and show x11
decay except for Dt 4 0.8. The high-frequency spectrum for
Dt 4 0.8 is independent of x. Figure 7b depicts the source
time functions, which show the larger high-frequency fluc-

394

Y. HisadaFigure 4.

The incoherent rupture time Dtr for two cases (top), and their source spectra (middle) and corresponding source time functions (bottom). The rupture stops
abruptly at the center of the fault for Dt second for case 1, and the rupture stops andstarts twice for case 2. The amplitudes of the source spectra are multiplied by x, andthus the constant amplitude represents the x11 decay. Gray lines indicate the case forthe coherent rupture time (Case 0; Dt 4 0).

A Theoretical Omega-Square Model Considering the Spatial Variation in Slip and Rupture Velocity

395

Figure 5.

The normalized amplitudes of the
k-inverse, k-square, and k-cube models for the
Dtr distributions.

tuations for the larger Dt. In particular, the functions for Dt
4 0.4 and 0.8 show unacceptably incoherent fluctuations.
Thus, a Dt between 1/40 and 1/20 of the total duration may
be appropriate.

Discussion
From the results presented here, it appears that the mostappropriate x-squared model consists of the slip-velocity
model with k-squared spatial distributions in slip and incoherent rupture time, Dtr . I call this model the x-inversesquared model, because the x-squared source spectra consistof the two x-inverse models due to the slip velocity and thespatial variations, as discussed below.
Figure 8a and 8b shows schematic explanations formodeling the x-squared source spectra using the k-squaredmodel of Bernard et al. (1996) and by the x-inverse-squaredmodel, respectively. There are two main differences between
the two models. The first is the modeling of slip velocity.
The slip velocity of the former consists of the scale-dependent rectangular boxcars. In addition, this model allows theduration of boxcars to be infinitely small at the highest frequency; this is equivalent to the Dirac delta function, whose
Fourier amplitude is constant with x. Consequently, theFourier amplitude of the slip velocity becomes independentof x. On the other hand, the slip velocity of the x-inversesquared model consists of triangles. As discussed earlier, thefinite acceleration amplitudes of the triangles are physicallymore appropriate than the boxcars. In addition, this model
introduces a minimum duration among the triangles, whose
reciprocal represents the source-controlled f max. Thus, theFourier amplitude of our slip velocity model falls off as theinverse of x up to f max, and as the inverse of x-squared at
frequencies higher than f max. Even though it is still a matterfor argument whether f max is caused by the source or site
effects (Hanks, 1979, 1981; Anderson and Hough, 1984; Papageorgiou and Aki, 1985), it seems natural to have a cornerfrequency controlled by smoothness in the slip velocity; otherwise the slip acceleration could be unacceptably large. The
x11 decay of the source spectra of the slip velocity agrees
with the empirical model derived by Dan and Sato (1999),
although the latter is restricted to rather longer periods.
The other difference between the k-squared model andthe x-inverse-squared model is the modeling of rupture velocity. The rupture velocity of the former is constant,whereas the latter is variable. As discussed earlier, the variable rupture velocity is more realistic, and it generates highfrequency waves efficiently. To model those effects, I introduced the incoherent rupture times Dtr, and found that thek-squared model for Dtr distribution seemed the most plausible among the k-inversed, k-squared, and k-cubed models.
The similar k-squared distributions for slip and Dtr may suggest some physical relationship between them.
As shown in the Figure 8b, the source spectrum of thex-inverse-squared model falls off as the inverse of x under
the assumption of the instantaneous slip (delta function as
velocity) with the k-squared slip distributions. Since thesource spectrum is expressed by the convolution of the slip
velocity and the source spectrum with instantaneous slip, thesource spectrum of this model falls off as the inverse of xsquared up to f max (namely, the x-squared model).
Finally, Figure 9 shows the comparison of source acceleration spectra among the k-squared model, the xinverse-squared model, and an empirical x-squared modelusing the same source parameters in the previous section (LVr 4 2.5 km/
4 10 km, Dm 4 1 m, Cd 4 1, NL 4 2000, ¢s, M 4 500, and fmax 4 10 Hz). All the spectra are normalized by the seismic moment. For the k-squared model, I
use equations (13) and (15) with a 4 0.5 and L0 4 0.2L (abroad-pulse model; Bernard et al., 1996), whose corner fre¢r /L0) is 0.625 Hz. For the x-inverse-squared
quency (f 0 4 aV
model, I show one example among many choices for the
combinations for the slip velocities and the Dtr distributions,because the all results show similar x-squared source spec-

396

Y. HisadaFigure 6.

(a) The incoherent rupture time Dtr for the k-inversed, k-squared, and kcubed models, and (b) their source spectra and (c) the source time functions. The
amplitudes of the source spectra are multiplied by x, and gray lines indicate the casefor the coherent rupture time (Dt 4 0).

tra. The parameters for our model are Ar 4 !2, Nr 4 4(smax 4 0.8 sec and f 1 40.63 Hz), and Dt 4 0.2 sec. As
for the k-squared Dtr distributions, I generate one hundred
sets of random phases in equation (28d), and show the average of source spectra with the standard deviations in the
figure. For the empirical x-squared model, I use Brune’smodel (1970) with f max 4 10 Hz (Boore, 1983) and thecorner frequencies fc 4 0.147 Hz (for Dr 4 50) and 0.234
Hz (for Dr 4 200) under the values of W 4 L/2, Vs 4 3.8
km/sec, and Mo 4 1.01 2 1026 dyne cm. As compared with
Brune’s model, the k-squared model and the x-inverse-

squared model show rather larger amplitudes at lower andhigher frequencies, respectively. However, all the spectra
shows similar x-squared amplitudes at high frequencies,
overall.
In summary, I believe that no matter which models areused, the far-field acceleration waveforms would be similar.
However, the broad-band strong motions, especially in nearfield, would be greatly different among the various models;the unphysical assumptions may cause implausible results.
The advantage of the x-inverse-squared model is that it hasa clear physical basis for the slip velocity and the spatial

A Theoretical Omega-Square Model Considering the Spatial Variation in Slip and Rupture Velocity

397

Figure 7.

(a) The source spectra of the ksquare models for Dtr for the cases of Dt 4
0.1, 0.2, 0.4, and 0.8 seconds, and (b) the corresponding source time functions. The amplitudes of the source spectra are multipliedby x.

variations in the slip and rupture time. This model could
simulate the broadband strong motions from the near-field
displacements (coherent and low-frequency waves) to the
far-field accelerations (incoherent and high-frequency
waves).

Conclusions
I investigated the theoretical basis of the x-squaredmodel by modifying the k-squared model of Bernard et al.
(1996), and proposed the x-inverse-squared model by considering the spatial variation in slip distribution, slip velocity, and rupture velocity. For the slip velocity, I proposed a
Kostrov-type model by superposing equilateral triangles,
where the two corner frequencies are introduced. The firstcorner frequency f 1 corresponds to the slip duration, while
the second is the source-controlled f max. The Fourier ampli-

tude of this model falls off as the inverse of x between f 1
and f max. On the other hand, for modeling the effects of thespatial variation in rupture velocity, I introduced the incoherent rupture time (Dtr), namely, the difference between theactual rupture time and the coherent (average) rupture time.
After checking various models including the k-inversed, ksquared, and k-cubed distributions for Dtr, I found that thek-squared model was the most plausible. I also found thatits source spectrum fell off as the inverse of x, when the slip
velocity is Delta function. Finally, I proposed the x-inversesquared model, which consists of the combination of the slip
velocity proposed earlier and the k-squared distributions inslip and Dtr, and confirmed that it is consistent with the empirical x-squared models. Since this model provides thetheoretical basis for constructing realistic source models at
broadband frequencies, it would be useful for predicting
strong ground motions including the near-field.

398

Y. HisadaFigure 8.

Schematic explanations for modeling of the x-square model using (a) thek-square model (Bernard et al., 1996) and (b) the x-inverse-squared model proposedin this article. The left figures in (a) and (b) illustrate the slip velocity models and their
Fourier spectra. The middle figures show the modeling of rupture time, and their source
spectra considering the k-squared slip distributions with the instantaneous slip. As
shown in the bottom figures, the final source spectra are obtained by the convolutionsof the slip velocity models and the source spectra with the instantaneous slip. Note thatthe x-inverse-squared model consists of the two x-inverse models.

A Theoretical Omega-Square Model Considering the Spatial Variation in Slip and Rupture Velocity

399

Figure 9.

Comparisons of the acceleration source spectra among the k-squaremodel, the x-inverse-squared model, and the empirical x-square model by Brune(1970) with fmax 4 10 Hz (Boore, 1983). As for the x-inverse-squared model, theaverage value with the standard deviation are plotted using one hundred sets of random
phases in equation (28d).

Acknowledgments
I am grateful to Susan Hough, Steven Day, and an anonymous reviewer for helpful review and comments. Comments and suggestions by
Jacobo Bielak, Sumio Sawada, and Stewart Koyanagi are also greatly improved this manuscript. This research was partly supported by a special
project of “US-Japan Cooperative Research for Urban Earthquake Disaster”
by the Ministry of Education, Science, Sports and Culture, and by Disaster
Prevention Research Institute of Kyoto University.

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Department of Architecture
Kogakuin University
Nishi-Shinjuku 1-24-2
Tokyo 163-8677, Japan
hisada@cc.kogakuin.ac.jp
Manuscript received 15 June 1999.