Bulletin of the Seismological Society of America, 91, 4, pp. 651–666, August 2001

A Theoretical Omega-Square Model Considering Spatial Variation in Slipand Rupture Velocity. Part 2: Case for a Two-Dimensional Source Modelby Yoshiaki Hisada

Abstract The theoretical basis of the x-squared model and the characteristics ofnear-source broadband strong ground motions are investigated using a 2D source
model with spatial variations in slip and rupture velocity. This is an extension of astudy by Hisada (2000a), who used 1D source models for the same purpose. First,
Hisada’s slip-velocity function (2000a) is modified by superposing scalene triangles
to construct Kostrov-type slip-velocity functions with arbitrary combinations for thesource-controlled f max and the slip duration. Then, it is confirmed that the Fourier
amplitudes of these slip velocities fall off as the inverse of x at frequencies lower
than f max (Hisada, 2000a). Next, the effects of 2D spatial distributions of slip andrupture time on the source spectra are investigated. In order to construct a realistic
slip distribution, the hybrid slip model is proposed, which is the combination of theasperity model at lower wavenumbers and the k-squared model (Herrero and Bernard,1994) at higher wavenumbers. The source spectra of the proposed 2D models, whichhave the k-squared distribution for slip and rupture time, fall off as the inverse of x,
when the slip is instantaneous. This result also agrees with Hisada (2000a). Therefore,
the x-inverse-squared model, which consists of the combination of the Kostrov-typeslip velocity proposed here and the k-squared distributions for both slip and rupture
time, is not only consistent with the empirical x-squared model, but also provides
the theoretical basis for constructing realistic 2D source models at broadband frequencies. In addition, it is confirmed that the proposed source model successfully
simulates most of the well-known characteristics of the near-fault strong groundmotions at broadband frequencies, that is, permanent offsets in displacements, longperiod pulses in velocities, and complex randomness in accelerations. The nearsource directivity effects are also confirmed; the fault-normal components are dominant over the fault-parallel components, especially at the forward rupture direction.
However, the ratio between the fault-normal and fault-parallel components is roughly
independent of frequency, which is contradictory to empirical models. This suggeststhat a 3D faulting model is necessary to represent more realistic near-source strong
motions at broadband frequencies.
Introduction
Hisada (2000a) proposed the x-inverse-squared modelby modifying the k-squared model of Herrero and Bernard(1994) and Bernard et al. (1996) to construct x-squared
source models (Aki, 1967, 1972). The k-squared model isbased on the three assumptions that (1) the spatial wavenumber spectrum of the slip distribution falls off as the inverse of the wavenumber squared (k-squared), (2) the Fourier amplitudes of the slip velocity are independent of x at
high frequencies, and (3) the rupture velocity is constant.
Hisada (2000a) modified the last two assumptions. First,
as a more realistic slip velocity, a Kostrov-type slip velocity
model (Kostrov, 1964; Nakamura and Miyatake, 2000),

which has a sharp rise and a relatively smooth decay, wasproposed by superposing isosceles triangles with different
durations (I used the term equilateral triangles in Hisada(2000a) by mistake). After checking the Fourier amplitudes
of various slip models, Hisada (2000a) found that (1) thismodel had two corner frequencies, the first (f 1) corresponding to the total slip duration (roughly the reciprocal of twice
the total duration) and the second to the source-controlled
f max (roughly the reciprocal of the minimum duration among
triangles), and (2) the Fourier amplitude of the slip velocity
fell off as the inverse of x between f1 and fmax.
Second, in order to investigate the effects of variable
651

652

Y. Hisadarupture velocity on source spectra, Hisada (2000a) introduced the incoherent rupture time (Dtr), namely, the difference between the actual rupture time and the coherent
rupture time. Using 1D unilateral source models, Hisada(2000a) found that (1) the k-squared distribution for Dtr was
the most plausible among k-inversed, k-squared, and k-cubed
distributions, and (2) the source spectra fell off as the inverseof x when combining those models with the instantaneous
slip and the k-squared slip distribution.
Finally, Hisada (2000a) proposed the x-inverse-squaredmodel, which consists of the combination of the two xinverse models, the slip velocity proposed previously andthe k-squared spatial distributions in slip and Dtr, and confirmed that it was consistent with the empirical x-squaredmodel.
In this article, I improve my previous model (Hisada,
2000a) using 2D source models and more flexible slipvelocity functions. First, I briefly introduce the formulationof the source spectra using 2D fault models. Second, I generalize Hisada’s slip-velocity model by superimposing scalene triangles, and I check its Fourier amplitude. I introduce
empirical relations to find appropriate relations among theparameters for constructing the slip velocities. Third, I investigate the effects of the spatial variation of slip and rupture time on the source spectra using 2D source models.
Fourth, I simulate broadband strong motions near a fault tocheck waveforms (accelerations, velocities, and displacements) and, especially, the directivity effects. The main effect of the rupture directivity is the increase of the longperiod ground motion in the direction normal to the fault
plane (e.g., Aki, 1968). However, empirical models show
this effect disappears at intermediate to high frequencies(e.g., Somerville et al., 1995). Thus, it is interesting to see
whether the proposed source model explains this phenomenon or not. Finally, I discuss the validity and the merits anddemerits of the x-inverse-squared model.

Formulation of 2D Source Models
and Source Spectra

W

L

0

0

冮 冮 {lD(e nk j* • exp(ix • tr)}dx1dx2 ,Ⳮ ej nk)Uik,jGeometry of a 2D source model and thecoordination system, together with an observation
point.

(2000a), the far-field displacements in a homogeneous fullspace can be simplified as follows:
Ui (Y; x) ⳱Ri
M0(x),
4pqrc3(2)where the subscript i denotes the ith component in the spherical coordinate system, and Ri is the radiation pattern for theith component. The variable c is the medium velocity (VP
for the radial and VS for the transverse and vertical directions), q is the density, and r is the distance from the rupture
front (x1, x2) to Y. M0 in equation (2) is the source spectrum
and is expressed as
M0(x) ⳱ l

The displacements (Ui ) at an observation station (Y)from a source model can be written as follows using a 2D
source model (e.g., Aki and Richards, 1980),
Ui (Y; x) ⳱Figure 1.

W

L

0

0

冮 冮 D(x , x )F (x , x ; x)
1

2

V

1

2

(3)

exp{ix(tc Ⳮ tr)}dx1dx2 ,where FV is the Fourier transform of the slip-velocity function, whose integration corresponds to the unit dislocation,and tc is the arrival time of the seismic waves from the rupture front to the observation point.
The arrival time and the rupture time are given as(1)rcr0 ⳮ x1 sin h cos u ⳮ x2 • cos h(x1, x2 K r0),ctc(x1, x2) ⳱
where Y is an observation point, x is the circular frequency,
W and L are the width and length of the fault, respectively,
l is the rigidity (⳱ 冪VS /q), D is the slip distribution, ek and*nj are the slip- and fault-normal vectors, respectively, Uik,jis the derivative of Green’s function along the jth direction,and tr is the rupture time (see Fig. 1). Following Hisada

(4)
and

653

A Theoretical Omega-Square Model. Part 2: Case for a Two-Dimensional Source Modeltr(x1, x2) ⳱withrSⳭ Dtr(x1, x2),
Vr

(5)

rS(x1, x2) ⳱ 冪(x1 ⳮ xS1)2 Ⳮ (x2 ⳮ xS2)2,(6)

where r0 is the distance from the origin of the fault plane toY, h and u are the location of Y in spherical coordinates, ¢Vris the constant rupture velocity, and xS1 and xS2 are the locations of the hypocenter (the starting point of the rupture).
Following Hisada (2000a), I divide the rupture time tr into¢r) and the incoherent rupture
the coherent rupture time (rs /Vtime (Dtr). I introduce Dtr to represent the fluctuations at
rupture front due to the variable rupture velocity.
Substituting equations (4), (5), and (6) into equation (3),the following source spectrum is obtained:冢r
M0(x) ⳱ l exp ix 0c

冣 冮 冮 D(x , x )F (x , x ; x)
W

L

0

0

1

2

V

1

2

r
Ⳮ Dt 冣冧dx dx ,冦 冢VrCexp ix

S

r

1

(7)

2

where NV is the total number of triangles, and Ar is the ratioof the area of the j Ⳮ 1th triangle with respect to the areaof the jth triangle (i.e., Ar ⳱ AjⳭ1/Aj).
In equation (9), ḟj is the scalene triangle of the jth element,冦

A1j • t,ḟj (t) ⳱ A2j • (sj ⳮ t),
0,

with

(0 ⱕ t ⱕ s1j)
(s1j ⱕ t ⱕ sj),
(sj ⱕ t)(10)

A1j ⳱ Vj /s1j , A2j ⳱ Vj /s2j and Vj ⳱ 2/sj ,(11)where A1j, A2j, and Vj are the absolute values of the maximum and minimum accelerations and maximum velocity of
the jth triangle, respectively. The variables s1j, s2j, and sj are
the first-half, the second-half, and the total durations of thejth triangle, respectively. They are defined as follows:dwhere, Cd is the directivity coefficient of the 2D source
model,
Cd ⳱

1
.
¢
x2
Vr x1
1ⳮ
cos h Ⳮsin h cos u
c rS
rS

冢

冣

Following Hisada (2000a), I express the slip-velocity
function (Fv) as a function with a sharp rise and a relatively
smooth decay, which can be obtained by superimposing triangles with different durations. The combination of triangles
from short to long durations guarantees broadband frequency contents. While Hisada (2000a) used isosceles triangles as element triangles for the simplicity, I will generalize the formulation using scalene triangles to utilize
arbitrary combinations of slip durations and f max.
In the time domain, the slip-velocity function superposing scalene triangles can be written as follows (see Fig. 2):
NV

兺

NV

with A ⳱兺 Ar jⳮ1,

j⳱1

Tr NVⳮ1fmax(13)

with smin  s1 ⳱Fourier Amplitudes of Slip-Velocity Functions

1
Ar jⳮ1 • ˙fj(t)
A i⳱1

(12)

(8)

When h ⳱ 90, equation (8) becomes the directivity coefficient of the 1D source model (Ben-Menahem, 1961).
As seen in (7), the source spectra are controlled by thespatial distributions of the static slip, the slip-velocity function, and the rupture time. In the following, I first check the
Fourier amplitudes of the slip velocity function and then theeffects of the spatial variations of the slip and the rupture
time on the source spectra.

FV(x1, x2; t) ⳱

si
, and
Tr
s2j ⳱ sj ⳮ s1j, (j ⳱ 1, 2, 3, . . . , NV)

sj ⳱ smin • Tr jⳮ1, s1j ⳱

(9)

1
fmax

, and smax  sNV ⳱

where Tr is the ratio of sjⳭ1 with respect to sj and is also
the ratio of sj to s1j (i.e., Tr ⳱ sjⳭ1/sj ⳱ sj /s1j; see Fig. 2).
The reciprocal of the minimum duration (s1) corresponds tothe source-controlled f max (Hisada, 2000a), and smax is thetotal duration of the slip. For the case of the isosceles triangle, Tr is 2 (see equation 19 of Hisada [2000a]).
The slip displacement and acceleration are obtained byreplacing ḟj in equation (9) by
t2
,
sjs1j

fj (t) ⳱冦(0 ⱕ t ⱕ s1j)

s1js2j Ⳮ (tⳮs1j )(s2jⳭsjⳮt) (s1j ⱕ t ⱕ sj),,
sjs2j
(sj ⱕ t)1,

(14)

for displacement, and

冦

A1j ,
f̈j (t) ⳱ ⳮA2j ,0(0 ⱕ t ⱕ s1j )(s1j ⱕ t ⱕ sj )(sj ⱕ t)(15)

for acceleration.
The Fourier transform of equation (9) can be obtained
analytically:

654

Y. HisadaFigure 2.

The slip-velocity function (black line) and the corresponding slip displacement(thick gray line). The velocity function consists of several isosceles triangles with various
durations (six triangles in thin lines in this case).
N

FV (x1, x2; x) ⳱

1 VAr jⳮ1 • ˙fj(x),
A j⳱1

兺

(16)

(22)

On the other hand, Somerville et al. (1999) derived thesimilar relations using more recent source inversion results,
as follows:whereḟj (x) ⳱ i

withsmax (sec) ⳱ 0.0726L (km)冢冣

exp(iv1j) sin v1j
sin v2jⳮexp(ivj) ,
vj
v1j
v2j

x • sjx • s1jx • s2j
vj ⳱, v1j ⳱, v2j ⳱
,
2
2
2

(17)

D̄(cm) ⳱ 1.56 • 10ⳮ7 • M01/3,(23)

smax (sec) ⳱ 2.03 • 10ⳮ9 • M01/3

(24)

(18)

and i is the imaginary number.
Although the parameters f max (⳱1/s1), Ar, Tr, and NV
in the slip-velocity function can be arbitrary, I roughly estimate their relations for a given magnitude by considering
empirical relations and then check the Fourier amplitudes ofthe plausible slip velocities.
First, according to Sato (1979), who compiled thesource parameters of Japanese earthquakes, the following
empirical relations were obtained between magnitude, M,
and fault length, L, and between M and average slip, D̄:
log10 L(km) ⳱ 0.5M ⳮ 1.88,(20)¯
log10 D(cm)⳱ 0.5M ⳮ 1.4 .

(21)

In addition, Geller (1976) obtained the following empirical
relation between fault length and slip duration (rise time).

where M0 is the seismic moment. Therefore, when we combine the previous equations with a magnitude–seismic moment relation, such as that of Hanks and Kanamori (1979),M⳱2
log10 M0 ⳮ 10.7,
3(25)

we obtain the other set of empirical relations among magnitude, the average slip, and the slip duration.
The left columns in Table 1 show relations for various
M values vs. the corresponding combinations of smax and D̄
using equations (20)–(22) by Sato and Geller and equations(23)–(25) by Somerville et al. (1999). As for f max, I tested
three cases: 10, 5, and 1 Hz. Although D̄ values are comparable between the relations by Sato (1979) and Somervilleet al. (1999), smax values obtained by Geller (1976) are about
two times larger than those obtained by Somerville et al.
(1999). This is probably caused by the different datasets andfrequency bands between the two relations; especially

655

A Theoretical Omega-Square Model. Part 2: Case for a Two-Dimensional Source Model

Table 1
Maximum Slip Velocities and Accelerations Using Various Magnitudes (M), fmax Values, and Ar Values
fmax ⳱ 10 Hz and Tr ⳱ 1.77
Sato et al.
Ms (sec)

D(cm)

4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0

0.1
0.2
0.3
0.6
1.0
1.7
3.1
5.4
9.6

4.0
7.1
13
22
40
71
126
224
398

Ar ⳱ 1.2

Somerville et al.
s (sec)

0.05
0.08
0.14
0.26
0.45
0.81
1.4
2.6
4.5

Ar ⳱ 1.4

Ar ⳱ 1.6

Ar ⳱ 1.77

Ar ⳱ 2

D(cm)

Nv

Vmax

Amax

Vmax

Amax

Vmax

Amax

Vmax

Amax

Vmax

Amax

3.5
6.2
11
20
35
62
110
196
349

1
2
3
4
5
6
7
8
9

78
87
106
127
168
225
308
429
604

1416
1580
1934
2301
3059
4090
5591
7786
10980

78
83
96
105
127
153
187
231
288

1416
1515
1738
1903
2297
2761
3369
4161
5179

78
80
88
89
99
107
118
129
443

1416
1460
1580
1603
1767
1919
2097
2307
2545

78
78
82
79
81
82
82
82
83

1416
1420
1469
1404
1443
1447
1451
1457
1463

78
76
75
68
67
67
66
66
66

1416
1373
1345
1197
1129
1029
929
836
749

Nv

Vmax

Amax

Vmax

Amax

Vmax

Amax

Vmax

Amax

Vmax

Amax

1
2
3
4
5
6
7
8

71
82
94
120
157
209
287
399

618
718
817
1047
1362
1822
2496
3475

71
79
85
100
118
142
174
215

618
689
736
870
1028
1237
1513
1869

71
76
77
84
91
99
109
120

618
665
671
735
795
865
949
1044

71
75
73
76
77
79
81
82

618
650
632
661
674
687
702
717

71
72
66
64
63
64
65
66

618
626
573
553
513
470
427
386

D(cm)

Nv

Vmax

Amax

Vmax

Amax

Vmax

Amax

Vmax

Amax

Vmax

Amax

35
62
110
196
349

1
2
3
4
5

80
89
106
133
173

ⳮ185
158
187
234
305

80
84
91
101
114

ⳮ185
148
161
178
201

80
86
96
110
130

ⳮ185
151
169
194
229

80
81
81
82
83

ⳮ185
142
143
145
146

80
78
74
70
68

ⳮ185
137
131
123
113

fmax ⳱ 5 Hz and Tr ⳱ 1.74
Sato et al.
Ms (sec)

4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0

0.2
0.3
0.6
1.1
1.8
3.2
5.6
9.7

D(cm)

7.1
13
22
40
71
126
224
398

Ar ⳱ 1.2

Somerville et al.
s (sec)

0.08
0.14
0.26
0.45
0.81
1.4
2.6
4.5

D(cm)

6.2
11
20
35
62
110
196
349

Ar ⳱ 1.4

Ar ⳱ 1.6

Ar ⳱ 1.74

Ar ⳱ 2

fmax ⳱ 1 Hz and Tr ⳱ 1.76
Sato et al.
Ms (sec)

D(cm)

6.0
6.5
7.0
7.5
8.0

1.0
1.8
3.1
5.5
9.6

40
71
126
224
398

Ar ⳱ 1.2

Somerville et al.
s (sec)

0.45
0.81
1.4
2.6
4.5

Ar ⳱ 1.4

Ar ⳱ 1.6

Ar ⳱ 1.76

Ar ⳱ 2

The average slip (D) and the slip duration (s) are empirically derived by Sato (1979) and Geller (1976). As a reference, the more recent results bySomerville et al. (1999) are also shown. The lightly shaded areas indicate the maximum values within realistic ranges. In particular, the bold values in theheavily shaded areas indicate values used to calculate Fourier amplitudes of the slip accelerations, as shown in Figure 4.

Geller’s relations were based on mainly longer-period seismograms from large-scale offshore earthquakes, whereas
those of Somerville et al. (1999) were based on mainly
shorter-period strong motions from small- to medium-scaleinland earthquakes. As shown in Figures 2 and 3, the slipvelocity functions used in this study are broadband; the beginnings of the slips excite the high-frequency waves, andthe subsequent slips are coda with decreasing amplitudes,
which contribute only to low-frequency waves. Therefore, I
use the relations by Sato (1979) and Geller (1976) as slipvelocity functions in this study. In this case, the values of
Tr are similar from 1.74 to 1.77 among the three f max values,
as shown in Table 1.
The right columns of Table 1 show the maximum (orminimum) values of slip velocities and accelerations. As forAr, several values between 1.2 and 2 are tested. The shaded
areas in Table 1 indicate the maximum accelerations andvelocities within realistic ranges, that is, about 50 to 200(cm/sec) for peak velocities, and 500 to 2000 (gal) for peakaccelerations. The values of 200 (cm/sec) and 2000 (gal) for
peak velocities and accelerations, respectively, may seem
too high for using the average slips. However, it should be
noted that the durations for the peak values are extremely
short, that is, instantaneous for the velocities and s11(0.5/
f max) for the accelerations. In addition, they are raw values
without material damping, scattering, and frequency-band
limits.
Figure 3a and b shows the slip velocities and the Fourier
amplitudes of the corresponding slip accelerations. Only results for f max ⳱ 5 Hz are shown because the results for all
the other cases are similar. The slip velocities are Kostrovtype functions (namely, a sharp rise and a smooth decay) forTr  Ar, scalene triangles for Tr ⳱ Ar (⳱1.74), and smoothfunctions without sharp peaks for Tr  Ar. Accordingly, theFourier amplitudes for the larger Ar show the smaller amplitudes at high frequencies. Similar to Hisada (2000a), theFourier amplitudes have two corner frequencies; the first is
nearly equal to the reciprocal of twice of the slip duration,

656

Y. HisadaFigure 3.

(a) The slip velocities and (b) the Fourier amplitudes of the corresponding
slip accelerations for the case of fmax ⳱ 5 Hz and Tr ⳱ 1.74.

657

A Theoretical Omega-Square Model. Part 2: Case for a Two-Dimensional Source Modelf1  1/2smax.

(26)

Thus, f 1  1.7, 0.45, 0.16, and 0.052 Hz for M ⳱ 5, 6, 7,
and 8, respectively. The second corner frequency is f max(⳱5 Hz).
Figure 4 shows the Fourier amplitudes of the slip accelerations of various magnitudes using the combinations of
the plausible peak velocities and accelerations, whose parameters are shown as the bold values in Table 1. Note that
their amplitudes are about flat between f 1 and f max in allcases, meaning the Fourier amplitudes of slip velocities fall
off as the inverse of x (i.e., a x-inversed model). This conclusion is the same as in Hisada (2000a).
In this article, I assume that the x-inversed model of theslip velocity is true along the whole fault plane. However,this may be modified for some cases, especially a source
model with surface faulting. The strong-motion records near
the 1999 Chi-Chi, Taiwan, earthquake provided an excellent
opportunity to investigate the slip-velocity function at various locations on the fault plane. In particular, it is interestingthat the strong motions near the surface rupture (i.e.,
TCU052 and TCU068) lack large high-frequency waves, ascompared with the regular strong motions in the epicentral
area. Seemingly, the shallower parts of the fault plane (say,less than 5 km) did not excite large high-frequency waves.
To interpret these effects using the slip model proposed here,
f max is probably very low and/or Ar may be larger than Tr.
One of the advantages of the proposed model is very easy
to include these effects by using variable f max and/or Ar depending on depth.

Figure 4.

Source Spectra Considering Variable Slipand Rupture Velocity
In order to investigate the effect of the 2D spatial variation of slip and rupture velocity on the source spectra, I
evaluate the source spectra of 2D sources by assuming temporarily the instantaneous slip (Fv(x1, x2, x̄ ) ⳱ 1) in equation(7). Neglecting the term lexp(ixR0 /c) in equation (7), thesource spectra can be expressed as:
S(x) ⳱

冮 冮 D(x , x ) exp冦ix冢VrC Ⳮ Dt 冣冧dx dx .
W

L

0

0

rs

1

2

r

1

2

(27)

d

In the following, I will check the source spectra using various types of spatial variation of slip and rupture velocity.
First, I will assume nondirectivity cases (i.e., h ⳱ u ⳱ 90
in equation 8) and then check its effects on the source
spectra.
Source Model with Rectangular Asperities
and Constant Rupture Velocity
One of the simplest asperity model would be a slip
model consisting of rectangular subfaults with locally constant slips, which is commonly used in source inversion studies. As an example, I use hereafter the inversion result of
the 1995 Hyogoken-Nanbu, Kobe, earthquake obtained by
Yoshida et al. (1996). Although this earthquake consisted oftwo main faults (the Kobe- and Awaji-side faults), I use only
the Kobe-side fault for simplicity. Figure 5 shows the slip
distribution of the Kobe-side fault; the fault length and width

The Fourier amplitudes of the slip accelerations of various magnitudes
using the combinations of the plausible peak velocities and accelerations, whose parameters are shown as the bold values in Table 1. Note that the velocity spectra fall
off as the inverse of x between f1 and fmax.

658

Y. HisadaFigure 5.

The slip distribution of the Kobe-side
fault model by Yoshida et al. (1996). The fault length
and width are 36 and 16 km, respectively, and thesubfault size is 4 ⳯ 4 km2.

are 36 and 16 km, respectively, and the subfault size is 4 ⳯
4 km2. I will assume here that the rupture velocity is constant(Vr ⳱ 2.8 km/sec and Dtr ⳱ 0).
To compute the source integral in equation (27), I will
use two different models. One model uses one integration
point per subfault (model 1a); this simple integration scheme
is still widely used not only in source inversion studies, butalso in strong-motion simulations. The integration can bedone as follows:
NL

S(x) ⳱NW兺 兺 Dij exp冦ix VrSij冧DLDW,ri⳱1 j⳱1(28)

where, NL and NW are the numbers of subfaults along thelength and width (NL ⳱ 9 and NW ⳱ 4 in this model), Dij
is the slip of the (i, j) subfault, and rSij is the distance fromthe hypocenter to the center of the (i, j) subfault. Figure 6ashows the rupture time distributions for model 1a. Since therupture velocity is infinite within subfaults, sharp discontinuities exist among adjacent subfaults in this model.
The other model uses much denser integration points torepresent a continuous rupture front (model 1b),
NL

NW

冦

ML

MW

冢 rVr 冣冧DLDW, (29)S(x) ⳱ 兺 兺 Dij 兺 兺 exp ix
i⳱1 j⳱1k⳱1 l⳱1

Sijkl

where ML and MW are the numbers of the divided integration
points on subfaults, and rSijkl is the distance from the hypocenter to the integration points. Figure 6b shows the rupture
time distributions of model 1b, where ML ⳱ MW ⳱ 40 are
used. Since the rupture velocity is constant and dense integration points are used, the rupture time is smoothly distributed in this model.

Figure 6.

The rupture time distributions for (a)model 1a and (b) model 1b. In model 1a, the integration is carried out by one point per sub-fault. Since
the rupture velocity is infinite within subfaults in thisscheme, sharp discontinuities exist among adjacentsubfaults. On the other hands, model 1b uses muchdenser integration points to represent a continuousand smooth rupture front.

Figures 7 and 8 show the source spectra and source–
time functions of models 1a and 1b, respectively. The amplitudes of the spectra are multiplied by x, and thus theconstant amplitude represents a xⳮ1 decay. The spectrum ofmodel 1a is roughly independent of x showing extremely
strong high-frequency amplitudes. The corresponding
source–time function (i.e., the far-field displacement) shows
unrealistically fluctuated high-frequency waveforms. On theother hand, the spectrum of model 1b falls off as xⳮ1. However, its source–time function shows stopping and starting
phases generated at the edges of subfaults, which also seems
unrealistic.
To summarize, both models seem far from reality; theslip should not be discontinuously distributed as shown in
Figure 5. Therefore, one should be careful in using this typeof simple asperity models (i.e., rectangular subfaults withlocally constant slips). Next I will check source models withmore realistic and continuous slip distributions.

659

A Theoretical Omega-Square Model. Part 2: Case for a Two-Dimensional Source Modelbutterworth function with random phases (e.g., Somerville,
et al., 1999),
N

D(x1, x2) ⳱Mcos冢2p兺 兺
22 2n⳱1 m⳱1 1 Ⳮ (m Ⳮ n )
D̄

冪

冣

冢

冣

x1
x2
• m • Ⳮ hmn cos 2p • n • Ⳮ hn ,L
W

Figure 7.

The source spectra of model 1a (blackline) and model 1b (gray line). The amplitudes of thespectra are multiplied by x. Therefore, the spectrumof model 1a is roughly independent of x, whileas thatof model 1b shows a xⳮ1 decay.

Source Model with k-Squared Slip Distribution
and Constant Rupture Velocity
I will check the source spectra of the 2D k-squaredmodel with a constant rupture velocity, where its spatial
wavenumber spectrum of the slip distribution falls off as theinverse of the wavenumber squared (k-squared) (Herrero andBernard, 1994).
Here, I propose a hybrid k-squared slip model in orderto obtain a realistic slip distribution for the Kobe earthquake
model (Fig. 9). First, at wavenumbers lower than the Nyquist
wavenumbers of the subfault dimensions (k ⱕ 1/2DL for thelength and k ⱕ 1/2DW for the width), I use a smooth slip
distribution obtained from the source inversion result. Since
the original Kobe model has an unrealistic discontinuous slip
distribution (Fig. 9a), the slip is linearly interpolated (Fig.
9b), then smoothed out using a bicubic Spline intepolation(Fig. 9c).
Second, at wavenumbers higher than the Nyquist wavenumbers, I use the k-squared slip model based on the 2D(30)

where n and m are the wavenumbers normalized by the fault
length and width, respectively, hmn and hn are random phases
between 0 and 2p (radians), and D̄ is the adjustable static
slip; the total sum of the final slip over the fault plane (Fig.
9d) must be the same as the sum of the original slip (Fig.
9a) to conserve the seismic moment. The variables M and N¢r /L and NV¢r /are the maximum values for m and n, and MV
W estimate roughly the limits of resolution in frequency.
Finally, the hybrid k-squared slip model is obtained bysuperposing the two models, as shown in Figure 9d. The
source spectra of this model are calculated as follows:
NL

NW

S(x) ⳱ 兺 兺i⳱1 j⳱1MLMW冦 兺 兺 D exp冢ix rVr 冣冧DLDW, (31)Sijkl

k⳱1 l⳱1

ijkl

where Dijkl is the slip at the (i,j,k,l) integration point.
The thick black line in Figure 10, as indicated by Dt ⳱
0.0 sec, shows the source spectrum of the hybrid model as¢r ⳱ 2.8 km/sec). I
suming the constant rupture velocity (Vgenerated one hundred sets of the k-squared model modifying the random phases in equation (30) and show theiraverage values in the figure. The amplitudes of the spectrum
are multiplied by x, and thus, the spectrum of this model
falls off as the inverse of x-squared, as expected by theoriginal k-squared model (Herrero and Bernard, 1994).
Therefore, when I combine this slip model with the slipvelocity model having the xⳮ1 decay, as shown earlier, thefinal source spectrum falls off as the inverse of x-cubed. The
reason for the lack of high-frequency waves should be due
to having neglected the spatial variation of the rupture velocity (or the rupture time) (Hisada, 2000a).

Figure 8.

The source–time functions of(a) model 1a and (b) model 1b. Model 1ashows extremely strong high-frequency waves,
whereas model 1b shows stopping and starting
phases generated at the edges of subfaults.

660

Y. HisadaFigure 9. The construction of the hybrid k-squared slip model for the Kobe earthquake: (a) the original discontinuous slip distribution by Yoshida et al. (1996); (b) the
linearly interpolated slip model; (c) the smoothed-slip model using the bicubic Splineinterpolation; and (d) the hybrid k-squared slip model, which is the combination ofmodel c and the k-squared model with random phases at the wavenumbers higher thanNyquist wavenumbers of the subfault dimensions.

Figure 10. The source spectra of the hybrid
k-squared models for the cases of Dt ⳱ 0.0,
0.5, 1.0, and 2.0 sec. The amplitudes of thespectra are multiplied by x. Therefore, thespectrum for Dt ⳱ 0.0 (smooth rupture without
the incoherent rupture time) falls off as the xsquared, whereas the other spectra show xⳮ1
decays.

A Theoretical Omega-Square Model. Part 2: Case for a Two-Dimensional Source Model

661

Source Model with k-Squared Distributionsof Slip and Rupture Time
Here, I will check the source spectra of the hybrid ksquared slip model with variable rupture velocities by introducing the incoherent rupture time (Dtr),
NL

S(x) ⳱

NW

ML

MW

Sijkl
兺 兺 冤 兺 兺 Dijkl exp冦ix冢 Vri⳱1 j⳱1 k⳱1 l⳱1r

冣冧冥DLDW, (32)Ⳮ Dtrijkl

where Dtrijkl is Dtr, at the (i, j, k, l) point.
I assume the k-squared distribution for Dtr, as discussed
in Hisada (2000a). Since I already have the k-squared distribution of the slip, I simply assume that the Dtr distribution
is reciprocally proportional to the slip distribution. The numerical results using dynamic source models (i.e., Day,
1982) and the observational results (i.e., Archuleta, 1982)
have shown that spatial variations of peak slip velocity arestrongly coupled to those of rupture velocity. Generally, one
finds that the larger the slip, the larger the slip velocity. In
addition, the larger Dtr is, the slower the rupture velocity.
Therefore, the model, whose Dtr distribution is reciprocally
proportional to the slip distribution, is probably better thanthe model whose Dtr distribution is proportional to the slip.
In fact, I have checked the latter cases, and have obtained
similar results.
Figure 11 shows the rupture time distributions for thecases where the average incoherent rupture times (Dt) are
0.5, 1.0, and 2.0 sec. As compared with the model for Dt ⳱
0 (Fig. 6b), the rupture times in Figure 11 are fluctuatedrapidly. The model with Dt ⳱ 2.0 is probably not realistic
because the rupture velocity exceeds locally VS and VP .
Figure 10 shows the source spectra using the hybrid ksquared slip model with the rupture time distributions shownin Figure 11. The amplitudes of the spectra are multipliedby x. Similarly to the case for Dt ⳱ 0, I generated 100 setsof the models for Dt by changing the random phases, andtheir average spectra are shown in the figure. When Dt ⬆ 0,
all the spectra show roughly x-inverse decays. In addition,
it is seen that the larger Dt is, the larger the high-frequency
waves become. These conclusions are the same as those of
the 1D source model (Hisada, 2000a).
The Effect of the Directivity on the Source Spectra
I will check the directivity effects on the source spectra
and the corresponding source–time functions. Since I haveobtained the same conclusion for every model I tried, I willshow here only one example for the case Dt ⳱ 1.0 (sec),
whose Dtr distribution is reciprocally proportional to the hybrid k-squared slip model.
Figure 12a shows the source spectra for h ⳱ 90 andu ⳱ 90 (no directivity), u ⳱ 0 (forward directivity), andu ⳱ 180 (backward directivity), whose amplitudes are multiplied by x. As expected (e.g., Bernard et al., 1996), theFigure 11. The rupture time distributions for thecases of the average incoherent rupture times (Dt) are(a) 0.5, (b) 1.0, and (c) 2.0 sec. I assumed that the Dtr
distribution is reciprocally proportional to the slip distribution (Fig. 9d).

amplitude is larger for the forward model and smaller forthe backward model. Note that all the spectra show x-inversed decays.
Figure 12b shows the corresponding source–time functions. As compared with the model of u ⳱ 90, the forward

662

Y. HisadaFigure 12. (a) The source spectra for u ⳱90 (no directivity), 0 (forward directivity),and 180 (backward directivity) for the case of
Dt ⳱ 1.0 sec, and (b) the corresponding
source–time functions.

Figure 13. The fault model and five observation points for simulating near-source strong
motions. The fault length (36 km) and width(16 km), and the slip distribution (the contours
in the figure) are the same as those of Figure
9d. The dot and arrows on the fault plane indicate the hypocenter and the direction of theright-lateral slip.

model exhibits the larger amplitude and shorter duration,
while the backward model shows smaller amplitude andlonger duration.

Strong Ground Motions near Fault and Rupture
Directivity Effects
Finally, I simulate broadband strong motions near afault using equation (1) and check the rupture directivity
effects using the proposed source model.
Figure 13 shows the fault model and five observation
points. The fault length and width and the slip distributionof the hybrid k-squared model are the same as those of Figure 9d, which are originally derived from the Kobe earthquake. As shown in the figure, it is a right-lateral strike-slip
fault with the dip angle of 90 and the rake angle of 210.
Its shallowest edge is located at the 4-km depth from the freesurface. The contours on the fault show the k-squared slip
distribution, and the black dot on the fault plane indicatesthe starting point of rupture. The five observation points line
up at 1 km apart from the fault plane and at regular 10-km
intervals on the free surface. Point 1 corresponds to the direction of the backward directivity, while points 3 to 5 correspond to the forward direction. As for the slip-velocity
function, I use the parameters listed as the bold values forM ⳱ 7 in Table 1: f max ⳱ 5 Hz, Tr ⳱ 1.74, NV ⳱ 6, andAr ⳱ 1.4.
As for Green’s function in equation (1), I use that of thehomogenous full-space and multiply the factor of 2 to take
into account the free surface condition. The material properties of the medium are q ⳱ 2.8 t/m3, VP ⳱ 6.0 km/sec,
VS ⳱ 3.5 km/sec, QP ⳱ 500, and QS ⳱ 250. First, I calculate
strong motions up to 5 Hz in the frequency domain and thencalculate time histories using the inverse fast Fourier transform algorithm and the low-pass filter with a flat range from0 to 4 Hz.
Figures 14a and b show the simulated accelerations, velocities, and displacements for the fault-normal and fault-

A Theoretical Omega-Square Model. Part 2: Case for a Two-Dimensional Source Model

parallel components, using the smooth rupture time model(Dtr ⳱ 0) (see Fig 6b) and the k-squared rupture time model(Dtr ⳱ 1.0 sec; see Fig. 11b), respectively. It should be notedthat the amplitude scales of the fault-normal components are
10 times larger than those of the fault-parallel components
in all the figures. This indicates strong directivity effects at
broad frequencies. The displacements in Fig. 14a and b show
similar simple waveforms: smoothed step functions for thefault-parallel components and long-period pulses for thefault-normal components, whose amplitudes grow in the forward direction, as expected theoretically. On the other hand,the velocities and accelerations are very different between
the two models. In particular, the accelerations in Figure 14ashow simple waveforms consisting of a couple of directivity
pulses. Each pulse corresponds to the asperity existing fromthe rupture front to the corresponding observation point (seeFig. 13). Their waveforms seem too simple to be accelerations, even though the complex k-squared slip distribution
was considered. On the contrary, the accelerations in Figure
14b show complex random characteristics at high frequencies, which seem much more realistic. Therefore, the ksquared distribution of the rupture time is much more important and essential for representing realistic broadband
waveforms than the k-squared slip distribution.
Figure 15 shows the Fourier amplitudes of the accelerations of the fault-parallel and -normal components and theiramplitude ratios at three observation points (1, 3, and 5).
The top three and the bottom three graphs represent the results for the smooth rupture model (Dtr ⳱ 0) and the ksquared rupture model (Dtr ⳱ 1 sec), respectively. As expected from the previous results, the Fourier amplitudes of
the tops fall off as x-cubed at the frequencies higher than
the first corner frequency, whereas those of the bottoms
show x-squared models. All the amplitudes of the faultnormal components are several times larger than the corresponding parallel components, and their ratios increase fromthe backward rupture direction (point 1) to the forward direction (points 3 and 5). Interestingly, the amplitude ratios
are roughly constant at the frequencies higher than the firstcorner frequency. This is contradictory to the empirical directivity models (e.g., Sommerville et al., 1995), where theratios decrease to 1 at higher frequencies. Undoubtedly, thescattering path effects should be one of the reasons for thiscontradiction. However, the main reason is probably due tothe limitation of the 2D fault model, whose plane is flat andits radiation pattern, eknj Ⳮ ej nk in equation (1), is independent of frequencies and/or wavenumbers. To represent more
realistic radiation patterns and rupture directivity effects, I
probably need to incorporate an empirical frequencydependent radiation pattern (e.g., Pitarka et al., 2000) or to
introduce a more realistic 3D faulting model. In particular,
the 3D model should be wavenumber dependent, that is, anearly flat plane at longer wavenumbers and highly fluctuated surface or volume at shorter wavenumbers.

Figure 14. The simulated near-source accelerations, velocities, and displacements for the faultnormal and fault-parallel components, using (a) thesmooth rupture time model (Dtr ⳱ 0; see Fig. 6b),and (b) the k-squared rupture time model (Dtr ⳱ 1.0
sec; see Fig. 11b).

663

664

Y. HisadaFigure 15. The Fourier amplitudes of accelerations (unit: m/sec) of the fault-paralleland fault-normal components and their amplitude ratios at three observation points (1,3, and 5). The top three and the bottom three graphs represent the results for the smooth
rupture model (Dtr ⳱ 0) and the k-squared rupture model (Dtr ⳱ 1 sec), respectively.

Discussion
I obtained the same conclusions as Hisada (2000a) using
2D source models. The x-squared model consists of two xinversed spectra models; one is the slip velocity function,and the other is the k-squared model for the spatial distributions of slip and rupture time. Therefore, following Hisada(2000a), I shall call this model the x-inverse-squared model.
Regarding the slip-velocity function, I generalized theprevious results of Hisada (2000a) by superimposing scalene
triangles to model arbitrary Kostrov-type slip velocities (Fig.
2). It was important to superimpose triangles with various
durations to guarantee broadband frequency contents.
Among various combinations of the parameters, I estimatedthe plausible combinations using the empirical relations for
magnitude, average slip, and slip duration, obtaining thesame conclusion as Hisada (2000a). Namely, the Fourier amplitude of the slip velocity has two corner frequencies: oneis f max and the other is f 1. f max is the reciprocal of the minimum duration among the scalene triangles, and f 1 is nearly
equal to the reciprocal of twice the total slip duration. Finally, I confirmed that the Fourier amplitude fell off as theinverse of x between f max and f 1 (see Fig. 4).

Concerning the effects of the spatial variations on slip
and rupture time, I investigated the conventional asperity
models and the k-squared models. The conventional asperity
model, in which slips are locally constant within rectangularsubfaults, is not appropriate for simulating ground motions
because the discontinuities of slip along the subfault boundaries generates artificial starting and stopping phases. Thus,
I proposed the hybrid k-squared slip model (see Fig. 9),
which is the combination of the smoothed asperity model at
low wavenumbers and the k-squared slip model at highwavenumbers. On the other hand, since it is not well known
how to distribute the incoherent rupture time (Dtr) on fault
planes, the Dtr distribution was simply assumed to be reciprocally proportional to the slip distribution. This led to thesame conclusion as Hisada (2000a); namely, the k-squared
distributions of slip and rupture time seem the most appropriate. In addition, the source spectra of these models fall
off as the inverse of x, when the slip velocity is the delta
function.
Next, I discuss the merits and demerits of the proposed
x-inverse-squared model from the practical point of simulating strong ground motions. The advantage of this modelis that it provides a sound theoretical basis for constructing

665

A Theoretical Omega-Square Model. Part 2: Case for a Two-Dimensional Source Modelthe conventional x-squared model. Even though the subfault
sizes used in source inversion studies are usually fairly large,
and the corresponding resolutions are limited at longer periods (i.e., a few seconds for M 7, and several seconds forM 8 earthquakes), I can easily extend the limit of resolutions
by assuming the self-similarity rule (i.e., the k-squaredmodel). Therefore, it is possible to generate realistic broadband strong motions including not only the long-period coherent waves (i.e., the permanent offset and the forward directivity pulses near faults), but also the short-period random
waves (i.e., accelerations), as demonstrated in Figure 14b.
Even some peculiar strong motions, such as the accelerations
observed in the area near the surface rupture during the 1999
Chi-Chi earthquake (i.e., TCU052 and TCU068) whose
spectra are not apparently x-squared, can be easily reproduced by using the slip-velocity function with a small f max
and/or a large Ar, as discussed earlier.
On the other hand, a disadvantage of this method is thatit takes a lot of computational time at higher frequencies
because very dense integration points are necessary to represent a continuous slip and rupture front. Numerical experiments show that the distance between integration points(Dd, namely, DL/ML and DW/MW in equation 32) should be
Dd ⱕ minimum wavelength/5  10.
Especially, denser integration points are necessary at
backward directions than forward directions to avoid artificial excitations at high frequencies. In the case of the Kobe
model presented earlier, for example, I distributed 57,600
integration points (9 ⳯ 4 subfaults multiplied by 40 ⳯ 40
points) on the relatively small fault plane (16 ⳯ 36 km2) tocalculate source spectra up to 5 Hz. Thus, it is not easy toincorporate this source model with more realistic Green’s
function, such as those of flat-layered structures, on larger
fault planes. In addition, this model does not reproduce thefrequency-dependent rupture directivity effects, which are
empirically known. Therefore, the fault-normal componentsof this model are always much larger than the parallel components at any frequencies. However, those problems would
be easily remedied by incorporating this model with stochastic source methods at high frequencies and/or empiricalfrequency-dependent radiation patterns (i.e., Kamae et al.,
1998; Hisada, 2000b; Pitarka et al., 2000).

whose slip distribution consisted of the smoothed asperity
model at low wavenumbers and the k-squared slip model athigh wavenumbers. In addition, the distribution of the incoherent rupture time (Dtr) was simply assumed to be reciprocally proportional to the slip distribution. This also led me
to confirm the same conclusions of Hisada (2000a); namely,
the source spectra of the hybrid k-squared models fell off as
the inverse of x, when the slip velocity was the delta function. Therefore, the x-inverse-squared model, which is the
combination of the proposed slip velocity and the hybrid ksquared distributions in slip and Dtr, provides the theoretically solid basis of the x-squared model.
In addition, this model successfully simulated most of
the well-known characteristics of the near-fault strong
ground motions at broadband frequencies: the permanent
offset in displacements, the long-period directivity pulses invelocities, and the complex randomness in accelerations. In
particular, I found that the k-squared distribution of the rupture time is much more important and essential not only for
exciting high-frequency waves, but also for generating realistic randomness in acceleration waveforms. On one hand,
this model did not reproduce the frequency-dependent rupture directivity effects (i.e., the fault-normal components are
dominant over the parallel components only at longer periods). This is probably caused by the flat fault plane andthe frequency (or wavenumber)-independent radiation pattern. This suggests that a 3D faulting model would be necessary to obtain more realistic broadband strong motions infuture study.

Acknowledgments
I am grateful to Dr. Steven Day and an anonymous reviewer for helpful review and comments. Comments and suggestions by Drs. Jacobo Bielak, Sumio Sawada, Stewart Koyanagi, Tomotaka Iwata, Kojiro Irikura,
Takashi Miyatake, Robert Graves, Arben Pitarka, and Paul Somerville arealso greatly improved this manuscript. This research was partly supported
by a special project of U.S.–Japan Cooperative Research for Urban Earthquake Disaster Mitigation (No. 11209201), Grant-in-Aid for Scientific Research on Priority Area (Category B), Ministry of Education, Culture,
Sports, Science, and Technology of Japan (principal investigator: Dr.
Tomotaka Iwata).

References
Conclusions
I proposed the x-inverse-squared model to construct thex-squared model by considering the spatial variation in slip,
slip velocity, and rupture time on 2D fault models. For theslip velocity, I generalized Hisada’s model (2000a) by superposing scalene triangles to represent Kostrov-type slip velocities with arbitrary combinations of the source-controlled
f max and slip durations. Then, I obtained the same conclusions of Hisada (2000a); namely, the Fourier amplitude ofthis model falls off as the inverse of x between f 1 and f max.
On the other hand, for modeling the spatial variation in slip
and rupture time, I proposed a hybrid k-squared model,

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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 23 June 2000.