155
Bulletin of the Seismological Society of America, Vol. 94, No. 1, pp. 155–170, February 2004
Effects of Fault Dip and Slip Rake Angles on Near-Source Ground Motions:
Why Rupture Directivity Was Minimal in the 1999 Chi-Chi, Taiwan,
Earthquake
by Brad T. Aagaard,* John F. Hall, and Thomas H. Heaton
Abstract
We study how the fault dip and slip rake angles affect near-source
ground velocities and displacements as faulting transitions from strike-slip motion
on a vertical fault to thrust motion on a shallow-dipping fault. Ground motions are
computed for five fault geometries with different combinations of fault dip and rake
angles and common values for the fault area and the average slip. The nature of the
shear-wave directivity is the key factor in determining the size and distribution of
the peak velocities and displacements. Strong shear-wave directivity requires that
(1) the observer is located in the direction of rupture propagation and (2) the rupture
propagates parallel to the direction of the fault slip vector. We show that predomi-
nantly along-strike rupture of a thrust fault (geometry similar in the Chi-Chi earth-
quake) minimizes the area subjected to large-amplitude velocity pulses associated
with rupture directivity, because the rupture propagates perpendicular to the slip
vector; that is, the rupture propagates in the direction of a node in the shear-wave
radiation pattern. In our simulations with a shallow hypocenter, the maximum peak-
to-peak horizontal velocities exceed 1.5 m/sec over an area of only 200 km
2
for the
30
-dipping fault (geometry similar to the Chi-Chi earthquake), whereas for the 60
-
and 75
-dipping faults this velocity is exceeded over an area of 2700 km
2
. These
simulations indicate that the area subjected to large-amplitude long-period ground
motions would be larger for events of the same size as Chi-Chi that have different
styles of faulting or a deeper hypocenter.
Introduction
In the past decade, several earthquakes near large urban
areas have caused considerable damage, including the 1994
Northridge, California, the 1995 Hyogo-Ken Nanbu (Kobe),
Japan, the 1999 Izmit and Duzce, Turkey, and the 1999 Chi-
Chi, Taiwan, earthquakes. These earthquakes and their as-
sociated ground-motion records increased the awareness of
the destructive capability and characteristics of near-source
ground motions (e.g., see Olsen and Archuleta, 1996; Som-
erville
et al.
, 1997; Hisada
et al.
, 1998; Kamae and Irikura,
1998; Pitarka
et al.
, 1998; Huang
et al.
, 2000; Oglesby
et
al.
, 2000; Ouchi
et al.
, 2001; Chen
et al.
, 2001). Two factors
control the amplitude of near-source ground motions: rup-
ture directivity and proximity to the fault. At all locations in
the near field, the directivity of the rupture affects the am-
plitude of the motion. Additionally, at locations very close
to the fault trace, the static offset also directly contributes to
the amplitude of the motion. This means that for events of
*Present address: USGS, MS977, 345 Middlefield Rd., Menlo Park, Cali-
fornia 94025.
the same size, which have similar static offsets, changes in
the rupture directivity largely determine the variations in the
amplitude of the near-source ground motions from one event
to another. The primary factor controlling the size of the
directivity effect is not simply the distance the rupture prop-
agates toward a location, but the distance the rupture prop-
agates toward a location while it is parallel to the direction
of slip. Consequently, the dimensions and the dip angle of
the fault, the direction of slip (slip rake angle), and the lo-
cation of the hypocenter all play critical roles in determining
the character and amplitude of near-source ground motions.
Early efforts aimed at understanding near-source ground
motions focused on simple numerical models (e.g., Haskell,
1969; Archuleta and Frazier, 1978; Archuleta and Hartzell,
1981). More recently, some researchers have focused on spe-
cific ground-motion recordings (Iwan and Chen, 1994) or
damage near a surface rupture (Allen
et al.
, 1998) to infer
the characteristics of near-source ground motions. Three-
dimensional simulations have been used to understand the
physics of near-source ground motions for specific events
156
B. T. Aagaard, J. F. Hall, and T. H. Heaton
(e.g., Olsen
et al.
, 1997; Nielsen and Olsen, 2000; Oglesby
and Day, 2001). Researchers have also used them to examine
hypothetical scenarios. Olsen
et al.
(1995) and Graves
(1998) examined near-source ground motions for ruptures
on the San Andreas fault, and Olsen and Archuleta (1996)
considered various scenarios within the Los Angeles area.
Whereas these previous studies help to explain patterns of
damage in particular earthquakes or what such patterns
might be for some future event, they generally do not shed
light on the fundamental characteristics of near-source
ground motions and how these vary with changes in the
seismic source parameters. Dynamic rupture models have
improved our understanding of how near-source ground mo-
tions develop from the basic features of the rupture process
(Olsen
et al.
, 1997; Inoue and Miyatake, 1998; Oglesby
et
al.
, 2000; Aagaard
et al.
, 2001b), but only a couple of these
studies (Oglesby
et al.
, 2000; Aagaard
et al.
, 2001b) have
systematically explored how the source parameters affect the
near-source ground motions.
We complement these two efforts and other work (Aa-
gaard
et al.
, 2001a) that used kinematic source models to
systematically examine source parameters and near-source
ground motions by focusing on an event of a specific size
and determining how changes in the style of faulting, in
particular the fault dip and slip rake angles, affect the near-
source ground motions. We examine the distribution of
shaking and the characteristics of the near-source ground
motions, as well as how these change in response to varia-
tions in the fault dip and slip rake angles for two hypocen-
ters. We consider several measures of the ground motions,
including the area where the ground motion exceeds a given
level and the mean maximum amplitude of the motion as a
function of distance from the fault. Furthermore, by selecting
a parameter space that includes a scenario that approxi-
mately matches the 1999 Chi-Chi earthquake in Taiwan, we
explore the role that the style of faulting played in deter-
mining the area subjected to high-intensity long-period
(2 sec and longer) shaking in the Chi-Chi earthquake.
These long-period ground motions place the greatest de-
mand on structures with similar elastic periods, such as tall
buildings. However, velocity pulses with amplitudes of
about 1 m/sec are large enough to cause yielding in struc-
tures with significantly shorter periods (Hall, 1995, 1997).
Once a structure begins to yield, the displacement amplitude
becomes very important, with larger displacements resulting
in greater yielding and damage. For example, using simu-
lations of steel-frame buildings, Hall, (1997) found that the
motion recorded at the Lucerne Valley station in the 1992
Landers earthquake with a velocity pulse duration of about
4 sec could, in addition to causing collapse of a 20-story
building (3.5-sec fundamental elastic period), also generate
considerable yielding in a 6-story structure (1.5-sec funda-
mental elastic period).
Observations from the Northridge earthquake support
these findings. Boatwright
et al.
(2001) noted that the dis-
tribution of red-tagged buildings (mostly low-rise structures
with periods shorter than 1 sec) following the 1994 North-
ridge earthquake closely resembled the distribution of peak
ground velocity for periods of around 1.5 sec. Likewise,
Wald
et al.
(1999) developed a linear regression between
peak velocity and modified Mercalli intensity for eight Cali-
fornia earthquakes. This implies that velocity, not accelera-
tion, correlates well with damage across the general popu-
lation of structures. Thus, although the bandwidth of the
simulations in our study is restricted to periods of 2 sec and
longer and is most applicable to structures with similar fun-
damental elastic periods, the amplitudes of these long-period
ground motions also play a role in the response of structures
with shorter periods.
Earthquake Scenarios
We compute the near-source ground motions for two
hypocenters for each of five different pairs of fault dip and
slip rake angles. We also select a realistic fault length-to-
width ratio for each dip angle. The scenarios vary from a
pure strike-slip rupture on a long, narrow fault to a pure
thrust rupture on a significantly shorter and wider fault.
Methodology
We follow the general methodology of our previous
work involving simulations of near-source ground motions,
so this section contains only a brief summary of the methods
used in the earthquake simulations. Aagaard (1999) and Aa-
gaard
et al.
(2001a) provided detailed discussions of the
methodology. As discussed later, this study improves upon
our previous characterizations of the seismic source; in par-
ticular, the length scales of the spatial heterogeneity in the
final slip are compatible with those found in kinematic
source inversions, and we allow the rupture speed to vary as
a function of the direction of propagation relative to the di-
rection of slip.
We discretize the three-dimensional domain using linear
tetrahedral finite elements. This transforms the three-
dimensional dynamic elasticity equation,
k
u
d
l
(
u
u
)
q
̈
u
,
(1)
k,kj ij
i,jj
j,ij
i
into a matrix differential equation,
[
M
]{
̈
u
(
t
)}
[
C
]{
̇
u
(
t
)}
[
K
]{
u
(
t
)}
{
F
(
t
)},
(2)
where [
M
] denotes the mass matrix, [
C
] denotes the damping
matrix, [
K
] denotes the stiffness matrix, {
F
(
t
)} denotes the
force vector at time
t
, and {
u
(
t
)} denotes the displacement
vector at time
t
. Our discretization of the finite-element
model limits the simulation to wave propagation for waves
with periods of 2.0 sec and longer.
Anelastic attenuation is not included because it has little
effect on long-period near-source ground motions, so the
only contribution to the damping matrix comes from the
absorbing boundaries on the lateral sides and bottom of
Effects of Fault Dip and Slip Rake Angles on Near-Source Ground Motions
157
40km
90km
40km
W
/2 cos
θ
Dip,
θ
E
as
t
Nort
h
160km
80k
m
L
W
–
Figure
1.
Geometry of the simulation domain. The fault has a dip angle of
h
,
a
length of
L,
and a width of
W
. The center of the fault lies 10 km south of the center
of the domain. The dotted line running east–west sits above the center of the fault,
whereas the dotted line running north–south intersects the fault trace.
Density (g/cm
3
), Velocity (km/s)
Depth (km)
v
p
v
s
ρ
0
1
2
3
4
5
6
7
8
40
30
20
10
0
Figure
2.
Dilatational wave speed (
m
p
), shear wave
speed (
m
s
), and mass density (
q
) as a function of depth.
the domain. These absorbing boundaries prevent waves from
reflecting off the truncated sides of the domain and contam-
inating the solution.
We model the earthquake by creating dislocations in the
finite-element model that mimic the slip on a fault. In the
scenarios discussed here, we specify the slip time history at
each point on the fault, where the time history follows the
integral of Brune’s far-field time function with the final slip
and peak slip rate as parameters.
Simulation Domain
In each of the scenarios, the domain is 160 km long,
80 km wide, and 40 km deep, as shown in Figure 1. The
material properties vary only as a function of depth, as il-
acteristics indicate this earthquake had a rupture length of
between 80 and 100 km, a rupture width of between 30 and
40 km, and a dip angle of 20
–30
(Huang
et al.
, 2000; Ma
et al.
, 2000; Johnson
et al.
, 2001; Ji
et al.
, 2001). By varying
the length-to-width ratios as we change the slip rake angle,
we create realistic fault geometry for each of the five fault
dip angles.
Based on the regression relations of Wells and Copper-
smith (1994) between fault geometry and average slip cou-
pled with these fault dimensions, we chose a common av-
erage slip of 2.9 m for the scenarios. This average slip is
smaller than the average slip estimated for the Chi-Chi earth-
quake, so our fault with a dip angle of 30
and our choice
of material properties has a moment magnitude of only 7.4
lustrated in Figure 2. This set of material properties repre-
sents the average variations in Taiwan and corresponds to
the variation in a region without a deep sedimentary basin
(Ma
et al.
, 1996, 2001).
Earthquake Source Parameters
Table 1 gives the five pairs of fault dip and slip rake
angles along with the fault lengths and widths for the dif-
ferent fault geometries. For the five fault geometries, we
chose fault dip angles uniformly distributed between 90
and
30
with rake angles uniformly distributed between 0
and
90
. Similarly, the lengths of the faults decrease linearly
from 120 to 80 km long while maintaining an area of 2400
km
2
. The 80-km-long and 30-km-wide fault, which has a dip
angle of 30
, roughly matches the geometry of the 1999 Chi-
Chi earthquake in Taiwan. Inversions for the source char-
158
B. T. Aagaard, J. F. Hall, and T. H. Heaton
Dist. Along Strike (km)
Dist. Down Dip (km)
0
10
20
30
40
50
60
70
80
90
100
20
10
0
Final Slip (m)
0.0
2.0
4.0
6.0
8.0
Figure
3.
Distribution of final slip for the fault that has a dip angle of 60
. The
distribution is a low-pass filtered random distribution. The distributions for the other
geometries are similar.
Table 1
Seismic Source Parameters for Each Pair of Fault Dip and Slip Rake Angles
Dip Angle
(deg)
Rake Angle
(deg)
Fault Length
(km)
Fault Width
(km)
Average Slip
(m)
Peak Slip Rate
(m/sec)
Rupture Speed
(%
m
s
)
90
0.0
120
20
75
22.5
110
22
60
45.0
100
24
2.9
2.0
85% parallel to slip
45
67.5
90
27
68% perpendicular to slip
30
90.0
80
30
The style of faulting smoothly transitions from pure strike-slip motion on a long, narrow fault to pure thrust
motion on a much shorter, wider fault. The rupture speed is set relative to the local shear-wave speed,
m
s
.
compared with the moment magnitude of 7.6–7.7 estimated
for the Chi-Chi earthquake. For each fault the slip rake an-
gles are uniform and remain constant during the rupture. We
also use a uniform peak slip rate of 2.0 m/sec in the slip time
history, which, for final slips of around 3 m, gives slip du-
rations compatible with those found in kinematic source in-
versions (Heaton, 1990; Somerville
et al.
, 1997).
We create the distributions of slip by low-pass filtering
random distributions. Starting with a uniform random dis-
tribution on a 1.0-km uniform grid (which is coarser than
the node spacing in the finite-element model), we low-pass
filter along the fault strike and then along the dip using a
first-order Butterworth filter. We also taper the slip along the
buried edges of the fault. Whereas the distributions for each
fault geometry all have an average slip of 2.9 m, the maxi-
mum slip ranges from 5.7 to 7.1 m because we start with
different random distributions for each fault geometry. Fig-
ure 3 shows the distribution of final slip for the fault with a
dip angle of 60
. The slip distributions for the other fault
geometries are similar. Although a power-law spectral fall-
off would create slip distributions that better match what is
found in kinematic source inversions (Somerville
et al.
,
1997; Mai and Beroza, 2002), the long-period near-source
ground motions are relatively insensitive to the distribution
of slip compared with the hypocenter location and the rup-
ture speed (Aagaard
et al.
, 2001b), so that the dominant fea-
tures of the ground motions do not change when using a
low-pass filtered random distribution compared with a
power-law filtered random distribution.
Figure 4 shows the two hypocenters we consider for
each dip angle of the fault. The shallow hypocenter sits mid-
depth at the southern quarter point of the fault and corre-
sponds to a highly unilateral rupture, whereas the deep hy-
pocenter sits 5.0 km up-dip from the bottom center of the
fault and corresponds to a bilateral case with more up-dip
rupture. For strike-slip faulting the ruptures propagate
mostly in the mode-II direction (parallel to slip and a local
maximum in shear-wave radiation pattern) for both hypo-
centers. On the other end of the spectrum, for pure thrust
faulting and the shallow hypocenter the rupture propagates
primarily in the mode-III direction (perpendicular to slip and
a node in the shear-wave radiation pattern) with very little
propagation in the mode-II direction; for pure thrust faulting
and the deep hypocenter, the amount of rupture in the mode-
II direction increases significantly.
The rupture speed determines when slip begins at each
point on the fault. Numerous dynamic rupture simulations
(e.g., Andrews, 1976; Day, 1982; Madariaga
et al.
, 1998;
Aagaard
et al.
, 2001b) indicate that ruptures propagate
slightly slower in the direction perpendicular to slip (mode-
III direction) compared with the direction parallel to slip
(mode-II direction). Therefore, instead of an isotropic rup-
ture speed, we independently specify the rupture speed to be
85% of the local shear-wave speed in the mode-II direction
(parallel to the slip direction) and 20% slower in the mode-
III direction (perpendicular to the slip direction), as illus-
trated in Figure 4.
Effects of Fault Dip and Slip Rake Angles on Near-Source Ground Motions
159
5km
Hypocenter
Shallow
Hypocenter
Deep
W/2
L/4
L/2
L
W
v
II
r
v
III
r
Slip
v
II
r
v
III
r
Figure
4.
Relative locations of the shallow and deep hypocenters on the fault sur-
face, which has a length of
L
and a width of
W
. The shallow hypocenter lies middepth
at a quarter point, and the deep hypocenter lies midway along strike 5 km up-dip from
the bottom of the fault. The shallow hypocenter corresponds to the general location of
the hypocenter in the Chi-Chi earthquake. The dashed ellipse identifies the rupture
front at some point in time propagating away from the shallow hypocenter and illus-
trates how the rupture speed is set independently in the mode-II ( ) and mode-III
II
m
r
(
) directions.
III
m
r
Scenario Nomenclature
The name of each scenario corresponds to the dip angle
of the fault, the slip rake angle, and the location of the hy-
pocenter. For example, Dip90Rk0HySh refers to the sce-
nario where the fault has a dip angle of 90
and a slip rake
angle of 0
and the rupture begins at the shallow hypocenter.
Similarly, Dip45Rk68HyDp refers to the scenario where the
fault has a dip angle of 45
and a slip rake angle of 67.5
and the rupture begins at the deep hypocenter.
Results
Overview of Rupture Behavior and Ground Motions
We begin by examining how the general characteristics
of the rupture behavior and resulting ground motions change
with the fault dip and slip rake angles. Additional results and
figures can be found in Aagaard
et al.
(2002). In scenario
Dip90Rk0HySh, the rupture propagates fastest along the
strike of the fault. As we vary the style of faulting across the
scenarios by decreasing the dip angle of the fault and in-
creasing the rake angle of slip so that it has a larger vertical
component, the rupture speed along the strike decreases
while the rupture speed up-dip increases. For the case of the
fault at a dip angle of 45
and a rake angle of 67.5
, the
fastest rupture speed occurs 22.5
off the up-dip and down-
dip directions. This creates asymmetry in the propagation of
the rupture for the centrally located deep hypocenter.
Due to the existence of surface rupture in the layered
medium, surface waves in the form of combinations of Love
and Rayleigh waves dominate the long-period ground mo-
tions. As the rupture propagates in scenario Dip90Rk0HySh,
large-amplitude Love waves with amplitudes approaching
2.0 m/sec form in the region where the propagation direction
generally coincides with the slip direction, which in this case
is north of the epicenter. The particle motion for these waves
is in the east–west direction (perpendicular to the fault trace).
The Love-wave amplitudes generally build along the length
of the fault as the rupture reinforces the waves and then
begin steadily decreasing upon reaching the northern tip of
the fault. The heterogeneous distribution of slip disrupts the
reinforcement of the Love waves, so that the amplitudes un-
dergo minor fluctuations as they grow.
As the dip angle of the fault decreases and the vertical
component of slip increases, the rupture generates Love
waves less effectively and becomes more effective at gen-
erating Rayleigh waves. The rotation of the slip direction
toward the dip direction results in reinforcement of the
SV
waves (shear waves with particle motion in the vertical di-
rection) emanating from an angle of 45
with respect to the
slip direction. These
SV
waves produce Rayleigh waves as
they hit the ground surface. Consequently, the largest Ray-
leigh waves (with amplitudes near 1.5 m/sec) occur north-
west of the epicenter; the particle motions are retrograde
with the largest horizontal component in the northwest–
southeast direction.
Maximum Displacements and Velocities
Choices for measuring the intensity of the shaking in-
clude the maximum amplitude of the motion and the maxi-
mum peak-to-peak amplitude of the motion, where we mea-
sure the peak-to-peak amplitude using consecutive peaks. In
practice, the greatest difference between the two occurs for
the case of double-sided, symmetric displacement or veloc-
ity pulses, as shown in Figure 5. A displacement ramp and
the corresponding single-sided velocity pulse roughly ap-
proximate the ground motion at a location with a static off-
160
B. T. Aagaard, J. F. Hall, and T. H. Heaton
Displacement
Time
Max PP = Max Amp
Max PP = Max Amp
Time
Displacement
Time
Velocity
Max PP = 2 Max Amp
Max PP = Max Amp
Time
Velocity
Time
Displacement
Max PP = 2 Max Amp
Time
Max PP = 2 Max Amp
Velocity
Figure
5.
Illustration of how the peak-to-peak displacement and velocity amplitudes
compare to the maximum amplitudes for three different types of ground motions: a
displacement ramp and corresponding single-sided velocity pulse (left), a single-sided
displacement pulse and corresponding double-sided velocity pulse (center), and a double-
sided displacement pulse and corresponding velocity pulses (right). Except for the case
of double-sided displacements, which are generally associated with surface waves, the
maximum peak-to-peak displacements (measured using consecutive peaks) match the
maximum displacements. On the other hand, the maximum peak-to-peak velocities ex-
ceed the maximum velocities, except for the case in which there is a large static offset.
set. In this case, there is no difference between the maximum
peak-to-peak amplitude and the maximum amplitude. A
single-sided displacement pulse and the corresponding
double-sided velocity pulse roughly approximate the ground
motion at a location without a static offset and no surface
waves. Whereas the maximum displacement equals the max-
imum peak-to-peak displacement, the maximum peak-to-
peak velocity exceeds the maximum velocity by up to a
factor of 2. Finally, a double-sided displacement pulse and
the corresponding velocity pulses roughly approximate the
ground motion at a location with large-amplitude surface
waves and no static offset. In this case, the maximum peak-
to-peak displacement and velocity amplitudes can both be
up to twice the maximum amplitudes. We will use the max-
imum magnitude of the displacement and the maximum
peak-to-peak velocity as measures of ground-motion inten-
sity because double-sided velocity pulses place a greater de-
mand on structures than single-sided velocity pulses (Hall
et al.
, 1995).
In scenario Dip90Rk0HySh, the maximum horizontal
displacements and maximum peak-to-peak horizontal veloc-
ities increase along the strike of the fault north and south of
the epicenter as shown in Figure 6. This effect is much more
pronounced north of the epicenter because the rupture ex-
tends much further in this direction compared with south of
the epicenter. The amplitudes also decay rapidly with dis-
tance away from the surface trace of the fault. The maximum
horizontal displacement is 2.6 m, and the maximum peak-
to-peak horizontal velocity is 2.6 m/sec.
As the dip of the fault decreases and the vertical com-
ponent of slip increases, the pattern of shaking becomes
much more asymmetric with two clear features: (1) the max-
imum horizontal displacements on the hanging wall (above
the fault) increase and follow the variations in the distribu-
tion of slip, and (2) the strongest shaking remains concen-
trated in the region with the maximum directivity. These
trends are visible in Figure 7, which displays the maximum
displacements and maximum peak-to-peak velocities for
scenario Dip60Rk45HySh. The steep dip of the fault and the
rake angle of 45
lead to large Love and Rayleigh waves
that propagate toward the northwest. As a result, a large
region emanating off to the northwest from the northern end
of the fault experiences maximum displacements exceeding
1.0 m and maximum peak-to-peak velocities exceeding
2.0 m/sec.
With the shallow hypocenter, as we transition from
strike-slip motion to thrust motion, the rupture continues to
propagate mostly along the strike of the fault so that the
rupture direction becomes less aligned with the slip direc-
tion. In other words, the rupture switches from propagating
in the mode-II direction, which is a local maximum in the
shear-wave radiation pattern, to propagating in the mode-III
direction, which is a node in the shear-wave radiation pat-
tern. In our limiting case where the fault dip angle is 30
with pure thrust faulting, the inability of the rupture to ef-
fectively reinforce Love or Rayleigh waves leads to much
smaller velocities at most locations on the ground surface
(Fig. 8). Although the maximum velocities do reach 1.4 m/
Effects of Fault Dip and Slip Rake Angles on Near-Source Ground Motions
161
Distance East (km)
Distance North (km)
Max. Horiz. Displacement
0
20
40
60
80
0
20
40
60
80
100
120
140
160
Displacement (m)
0.0
1.0
2.0
3.0
4.0
Distance North (km)
Max. P
–
to
–
P Horiz. Velocity
0
20
40
60
80
100
120
140
160
Velocity (m/s)
0.0
1.0
2.0
3.0
4.0
Figure
6.
Maximum amplitude of the horizontal displacements and maximum peak-
to-peak horizontal velocities on the ground surface for scenario Dip90Rk0HySh. The
thick solid line shows the surface trace of the fault, and the asterisk identifies the
epicenter. The maximum displacements and velocities generally increase along the fault
away from the epicenter and then decrease steadily past the ends of the fault.
Distance East (km)
Distance North (km)
Max. Horiz. Displacement
0
20
40
60
80
0
20
40
60
80
100
120
140
160
Displacement (m)
0.0
1.0
2.0
3.0
4.0
Distance North (km)
Max. P
–
to
–
P Horiz. Velocity
0
20
40
60
80
100
120
140
160
Velocity (m/s)
0.0
1.0
2.0
3.0
4.0
Figure
7.
Maximum horizontal displacements and maximum peak-to-peak hori-
zontal velocities on the ground surface for scenario Dip60Rk45HySh. The thick solid
line shows the surface trace of the fault, the thick dashed line indicates the surface
projection of the buried edges of the fault, and the asterisk identifies the epicenter. The
maximum displacements and velocities generally increase along the fault away from
the epicenter with a large region of intense shaking extending to the northwest.
Distance East (km)
Distance North (km)
Max. Horiz. Displacement
0
20
40
60
80
0
20
40
60
80
100
120
140
160
Displacement (m)
0.0
1.0
2.0
3.0
4.0
Distance North (km)
Max. P
–
to
–
P Horiz. Velocity
0
20
40
60
80
100
120
140
160
Velocity (m/s)
0.0
1.0
2.0
3.0
4.0
Figure
8.
Maximum horizontal displacements and maximum peak-to-peak hori-
zontal velocities on the ground surface for scenario Dip30Rk90HySh (whose geometry
is similar to the Chi-Chi earthquake). The thick solid line shows the surface trace of
the fault, the thick dashed line indicates the surface projection of the buried edges of
the fault, and the asterisk identifies the epicenter. The predominantly mode-III rupture
does not efficiently reinforce the Love and Rayleigh waves, which results in much
smaller ground motions relative to the other scenarios.
162
B. T. Aagaard, J. F. Hall, and T. H. Heaton
Table 2
Maximum Displacements and Maximum Peak-to-Peak Velocities in the East–West (Fault-
Perpendicular), North–South (Fault-Parallel), Horizontal, and Vertical Directions for Each Scenario
Max. Disp.
Max. P-to-P Velocity
Scenario
EW
(m)
NS
(m)
Horiz.
(m)
Vert.
(m)
EW
(m/sec)
NS
(m/sec)
Horiz.
(m/sec)
Vert.
(m/sec)
Dip90Rk0HySh
2.0
2.6
2.6
0.76
2.6
1.0
2.6
0.80
Dip90Rk0HyDp
1.8
2.6
2.6
0.74
2.0
1.1
2.0
0.72
Dip75Rk22HySh
3.1
3.1
3.3
1.4
3.6
1.5
3.7
1.2
Dip75Rk22HyDp
2.1
3.1
3.1
1.3
2.5
1.3
2.8
1.2
Dip60Rk45HySh
3.1
2.8
3.2
2.9
3.9
2.4
4.2
3.0
Dip60Rk45HyDp
2.2
2.8
2.8
2.5
2.4
2.2
2.8
2.2
Dip45Rk68HySh
2.1
2.2
2.2
3.5
2.0
2.5
2.7
3.9
Dip45Rk68HyDp
2.1
2.3
2.3
3.2
1.5
2.4
2.4
2.8
Dip30Rk90HySh (Chi-Chi)
3.1
1.4
3.1
2.5
1.5
1.7
1.7
3.0
Dip30Rk90HyDp
3.1
1.7
3.1
2.8
1.6
2.2
2.2
3.3
sec at one location, the maximum peak-to-peak velocity is
only 1.7 m/sec. On the hanging wall of the fault, the maxi-
mum displacements do remain large, although they are dom-
inated by the pseudo-static displacement, which is consistent
with equal values of 3.1 m for the maximum amplitude and
the maximum peak-to-peak amplitude.
However, moving the hypocenter toward the bottom
center of the fault increases the amount of mode-II rupture
for the thrust motion cases. Of course, it has the opposite
effect for the case of pure strike-slip motion on a vertical
fault where the distance over which the portion of the rupture
that propagates to the north can effectively reinforce waves
decreases by one-third compared with the shallow hypocen-
ter. Thus, for the steeply dipping faults with mostly hori-
zontal slip, the ground motions decrease in most locations
when the hypocenter moves to the deeper, more central lo-
cation, whereas for shallow-dipping faults with a large thrust
component of slip, the ground motions increase at many lo-
cations. For the 60
-dipping fault with a slip rake angle of
45
, the maximum peak-to-peak velocity is 33% smaller for
the deep, central hypocenter compared with the shallow hy-
pocenter. On the other hand, for the 30
-dipping fault with
a slip rake angle of 90
(pure thrust), the maximum peak-to-
peak velocity increases by 30% when the hypocenter moves
from the shallow location to the deep, central location.
Table 2 gives the maximum displacements and the max-
imum peak-to-peak velocities in the east–west (fault perpen-
dicular), north–south (fault parallel), and vertical directions
as well as the maximum in any horizontal direction for each
of the 10 scenarios. In all 10 scenarios the ground motions
are large, with the maximum displacements exceeding
2.2 m and the maximum peak-to-peak velocities exceeding
1.7 m/sec.
Area Subjected to Levels of Ground Motion
In order to gauge how the severity of shaking changes
on a large scale as we transition from pure strike-slip faulting
on a vertical fault to thrust faulting on a 30
-dipping fault,
we consider three aggregate measures of ground motion: the
area on the ground surface where a given level of displace-
ment or velocity is exceeded, how fast the displacements and
velocities on the ground surface decay with distance from
the fault, and the far-field radiated energy.
Figure 9 gives the areas on the ground surface where
the maximum displacements or maximum peak-to-peak ve-
locities exceed a given value for each of the five scenarios
with the shallow hypocenter (middepth at the southern quar-
ter point of the fault), where each scenario corresponds to a
different dip angle of the fault. In all five cases very large
areas (more than 1000 km
2
) receive long-period ground mo-
tions with displacements or peak-to-peak velocities greater
than 1.3 m or 1.0 m/sec. In accordance with the observations
noted earlier, the amount of rupture directivity toward the
surface controls the amplitude of the motion, so that the
largest areas subjected to strong shaking occur in the sce-
narios with a fault dip angle of 60
or 75
. Moreover, at the
strongest levels of shaking, these areas far exceed the cor-
responding areas for the other scenarios.
The case of pure strike-slip motion on a vertical fault
generally falls in the middle ground below the 60
- and 75
-
dipping fault scenarios and above the 45
- and 30
-dipping
fault scenarios. The curves relating area and maximum dis-
placements for scenarios Dip30Rk90HySh and Dip45-
Rk22HySh closely follow one another, but the curve relating
area and maximum peak-to-peak velocities for scenario
Dip45Rk22HySh lies well to the right of the one for scenario
Dip30Rk90HySh (similar geometry to the Chi-Chi earth-
quake). Thus, for the shallow hypocenter the case of pure
thrust motion on a 30
-dipping fault results in the smallest
area subjected to a given level of peak-to-peak velocity.
Compared with the scenarios with the shallow hypo-
center, those with the deep, central hypocenter (5.0 km up-
dip from the bottom center of the fault) exhibit much less
variation in the area experiencing a given level of shaking
for the various combinations of fault dip and slip rake angles
as illustrated in Figure 10. The more central hypocenter near
Effects of Fault Dip and Slip Rake Angles on Near-Source Ground Motions
163
Max. Disp. (m)
Area Exceeding Value (km
2
)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
100
500
1000
5000
10000
Dip 90
°
Dip 75
°
Dip 60
°
Dip 45
°
Dip 30
°
(Chi
–
Chi )
Max. Peak
–
to
–
Peak Velocity (m/s)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Figure
9.
Area on the ground surface where the maximum horizontal displacements
(left) and maximum peak-to-peak horizontal velocities (right) exceed a given value for
scenarios with the shallow hypocenter. In all five scenarios, areas greater than 1000 km
2
undergo displacements exceeding 1.3 m and peak-to-peak velocities exceeding 1.0 m/
sec, with much larger areas for scenarios Dip60Rk45HySh and Dip75Rk22HySh.
Max. Disp. (m)
Area Exceeding Value (km
2
)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
100
500
1000
5000
10000
Dip 90
°
Dip 75
°
Dip 60
°
Dip 45
°
Dip 30
°
Max. Peak
–
to
–
Peak Velocity (m/s)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Figure
10.
Area on the ground surface where the maximum horizontal displace-
ments (left) and maximum peak-to-peak horizontal velocities (right) exceed a given
value for scenarios with the deep, central hypocenter. Shifting the hypocenter to near
the bottom center of the fault leads to much smaller variations in the area subjected to
a given level of shaking across the five fault dip and slip rake angle pairs.
the bottom of the fault leads to less variation in the distance
the rupture propagates in the mode-II direction (direction
parallel to slip), because the amount of along-strike rupture
decreases while the amount of up-dip rupture increases. This
reduces the amount of mode-II rupture in the cases with
small slip rake angles, which were dominated by mode-II
rupture for the shallow hypocenter, and increases the amount
of mode-II rupture in the cases with large slip rake angles,
which were dominated by mode-III rupture for the shallow
hypocenter. The 60
-dipping fault with a slip rake angle of
45
generally continues to produce the largest areas sub-
jected to a given level of shaking with this different hypo-
center, but the curve for pure thrust motion on the 30
-dip-
ping fault shifts toward the middle ground. For some ranges
of moderate peak-to-peak velocities, the case of pure thrust
motion on the 30
-dipping fault has the largest areas where
these levels of motion are exceeded.
Comparing the two scenarios with the 30
-dipping fault,
we find that changing the hypocenter has only a small effect
on the curve for the maximum displacements, because the
slip distribution, which remains the same, largely controls
the amplitude of the displacements. On the other hand, mov-
ing the hypocenter from the shallow location to the deep,
central location shifts the curve for the peak-to-peak veloc-
ities toward larger velocities as a result of the increase in the
amount of mode-II (in this case up-dip) rupture.
164
B. T. Aagaard, J. F. Hall, and T. H. Heaton
Figure
11.
Illustration of how an infinite fault is created from the finite fault in
order to examine the decay in ground-motion amplitudes with distance from the fault.
Decay in Ground Motion with Distance
We want to characterize how the ground motions decay
with distance from the fault, while also including the effects
of rupture directivity and the unpredictability of the hypo-
center. We consider identical events occurring along an in-
finitely long fault and superimpose the distributions of the
maximum displacements and velocities for a given scenario
such that the ruptures lie end to end, as illustrated in Figure
11. At each location we select the largest values across all
of the overlapping domains and then average along the strike
of the fault to obtain the average motion on each side of the
fault at a given distance. For a discussion of how the dis-
placements and velocities decay with distance from the fault
for each scenario as well as comparisons with the Uniform
Building Code near-source factor, see Aagaard
et al.
(2002).
In Figure 12 we compare the mean maximum displace-
ments and mean maximum peak-to-peak velocities for sce-
narios with the shallow hypocenter across the five fault dip
angles. On the down-dip (east) side of the fault at distances
between 10 and 30 km, the mean values vary remarkably
little with the dip angle of the fault. At closer distances and
on the up-dip (west) side of the fault, the mean maximum
values span a larger range of values; they are lowest for pure
thrust motion on the 30
-dipping fault and up to 2 times
greater for oblique slip on the 60
- and 75
-dipping faults.
These differences arise from the large amount of rupture
directivity that occurs for the combination of the shallow
hypocenter with the steeply dipping fault geometries and the
small amount of rupture directivity that occurs for the same
hypocenter with the shallow-dipping fault geometries.
As we found with the area where the maximum motion
exceeds a given level, we find less variation in how the mean
maximum motion decays with distance from the fault for the
deep, central hypocenter (Fig. 13) than for the shallow hy-
pocenter (Fig. 12). Near the trace of the fault, the mean dis-
placements and mean peak-to-peak velocities exhibit only
small variations. Likewise, the mean peak-to-peak velocities
on the down-dip (east) side of the fault decay in nearly an
identical fashion for all five fault dip angles. However, on
the up-dip (east) side of the fault, the mean displacements
and mean peak-to-peak velocities decay at varying rates for
the different fault dip angles. The values decay rapidly for
the steeply dipping faults and significantly more slowly
for the shallow dipping faults, although the displacements
for the 30
-dipping fault drop dramatically from the hanging
wall (east side) to the footwall (west side) before decaying
slowly with distance.
Radiated Energy
The far-field radiated energy (Fig. 14) displays the same
general trends across the 10 scenarios as the velocity am-
plitudes on the ground surface. We compute the far-field
radiated energy by finding the energy dissipated through the
damping matrix, which corresponds to the energy in the seis-
mic waves that propagate out to the boundaries of the do-
main (far field). Consequently, the radiated energy does not
account for energy at periods shorter than 2.0 sec, which is
not present in our numerical simulations.
The two scenarios with the largest amplitude ground
motions (scenarios Dip60Rk45HySh and Dip75Rk22HySh)
also radiate the largest amount of energy (1.1
10
16
J).
This is, in large part, due to the greater amount of mode-II
rupture in these scenarios. Scenario Dip90Rk0HySh radiates
slightly less energy (9.8
10
15
J). As expected from the
amplitude of the ground motions, the smallest radiated en-
ergy of 5.5
10
15
J occurs in scenario Dip30Rk90HySh
(Chi-Chi). The slower rupture speed in the mode-III direc-
tion relative to the mode-II direction accentuates the rela-
tively smaller amount of radiated energy for the ruptures that
are dominated by mode-III rupture. This results from the fact
that mode-II propagation is more effective than mode-III
propagation in generating far-field shear waves, and these
far-field shear waves carry most of the radiated energy.
Moving the hypocenter to the deep, central location re-
sults in a 23% increase in the amount of radiated energy for
the 30
-dipping fault, while it decreases the radiated energy
for the other fault geometries. Note that this increase in the
radiated energy stems solely from the shift in the hypocenter
and the associated increase in the amount of mode-II rupture.
Scenarios Dip75Rk22HyDp, Dip60Rk45HyDp, and Dip-
45Rk22HyDp all radiate about 7.2
10
15
J. Consequently,
in accordance with the level of long-period shaking, the sce-
narios with the centrally located deep hypocenter exhibit
much less variation in the radiated energy than those with
the shallow hypocenter. Overall, we find that the variations
in the far-field radiated energy closely follow the trends in
the amount of rupture directivity and the relative velocity
amplitudes on the ground surface.
Effects of Fault Dip and Slip Rake Angles on Near-Source Ground Motions
165
EW Dist. from Top of Fault (km)
Max. Disp. (m)
–
40
–
20
0
20
40
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
EW Dist. from Top of Fault (km)
Max. P
–
to
–
P Velocity (m/s)
–
40
–
20
0
20
40
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Dip 90
°
Dip 75
°
Dip 60
°
Dip 45
°
Dip 30
°
(Chi
–
Chi )
Figure
12.
Mean maximum horizontal displacements and mean maximum peak-
to-peak horizontal velocities as a function of distance from the fault for scenarios with
the shallow hypocenter. Although the mean values do not dramatically differ on the
down-dip (east) side of the fault, they span a wide range of values both up-dip (west)
of the fault and near the fault trace.
EW Dist. from Top of Fault (km)
Max. Disp. (m)
–
40
–
20
0
20
40
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
EW Dist. from Top of Fault (km)
Max. P
–
to
–
P Velocity (m/s)
–
40
–
20
0
20
40
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Dip 90
°
Dip 75
°
Dip 60
°
Dip 45
°
Dip 30
°
Figure
13.
Mean maximum horizontal displacements and mean maximum hori-
zontal peak-to-peak velocities as a function of distance from the fault for scenarios
with the deep, central hypocenter. The mean maximum values generally fall within a
smaller range for the deep hypocenter compared with the shallow hypocenter. On the
up-dip (west) side of the fault, the mean velocities decay more slowly as the fault dip
becomes shallower.
Discussion
The 10 scenarios we have discussed illustrate how
changes in fault geometry (fault dimensions, fault dip angle,
and slip rake angle) determine the amount that a rupture
propagates in the mode-II direction, which in turn controls
the amount of rupture directivity and the associated area sub-
jected to large-amplitude velocity and displacement pulses.
By design, scenario Dip30Rk90HySh approximates the rup-
ture of the Chi-Chi earthquake. Although we do not attempt
to model the complex changes in the strike of the fault and
the slip rake angle or the physics of the sliding processes,
we do match the nominal dip and rake angles (30
and 90
,
166
B. T. Aagaard, J. F. Hall, and T. H. Heaton
90
75
60
45
30
0
2
4
6
8
10
12
Fault Dip Angle (degrees)
Radiated Energy (J) x10
15
Shallow Hypocenter
Deep Hypocenter
Chi
–
Chi
Figure
14.
Far-field radiated energy for each of the two hypocenters for each of
the five different fault geometries. For predominantly along-strike, unilateral ruptures
(shallow hypocenter) the steeply dipping faults radiate considerably more energy than
the shallow dipping faults. For predominantly up-dip, bilateral ruptures (deep, central
hypocenter) the radiated energies from the five scenarios exhibit much less variation.
respectively), the hypocenter location (approximately 15 km
down-dip about one-quarter of the distance along the strike
of the rupture), fault area (2400 km
2
), and rupture speed
(faster in the south where the rupture propagates predomi-
nantly up-dip in the mode-II direction and slower in the
north where the rupture propagates predominantly along
strike in the mode-III direction) (Huang
et al.
, 2000; Ma
et
al.
, 2000; Johnson
et al.
, 2001; Ma
et al.
, 2001; Ji
et al.
,
2001). We also do not attempt to match the distribution of
slip on the fault but instead use a low-pass filtered random
distribution in order to allow creation of the slip distributions
for all 10 scenarios using the same procedure. Because we
are interested in the rupture directivity, this difference is not
as important as matching the geometry.
We compare measured peak horizontal velocities from
eight sites surrounding the surface rupture in the near field
of the Chi-Chi earthquake with those from six of the simu-
lated scenarios. The sites (locations are shown in Fig. 15)
are within approximately 20 km of the surface rupture and
include a wide range of azimuths so that we can investigate
the distribution of shaking associated with rupture directiv-
ity. We avoid sites that lie extremely close to the surface
rupture because they are especially sensitive to the amount
of nearby slip, which we do not match. We apply the
m
0
baseline correction algorithm of Boore (2001) to the Chi-
Chi acceleration time histories (Lee
et al.
, 2001) before in-
tegrating to obtain velocities. The velocities are low-pass
filtered using a third-order Butterworth filter with a corner
frequency of 0.5 Hz. The sites for the six simulated scenarios
match those of the Chi-Chi stations, and the fault geometry
for the 30
-dipping fault approximates the geometry of the
Chi-Chi rupture. For the simulated scenarios we align the
centers of the surface traces. We also low-pass filter the sim-
ulated time histories using a Butterworth filter with the same
corner frequency of 0.5 Hz; this has little effect because the
seismic source and discretization by design are bandlimited
to target periods of 2.0 sec and longer.
The peak horizontal velocities for the 30
-dipping fault
and shallow hypocenter (top panel in Fig. 15) indicate that
scenario Dip30Rk90HySh matches the limited amount of
up-dip rupture directivity that occurred near the epicenter in
the Chi-Chi earthquake. The peak velocities in the simula-
tion follow the observed amplitudes reasonably well. The
simplifications in the fault geometry along with the uniform
rake angles prevent a better match. In the Chi-Chi earth-
quake, the rotation of the rake angle toward oblique motion
in the north increased the amount of rupture directivity (al-
though it remained rather small). As a result, at stations T104
and T120 the peak ground velocities for the Chi-Chi earth-
quake exceed those from scenario Dip30Rk90HySh, which
has a uniform rake angle of 90
.
Given the similarities in the pattern of peak velocities
associated with rupture directivity between the Chi-Chi
earthquake and scenario Dip30Rk90HySh, one can draw
some conclusions about the areas subjected to large-ampli-
tude displacement and velocity pulses for events with ge-
ometries that can be approximated by the nine other scenar-
ios. The five other scenarios included in Figure 15 illustrate
the basic trends.
Scenarios with strike-slip to oblique motion, such as
scenarios Dip60Rk45HySh, Dip60Rk45HyDp, Dip90Rk-
0HySh, and Dip90Rk0HySh, create significantly more rup-
ture directivity than the Chi-Chi earthquake and scenario
Dip30Rk90HySh. In addition to the closer alignment of the
slip vector to the predominant direction of rupture propa-
gation in these scenarios, these more steeply dipping faults
have a significantly greater rupture length, which also con-
tributes to the increase in rupture directivity and larger-
amplitude motions. The ruptures direct most of their energy
toward the north or northwest (stations T038 and T104), as
Effects of Fault Dip and Slip Rake Angles on Near-Source Ground Motions
167
T038
T104
T071
T120
T079
T122
C080
C029
Easting (km)
Northing (km)
240
260
280
300
2600
2620
2640
2660
2680
2700
2720
T038
T104
T071
T120
T079
T122
C080
C029
0.0
0.5
1.0
1.5
Peak Horiz. Velocity (m/s)
Shallow Hypocenter
Chi
–
Chi
Dip 30
°
Dip 60
°
Dip 90
°
T038
T104
T071
T120
T079
T122
C080
C029
0.0
0.5
1.0
1.5
Station
Peak Horiz. Velocity (m/s)
Deep Hypocenter
Figure
15.
Comparison of peak horizontal velocities across eight stations. The left panel shows
the stations (triangles) in the Chi-Chi earthquake used in comparison with the simulations. The thick
line shows the surface rupture of the Chi-Chi earthquake, and the asterisk indicates the epicenter.
We also overlay the surface projection of the fault (thin and dashed lines) for scenarios
Dip30Rk90HySh and Dip30Rk90HyDp with the epicenter for scenario Dip30Rk90HySh indicated
by the open circle. The right panel shows the peak horizontal velocities at these eight stations for
the Chi-Chi earthquake and simulated scenarios for three fault dip angles for the shallow hypocenter
(top) and the deep central hypocenter (bottom). The general agreement between the peak velocities
from scenario Dip30Rk90HySh and the Chi-Chi earthquake in the top panel demonstrate that this
scenario generates a similar distribution of peak velocities to the Chi-Chi earthquake. Other dip
angles and the deep, central hypocenter generate significantly more rupture directivity, which results
in larger peak velocities at sites in the forward direction, such as stations T038, T122, and C080.
illustrated by the larger peak velocities in Figure 15, but even
sites near the southern end of the fault experience rupture
directivity, as demonstrated by station C080 in scenario
Dip90Rk0HySh.
The greater rupture directivity increases the area sub-
jected to a given level of shaking as well as the mean max-
imum velocities as a function of distance from the fault. For
example, scenarios Dip60Rk45HySh and Dip90Rk90HySh
generate peak-to-peak velocities greater than 1.2 m/sec over
areas of 3400 and 2400 km
2
, respectively, compared with
700 km
2
for scenario Dip30Rk90HySh (Fig. 9). Similarly,
the mean maximum peak-to-peak velocities on the up-dip
side of the fault remain above 1.2 m/sec out to distances of
only a few kilometers for scenario Dip30Rk90HySh com-
pared with 25 and 10 km for scenarios Dip60Rk45HySh and
Dip90Rk0HySh, respectively (Fig. 12). Thus, the Chi-Chi
earthquake, with its shallow dip angle and predominantly
along-strike rupture, resulted in significantly less rupture di-
rectivity and area subjected to large-amplitude velocities
compared with other fault geometries with a similar hypo-
center.
Returning to Figure 15, scenario Dip30Rk90HyDp
(30
-dipping fault with the deep hypocenter) demonstrates
how a deeper, more centrally located hypocenter yields more
rupture directivity and leads to larger ground motions. The
velocities increase at most locations up-dip from the hypo-
center, particularly those near the central portion of the fault
(stations T104, T120, and T122). Although the greatest
mean maximum horizontal displacement is still 2.0 m, the
greatest mean maximum peak-to-peak horizontal velocity
increases from 1.3 to 1.5 m/sec. Furthermore, the area on
the ground surface subjected to a given level of peak-to-peak
168
B. T. Aagaard, J. F. Hall, and T. H. Heaton
R
up
tu
re
Hyp
o
ce
nte
r
Rupture
Rup
tu
re
Rupture
Hypocenter
Hypocenter
Slip
Hyp
o
ce
n
te
r
Slip
Slip
Slip
Minimal Directivity
Maximum Directivity
Moderate Directivity
Moderate Directivity
Pro
p
ag
a
tio
n
Propagatio
n
Pro
p
a
ga
tio
n
Propagatio
n
Directivit
y
Directivity
Directivity
D
irec
tivity
D
ire
ctiv
ity
Figure
16.
Summary of how the fault geometry and the slip rake angle affect the
amount and location of rupture directivity as indicated by the size and location of the
splotch on the ground surface. The top row illustrates two cases for a vertical fault with
oblique motion, and the bottom row illustrates two cases for a shallow-dipping fault
with thrust motion. The left column corresponds to predominantly along-strike rupture
(i.e., the shallow hypocenter), whereas the right column corresponds to predominantly
bilateral or up-dip rupture (i.e., the deep hypocenter).
velocity increases significantly for peak-to-peak velocities
greater than 0.5 m/sec (see Figs. 9 and 10). This suggests
that the minimal amount of rupture directivity in the Chi-
Chi earthquake confined the large-amplitude motions to lo-
cations very close to the fault trace. Hence, the ground mo-
tions decayed rapidly with distance from the fault trace. Had
the hypocenter been much deeper or more centrally located
along the strike of the rupture, we expect that the ground
motions would have decayed less rapidly with distance from
the fault and been large over a much greater area.
Conclusions
Owing to the presence of shallow slip in the magnitude
7.4 earthquake simulations considered here, Love and/or
Rayleigh waves dominate the ground motions; strike-slip
faulting tends to generate Love waves, and thrust faulting
tends to generate Rayleigh waves. The amount of rupture
toward the surface in the direction parallel to slip (mode-II
direction and a local maximum in the shear-wave radiation
pattern) controls the severity of the long-period shaking.
The different levels of long-period motion can be quantified
using various measures of the ground shaking, including
(1) the area where the displacements and peak-to-peak ve-
locities exceed a given level and (2) the mean maximum
displacements and mean maximum peak-to-peak velocities
at a given distance from the fault. For strike-slip faulting the
shaking is most severe for unilateral rupture, while for thrust
faulting the shaking is most severe for up-dip rupture from
a deep hypocenter. Figure 16 summarizes how the direction
of propagation and the slip rake angle affect the location and
degree of rupture directivity. The directivity effect is maxi-
mized in the regions where the rupture propagates parallel
to the slip vector.
This set of simulations suggests that the amount of rup-
ture directivity in the 1999 Chi-Chi earthquake was small
compared to other possible events of the same size that have
a deeper, more centrally located hypocenter or have a longer,
more steeply dipping fault with oblique motion. Although
the complex geometry at the north end of the fault did create
some directivity in addition to that which occurred up-dip
from the hypocenter, in general, the rupture propagated
along a node in the shear-wave radiation pattern, which lim-
ited the amount of rupture directivity. As a result, the large-
amplitude displacement and velocity pulses that are often
associated with near-source ground motions were confined
to regions very close to surface rupture or near the surface
rupture on the hanging wall. This implies that we should
expect severe long-period ground motions over a much
larger area when events of the same size occur with other
styles of faulting or deeper hypocenters.
Acknowledgments
Access to the Hewlett-Packard V-Class computer, located at the Cali-
fornia Institute of Technology, was provided by the Center for Advanced
Computing Research. This work was supported in part by the Pacific Earth-
quake Engineering Research Center through the Earthquake Engineering
Research Centers Program of the National Science Foundation under
Award Number EEC-9701568. We appreciate the helpful comments from
Ned Field, Ken Hudnut, Art McGarr, and two anonymous reviewers.
Effects of Fault Dip and Slip Rake Angles on Near-Source Ground Motions
169
References
Aagaard, B. T. (1999). Finite-element simulations of earthquakes, Tech-
nical Report 99-03, California Institute of Technology, Earthquake
Engineering Research Laboratory, Pasadena, California.
Aagaard, B. T., J. F. Hall, and T. H. Heaton (2001a). Characterization of
near-source ground motions with earthquake simulations,
Earthquake
Spectra
17,
no. 2, 177–207.
Aagaard, B. T., T. H. Heaton, and J. F. Hall (2001b). Dynamic earthquake
ruptures in the presence of lithostatic normal stresses: implications
for friction models and heat production,
Bull. Seism. Soc. Am.
91,
no. 6, 1765–1796.
Aagaard, B. T., J. F. Hall, and T. H. Heaton (2002). Effects of fault dip
and slip rake on near-source ground motions: why the 1999
M
7.6
Chi-Chi, Taiwan, earthquake was relatively mild, Technical Report
2002/12, Pacific Earthquake Engineering Research Center, University
of California, Berkeley, California.
Allen, C. R., J. R. Brune, L. S. Cluff, and A. G. Barrows (1998). Evidence
for unusually strong near-field ground motion on the hanging wall of
the San Fernando fault during the 1971 earthquake,
Seism. Res. Lett.
69,
no. 6, 524–531.
Andrews, D. (1976). Rupture velocity of plane strain shear cracks,
J. Geo-
phys. Res.
81,
no. 32, 5679–5687.
Archuleta, R. J., and G. A. Frazier (1978). Three-dimensional numerical
simulations of dynamic faulting in a half-space,
Bull. Seism. Soc. Am.
68,
no. 3, 541–572.
Archuleta, R. J., and S. H. Hartzell (1981). Effects of fault finiteness on
near-source ground motion,
Bull. Seism. Soc. Am.
71,
no. 4, 939–957.
Boatwright, J., K. Thywissen, and L. Seekins (2001). Correlation of ground
motion and intensity for the 17 January 1994 Northridge, California,
earthquake,
Bull. Seism. Soc. Am.
91,
no. 4, 739–752.
Boore, D. M. (2001). Effect of baseline corrections on displacements and
response spectra for several recordings of the 1999 Chi-Chi, Taiwan,
earthquake,
Bull. Seism. Soc. Am.
91,
1199–1211.
Chen, K.-C., B.-S. Huang, J.-H. Want, W.-G. Huang, T.-M. Change, R.-D.
Hwang, H.-C. Chiu, and C.-C. P. Tsai (2001). An observation of
rupture pulses of the 20 September 1999 Chi-Chi, Taiwan, earthquake
from near-field seismograms,
Bull. Seism. Soc. Am.
91,
1247–1254.
Day, S. M. (1982). Three-dimensional simulation of spontaneous rupture:
the effect of nonuniform prestress,
Bull. Seism. Soc. Am.
72,
no. 6,
1881–1902.
Graves, R. W. (1998). Three-dimensional finite-difference modeling of the
San Andreas fault: source parameterization and ground-motion levels,
Bull. Seism. Soc. Am.
88,
no. 4, 881–897.
Hall, J. (1995). Parameter study of the response of moment-resisting steel
frame buildings to near-source ground motions, Technical Report 95-
08, Earthquake Engineering Research Laboratory, California Institute
of Technology, Pasadena, California.
Hall, J. (1997). Response of steel frame buildings to near-source ground
motions, Technical Report 97-05, Earthquake Engineering Research
Laboratory, California Institute of Technology, Pasadena, California.
Hall, J. F., T. H. Heaton, M. W. Halling, and D. J. Wald (1995). Near-
source ground motion and its effects on flexible buildings,
Earthquake
Spectra
11,
no. 4, 569–605.
Haskell, N. (1969). Elastic displacements in the near-field of a propagating
fault,
Bull. Seism. Soc. Am.
59,
no. 2, 865–908.
Heaton, T. H. (1990). Evidence for and implications of self-healing pulses
of slip in earthquake rupture,
Phys. Earth Planet. Interiors
64,
no. 1,
1–20.
Hisada, Y., H. Bao, J. Bielak, O. Ghattas, and D. O’Hallaron (1998). Sim-
ulations of long-period ground motions during the 1995 Hyogo-Ken
Nanbu (Kobe) earthquake using a 3-D finite element method, in
2nd
International Symposium on Effect of Surface Geology on Seismic
Motion
, Yokohama, Japan, 1–3 December, 59–66.
Huang, B., K. Chen, W. Huang, J. Wang, T. Chang, R. Hwang, H. Chiu,
and C. Tsai (2000). Characteristics of strong ground motion across a
thrust fault tip from the September 21, 1999, Chi-Chi, Taiwan earth-
quake,
Geophys. Res. Lett.
27,
no. 17, 2729–2732.
Inoue, T., and T. Miyatake (1998). 3-D simulation of near-field strong
ground motion based on dynamic modeling,
Bull. Seism. Soc. Am.
88,
no. 6, 1445–1456.
Iwan, W., and X. Chen (1994). Important near-field ground motion data
from the Landers earthquake, in
Proc. of the 10th European Confer-
ence on Earthquake Engineering
, Vol. 1, Vienna, 28 August–2 Sep-
tember, 229–234.
Ji, C., D. V. Helmberger, T. Song, K. Ma, and D. J. Wald (2001). Slip
history and tectonic implication of the 1999 Chi-Chi earthquake,
Geo-
phys. Res. Lett.
28,
no. 23, 4379–4382.
Johnson, K., Y. Hsu, P. Segall, and S. Yu (2001). Fault geometry and slip
distribution of the 1999 Chi-Chi, Taiwan earthquake imaged from
inversion of GPS data,
Geophys. Res. Lett.
28,
no. 11, 2285–2288.
Kamae, K., and K. Irikura (1998). Source model of the 1995 Hyogo-Ken
Nanbu earthquake and simulation of near-source ground motion,
Bull.
Seism. Soc. Am.
88,
no. 2, 400–412.
Lee, W. H. K., T. Shin, W. Kuo, K. Chen, and C. Wu (2001). CWB free-
field strong-motion data from the 21 September Chi-Chi, Taiwan,
earthquake,
Bull. Seism. Soc. Am.
91,
1370–1376.
Ma, K., T. Song, S. Lee, and H. Wu (2000). Spatial slip distribution of the
September 20, 1999, Chi-Chi, Taiwan, earthquake (M(w) 7.6): in-
verted from teleseismic data,
Geophys. Res. Lett.
27,
no. 20, 3417–
3420.
Ma, K., J. Wang, and D. Zhao (1996). Three-dimensional seismic velocity
structure if the crust and uppermost mantle beneath Taiwan,
J. Phys.
Earth
44,
no. 2, 85–105.
Ma, K.-F., J. Mori, S.-J. Lee, and S. Yu (2001). Spatial and temporal dis-
tribution of slip for the 1999 Chi-Chi, Taiwan, earthquake,
Bull.
Seism. Soc. Am.
91,
no. 5, 1069–1087.
Madariaga, R., K. Olsen, and R. Archuleta (1998). Modeling dynamic rup-
ture in a 3-D earthquake fault model,
Bull. Seis. Soc. Am.
88,
no. 5,
1182–1197.
Mai, P., and G. Beroza (2002). A spatial random-field model to characterize
complexity in earthquake slip,
J. Geophys. Res. Solid Earth
107,
no. B11, 2308, doi 0.1029/2001JB000588.
Nielsen, S., and K. Olsen (2000). Constraints on stress and friction from
dynamic rupture models of the 1994 Northridge, California, earth-
quake, Pure Appl. Geophys.
157,
no. 11–12, 2029–2046.
Oglesby, D. D., and S. M. Day (2001). Fault geometry and dynamics of
the 1999 Chi-Chi (Taiwan) earthquake,
Bull. Seism. Soc. Am.
91,
1099–1111.
Oglesby, D. D., R. J. Archuleta, and S. B. Nielsen (2000). The three-
dimensional dynamics of dipping faults,
Bull. Seism. Soc. Am.
90,
no. 3, 616–628.
Olsen, K., and R. Archuleta (1996). Three-dimensional simulation of earth-
quakes on the Los Angeles fault system,
Bull. Seism. Soc. Am.
86,
no. 3, 575–596.
Olsen, K., R. Madariaga, and R. Archuleta (1997). Three-dimensional dy-
namic simulation of the 1992 Landers earthquake,
Science
278,
834–
839.
Olsen, K. B., R. J. Archuleta, and J. R. Matarese (1995). Three-dimensional
simulation of a magnitude-7.75 earthquake on the San-Andreas Fault,
Science
270,
no. 5242, 1628–1632.
Ouchi, T., A. Lin, A. Chen, and T. Maruyama (2001). The 1999 Chi-Chi
(Taiwan) earthquake: earthquake fault and strong motions,
Bull.
Seism. Soc. Am.
91,
966–976.
Pitarka, A., K. Irikura, T. Iwata, and S. Sekiguchi (1998). Three-dimen-
sional simulation of the near-fault ground motion for the 1995 Hyogo-
ken Nanbu (Kobe), Japan, earthquake,
Bull. Seism. Soc. Am.
88,
no.
2, 428–440.
Somerville, P. G., N. F. Smith, R. W. Graves, and N. A. Abrahamson
(1997). Modification of empirical strong ground motion attenuation
relations to include the amplitude and duration effects of rupture di-
rectivity,
Seism. Res. Lett.
68,
no. 1, 199–222.
Wald, D. J., V. Quitoriano, T. H. Heaton, and H. Kanamori (1999). Rela-
170
B. T. Aagaard, J. F. Hall, and T. H. Heaton
tionships between peak ground acceleration, peak ground velocity,
and modified Mercalli intensity in California,
Earthquake Spectra
15,
no. 3, 557–564.
Wells, D., and K. Coppersmith (1994). New empirical relationships among
magnitude, rupture length, rupture width, rupture area, and surface
displacement,
Bull. Seism. Soc. Am.
84,
no. 4, 974–1002.
U.S. Geological Survey
525 South Wilson Ave.
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Manuscript received 17 March 2003.