Spatial Distribution of Simulated Response for
Earthquakes, Part I: Ground Motion
Jacobo Bielak, Antonio Fernández, Gregory L. Fenves, Jaesung Park, and
Bozidar Stojadinovic
Corresponding author: Gregory L. Fenves
Mailing address:
Department of Civil and Environmental Engineering
University of California, Berkeley
Berkeley, CA 94720-1710
Phone: 510-643-8543
Fax:
510-643-5264
Email: fenves@ce.berkeley.edu
Submission date for review copies:
July 30, 2004
Submission date for camera-ready copies:
<<<Greg, even though I agree with the change you made on page 3 regarding finite
differences and finite elements, I prefer to put finite differences and finite elements on an
equal footing to avoid controversy. Otherwise we would have to add a more detailed
explanation since a number of people have developed efficient FD codes for forward wave
propagation in basins>>>
Bielak -- 1
Spatial Distribution of Simulated Response for
Earthquakes, Part I: Ground Motion
Jacobo Bielak,a) M.EERI, Antonio Fernández,b) M.EERI, Gregory L. Fenves,c)
M.EERI, Jaesung Park,d) and Bozidar Stojadinovice), M.EERI
The objective of this study is to examine, by computational simulation, the
spatial and temporal distribution of the earthquake ground motion near a causative
fault. An idealized 20 km by 20 km region is modeled as a layer on an elastic
halfspace with dense spatial sampling for frequencies up to 5 Hz. Two scenario
events are considered, a strike-slip fault and a thrust fault, by the use of finite
dislocation models. For the strike-slip fault event, the fault normal component
exhibits a very strong forward directivity effect with a strong pulse-type motion
approximately twice the amplitude of the motion in the fault parallel direction.
The dynamic effect in the fault parallel direction produces a pulse-type motion
near the epicenter. For the thrust fault event the greatest concentration of ground
displacement occurs near the corners of the fault opposite the hypocenter, in the
rake direction. In contrast with the strike-slip fault, the ground displacement in
the direction of the slip is greater by a factor of two than in the direction normal to
the slip. In a companion paper, we examine how the free-field ground motion
from the two scenario earthquakes influences the spatial and temporal distribution
of structural response of a family of single degree-of-freedom elastoplastic
systems.
INTRODUCTION
Large earthquakes in urban regions cause a highly variable spatial distribution of damage
to the built-infrastructure because of source effects, path effects, large-scale geological
structures such as sedimentary basins, site response effects, and the structural characteristics
of the buildings and other infrastructure components. The distribution of damage in the 1994
Northridge Mw=6.5 earthquake (Somerville, 1995) and the 1995 Hyogoken-Nanbu, Japan,
Mw=6.9 earthquake (Akai et al., 1995) raised profound questions about these effects, with
particular concern about the large velocity pulses at sites near the faults. Additional data
obtained from the 1999 Kocaeli, Turkey, Mw=7.4 earthquake (Rathje et al., 2000) and the
1999 Chi-Chi, Taiwan, Mw=7.6 earthquake (Li and Shin, 2001) provided further evidence
that the spatial distribution of ground motion in a region is related to the fault mechanism and
a)
Professor, Department of Civil and Environmental Engineering, Carnegie Mellon University, Pittsburgh, PA,
15213.
b)
Formerly, Graduate Student Researcher, Department of Civil and Environmental Engineering, Carnegie
Mellon University, Pittsburgh, PA, 15213; currently, Manager of International Projects, Paul C. Rizzo Assoc.,
520 Exposition Mall, Monroeville, PA, 15245.
c)
Professor, Department of Civil and Environmental Engineering, University of California, Berkeley, CA,
94720.
d)
Graduate Student Researcher, Department of Civil and Environmental Engineering, University of California,
Berkeley, CA, 94720.
e)
Associate Professor, Department of Civil and Environmental Engineering, University of California, Berkeley,
CA, 94720
Bielak -- 1
path effects. The large number of ground motion data recorded in these earthquakes
confirmed that near-fault pulse-type ground motion can be very damaging to buildings, as
first identified in the 1971 San Fernando earthquake (Bertero et al., 1978).
Researchers have examined the spatial distribution of ground motion based on the
extensive records obtained in the 1994 Northridge earthquake. Aftershock records obtained
by Meremonte et al. (1996) and Hartzell et al. (1997) showed that the amplitude of peak
ground velocity varied by a factor of two or greater at different sites with the same general
geological structure but separated by distances of only 200 m. Boatwright et al. (2001)
developed correlations between ground motion parameters and effects on structural response,
as measured by an intensity measure based on building tags. The correlations show that peak
ground acceleration (PGA) is a poor predictor of damage, peak ground velocity (PGV) is
much better, and that an averaged pseudo-velocity (between 0.3 and 3.0 seconds period),
which is related to PGV, is a good predictor. Contours of PGV and averaged pseudo-velocity
correspond to the spatial distribution of damage based on a tagging intensity. In another
study using 1994 Northridge earthquake strong motion data, Bozorgnia and Bertero (2002)
examined the spatial distribution of structural response parameters associated with damage.
The spatial distribution of ductility demand for single-degree-of-freedom (SDF) systems of
1- and 3-sec vibration periods using a code-specified strength are qualitatively similar to the
PGV contours. The large ductility demands are concentrated in the updip direction of slip for
the 1-sec case and the updip direction and epicentral area for the 3-sec case. Additional
results are presented for spatial distributions of structural damage measures that include the
effect of ductility and hysteretic energy demands and capacities.
An important feature of large-magnitude earthquakes is the large amount of seismic
energy from the rupture that is concentrated in the forward directivity zone near the fault.
This energy is generally manifested as a single large pulse of ground motion (e.g., Somerville
et al., 1997). Somerville (1998) defines the forward directivity effects occurring at sites
where the fault rupture propagates towards the site and the direction of fault slip is aligned in
the direction of the site. These conditions are met with strike-slip faults and dip-slip rupture
in thrust faults. Forward directivity effects in the fault normal direction occur in all locations
along a strike-slip fault. For thrust faults, however, forward directivity effects occur mostly
in the surface projection of the fault, updip from the hypocenter. Backward directivity and
neutral directivity effects generally produce longer period, but lower amplitude, motion in the
fault normal and fault parallel directions. Somerville et al. (1997) developed a procedure for
considering the effects of near-fault motion on the ground motion from strike-slip faults by
modifying the elastic response spectra. In this approach, the spatial effect of directivity
depends on geometric parameters on the direction of fault rupture, the location of the site
with respect to the epicenter, and the fraction of fault rupture between the epicenter and site.
To account for near-fault effects in seismic design of buildings, the 1997 Uniform
Building Code (ICBO, 1997) incorporated near-fault amplification factors for the
acceleration-sensitive and velocity-sensitive period bands of the elastic design response
spectrum. The factors depend on the fault mechanism and magnitude, and also the distance
to the fault (within 15 km). The 1997 UBC near-fault factors are comparable with the mean
values for spectrum modification factors from Somerville et al. (1997), but they do not
distinguish between fault normal and fault parallel ground motion, for which the former is
substantially greater in the forward directivity zone.
Although recent large earthquakes have produced extensive ground motion records as a
result of increased instrumentation in urban areas, the spatial resolution is not high enough to
Bielak -- 2
describe fully the distribution of ground motion. Thus, there is an important need for highresolution, realistic simulations of ground motion in a region to study the seismological,
geotechnical, and structural effects of urban earthquakes in detail. A number of studies have
been devoted recently to modeling earthquake ground motion in realistic basins (e.g.,
Frankel, 1993; Graves, 1993; Olsen et al, 1995; Olsen and Archuleta, 1996; Pitarka et al,
1998; Hisada et al., 1998; and Stidham et al., 1999). These simulations, however, were
generally limited to low frequencies (≤ 0.5 Hz). Motivated by the 1994 Northridge
earthquake, Hall et al. (1995) and Hall (1998) simulated the ground motion produced by an
earthquake on a thrust fault to examine the effect of near-fault ground motions on 20-story
steel frame buildings and long-period based isolated buildings. The fault rupture parameters
were calibrated to correspond to a scenario Mw=7.0 event on the Elysian Park buried thrust
fault in Los Angeles. Ground motions were simulated on an 11 by 11 grid with 5-km
spacing. The peak ground velocity was 1.8 m/s and large areas had velocities greater than 1
m/s. The near-source velocity pulses produced large drifts in buildings located at grid points
in the forward-directivity zones, and fracture of the moment-resisting connections has a
detrimental effect on drifts.
The pioneering work of Aki (1968) and Haskell (1969), in which a kinematic model of an
extended fault was used to simulate the earthquake source, provided new means to simulate
ground motion in a region, including near-fault effects. This was followed by comprehensive
parametric studies by Boore et al. (1971), Archuleta and Hartzell (1981), Anderson and Luco
(1983a, 1983b), and Luco and Anderson (1983), among others, In the past ten years
numerical modeling methods for anelastic wave propagation that take into consideration the
earthquake source, propagation path, and local site effects in realistic models of basins have
become increasingly available using several methods. Boundary element and discrete
wavenumber methods have been popular for moderate-sized problems with simple geometry
and geological conditions (e.g., Mossessian and Dravinski, 1987; Kawase and Aki, 1990;
Bielak et al., 1991; Hisada et al., 1993; Sánchez-Sesma and Luzón, 1995; Bouchon and
Barker, 1996). Finite differences (e.g., Frankel and Vidale, 1992; Frankel, 1993; Graves,
1993, 1996; Olsen et al., 1995; Pitarka, 1999; Stidham et al., 1999; Sato et al., 1999) and
finite elements (e.g., Lysmer and Drake, 1971; Toshinawa and Ohmachi, 1992; Bao et al.,
1998; Aagard et al., 2001) are better suited for large-size problems that involve realistic basin
models with highly heterogeneous materials because of their flexibility and simplicity.
Bao et al. (1998) developed one of the first scalable, parallel, finite element tools for
large-scale ground motion simulation in sedimentary basins with heterogeneous materials.
Aagaard et al. (2001) used parallel finite element simulations to investigate the sensitivity of
long-period ground motion to different seismic source parameters, on strike-slip fault and
thrust fault mechanisms, with particular emphasis on the large velocity pulses in the forward
directivity zones. Their simulations show that the peak ground motion occurs within the
zones for which the 1997 UBC specifies near-fault factors. A buried thrust fault simulation,
however, shows peak ground motions occurring in the updip direction over a larger area than
defined for the near-fault factor.
Previous research using recorded ground motion data and early simulations has examined
the spatial distribution of ground motion and its effects on structural response. Due to the
relatively few number of ground motion records and earlier limitations in computational
capabilities, there remain additional questions concerning the characterization of the spatial
effects, the systematic interpretation of such effects using structural dynamic concepts, and
the evaluation of building code approaches for accounting for the effects. Thus, the objective
of this paper is to use the simulation of two scenario earthquake events with frequencies up to
Bielak -- 3
5 Hz to obtain a dense spatial sampling of ground motion over a large region to elucidate the
effects of the source and path on the near-fault ground motions. A companion paper (Fenves
et al., 2004) then uses the dense sampling of ground motion to examine the distribution of
structural response near the faults in the two scenarios. The use of an idealized two-layer
model of a region and single degree-of-freedom inelastic systems to represent structural
response allows the papers to focus on fundamental trends.
DESCRIPTION OF MODEL AND SCENARIO EARTHQUAKE EVENTS
The model considered in this study consists of an elastic layer on an elastic halfspace, as
shown in Figure 1, to represent a 20 km by 20 km region of rock material. We consider a
buried strike-slip fault (Figure 1a) and a buried thrust fault (Figure 1b) to represent two of the
most common types of earthquakes. The top edge of the vertical fault is 1 km beneath the
interface between the layer and the halfspace, whereas the top edge of the inclined fault
touches the interface. Red circles in Figure 1 denote the hypocenters of the earthquakes. The
dip angle (40º) of the thrust fault is similar to the 1994 Northridge earthquake source. The
density,  , shear-wave velocity, vs , and P-wave velocity, v p , of the layer are 2.6 g/cm3, 2.0
km/s, 4.0 km/s, respectively, and the corresponding values for the halfspace are 2.7 g/cm3,
3.46 km/s, 6.0 km/s. Material attenuation is not considered in the simulations.
We assume kinematic rupture of the fault by imposing a dislocation across the fault (jump
in the tangential displacement and continuous normal displacement). The rupture propagates
radially from the hypocenter at a constant speed until it reaches the edges of the fault. The
slip direction of rake is 0º and 90º for the vertical and the inclined faults, respectively; that is,
the particle motion across the fault is along the length of the fault in both cases. For the
vertical fault the motion is right lateral strike-slip, and for the inclined fault the motion of the
hanging wall is upward and that of the footwall is downward (thrust slip). The variation of
the dislocation with time is defined by the slip function u(t)  Df (t) , where D is the
uniform slip, and,
 
 t 
t 
f (t)  1  1  exp     H(t)
 T0  
  T0 
where T0 is the slip rise time. There is a delay time at each location with respect to the onset
time at the hypocenter of r / vrupt , where r is the hypocentral distance from the particular
location, and vrupt is the rupture velocity. In the simulations, we assume vrupt =3.0 km/s and
T0 = 0.1 sec, which allows modeling seismic waves up to 5.0 Hz. The model for the strikeslip event is the same that has been used by a group of modelers, including some of this
paper’s authors, in verifying several finite difference and finite element codes for simulating
earthquake ground motion in large regions (Day, 2002).
The seismic moment, M 0 , of an earthquake can be expressed as (Aki, 1966), M 0   AD ,
where  is the shear modulus of the crustal volume that contains the fault, and A is the area
of the causative fault. The moment magnitude, M w , is related to the seismic moment in


dyne-cm (Kanamori, 1977): M w  0.67 log 10 M 0  10.7 . For the scenario events, Mw = 6.0
and 5.8 is selected for the strike-slip fault and the thrust fault earthquakes, respectively.
Bielak -- 4
These magnitudes correspond to slip D of 0.125 m and 0.0542 m, respectively, for the two
events.
Despite their simplicity, models of this type have been shown to capture the essential
nature of earthquake ground motion in the near-fault region. For example, in a recent
simulation of the 1992 Landers earthquake in southern California, Hisada and Bielak (2004)
modeled the crustal region in the vicinity of the epicenter as a single layer on a halfspace and
the causative fault by piecewise strike-slip planar surfaces, and calculated the ground motion
at the Lucerne Valley station using an integral representation technique (Hisada and Bielak,
2003), with satisfactory results. Similarly, results for the spatial distribution of ground
motion to be presented later for the thrust-fault event are qualitatively similar to those
observed during the 1994 Northridge earthquake.
COMPUTATIONAL METHOD FOR GROUND MOTION SIMULATION
To simulate slip on the fault and the resulting ground motion within the domain, we use
an elastic wave propagation, finite element code developed for modeling earthquake ground
motion in large sedimentary basins (Bao et al., 1998). The wave propagation code is built
using Archimedes, a software environment for solving unstructured-mesh finite element
problems on parallel computers (Bao et al., 1998). Archimedes includes two- and threedimensional mesh generators, a mesh partitioner, a parceler, and a parallel code generator.
We use standard, Galerkin linear tetrahedral elements for the spatial discretization of the
governing Navier equations of elastodynamics over the tetrahedral mesh. The element sizes
are tailored to the local wavelengths of the propagating waves, and the mesh generation
strictly controls the aspect ratio of every element such that it does not exceed a prescribed
value. The spatial discretization leads to a standard system of second-order ordinary
differential equations with constant coefficients. These equations are solved using the central
difference method, an explicit, conditionally stable, step-by-step algorithm in the time
domain. To avoid the need of solving a system of algebraic equations at each time step, a
lumped mass matrix is used. As a result, the only significant operation at a time step is a
matrix-vector multiplication.
Two important issues must be considered for solving earthquake wave propagation
problems in infinite domains by the finite element method. One is the requirement for a
finite domain of computation and the need to limit spurious reflections at the artificial
absorbing boundaries. This is accomplished by a sponge layer of viscous material near the
artificial boundaries (Israeli and Orszag, 1981) and by placing a set of nodal viscous dampers
directly on the artificial boundaries, along the lines described in Lysmer and Kuhlemeyer
(1969). The second issue is the need to represent the slip on the fault in the finite element
formulation. We do this, following Aki and Richards (1980), by expressing the slip in terms
of a set of double couples, which in turn are expressed as body forces. The body forces are
reduced to equivalent nodal forces through the Galerkin process for spatial discretization
(Bao, 1988). To determine the amplitude of each double couple, the fault is divided into a
number of subfaults. For the case of uniform slip, the double couple within each element
intersected by the fault is proportional to the seismic moment, based on the area of the fault
contained within the element divided by the total area of the fault. The finite element code
has been verified with several finite differences codes for idealized and realistic earthquakes
(Day, 2002), including the strike-slip event considered here, with satisfactory results.
Bielak -- 5
(a) Strike-slip fault scenario
(b) Thrust fault scenario
Figure 1. Model of faults in a homogeneous elastic layer on a homogeneous elastic halfspace.
RESULTS OF GROUND MOTION SIMULATIONS
The large-scale finite element solution of the governing equations described in the
previous section is applied to the two scenario earthquake events with the idealized domain.
In earlier studies, Bao (1998) and Bao et al. (1998) found that numerical dispersion could
be kept to less than five percent over distances of 50 km by using 8 to 10 linear tetrahedral
elements per wavelength. Thus, for the present application the mesh size is 40 m for the top
layer and 80 m for the halfspace to represent ground motion up to 5 Hz. The discretized
Bielak -- 6
equations of motion (N=42 million) were solved on 128 processors of the Cray T3E at the
Pittsburgh Supercomputing Center.
STRIKE-SLIP FAULT
Tracking the ground motion from its inception, we first examine the displacement field in
a buried horizontal plane that passes exactly through the hypocenter of the vertical strike-slip
fault (Figure 1a). Figure 2 shows the spatial distribution of the peak horizontal displacement
in the hypocentral plane in the fault parallel (FP) and fault normal (FN) directions. For the
right-lateral slip considered in this scenario, the FP motion north of the fault plane is to the
right, whereas the motion south of it is to the left. The finite element discretization requires a
continuous displacement field throughout the domain; yet, slip is nonetheless allowed to
occur across the fault, spread over a one-element wide band. The amplitude of the slip across
the fault plane is uniform along the length of the dislocation and is equal to the prescribed
value of D=0.125 m. As expected from antisymmetry considerations, there is no motion in
the FP direction on the fault plane beyond the ruptured zone. Thus, the displacement across
the fault conforms exactly to the prescribed slip, including the requirement that the FN
displacement be continuous across the fault (Figure 2 bottom). This pattern of behavior
confirms that representing the prescribed slip by body forces is equivalent to imposing
directly the slip on the fault. Away from the fault, the FP displacement in the hypocentral
plane is concentrated primarily in the regions north and south of the fault. Some asymmetry
is observed with respect to a plane perpendicular to the fault plane passing through the
middle of the dislocation, but it is minor.
Figure 2. Strike-slip earthquake. Spatial distribution of the peak horizontal displacement in the
hypocentral plane in (top) fault parallel and (bottom) fault normal directions.
Bielak -- 7
On the other hand, the motion in the FN direction is dramatically different from that in
the FP direction. The FN motion arises primarily as a consequence of the couples that are
generated in the north-south direction along the fault to balance the east-west couples that
produce the slip. As a result of the fault rupture propagation, a nodal line develops east of the
hypocenter; points east of this line move north while those to the west move south. The
nodal line is curved and the FN motion is not symmetric with respect to the nodal line
because of the propagating nature of the rupture. The FN displacement is much larger in the
direction of the rupture propagation than in the backward direction, and it affects a wide
region on and away from the fault. The FN motion reaches its maximum value precisely at
the east end of the dislocation, where there is a stress concentration. This evidence of the
forward directivity effect, as will be seen later, occurs even more prominently on the free
surface. The forward directivity effects are dynamic in nature and a direct consequence of
the constructive interference of the propagating waves that travel in the direction of the
rupture, as can be seen by comparing Figure 2 with Figure 3, which shows the residual,
permanent FP and FN components of displacement in the horizontal hypocentral plane. In
contrast with the dynamic components evident especially at the bottom of Figure 2, the FN
residual displacement is antisymmetric with respect to the midpoint of the dislocation. Also,
the peak amplitude of the FP residual component across the fault is quite similar to that
observed during the passage of the seismic wave, but there is a significant reduction in the
amplitude of the FN permanent component with respect to that of the total displacement,
indicating significant dynamic action. The dynamic effect is much smaller in the FP
direction.
Figure 3. Strike-slip earthquake. Residual (permanent) (top) fault parallel and (bottom) fault normal
components of displacement in the hypocentral plane.
Bielak -- 8
To examine the ground motion generated by the propagating waves on the free surface of
the layered system, we next consider the spatial distribution of the absolute values of the FP
and FN components of the free surface peak displacements and velocities, as well as the
maximum amplitudes of the corresponding resultant displacement and velocity vector fields,
as shown in Figure 4. The FP velocity clearly shows that most of the dynamic effect in the
FP direction occurs near the epicenter. As at the hypocentral plane, the FN component of
velocity) exhibits a very strong forward dynamic directivity effect. The peak value of FN
component is twice that of the FP component. This difference in amplitudes is obvious for
the resultant velocity, in which the contributions of both the FP and FN components are
apparent, but where the effect of the FN component is much stronger. The distribution of the
acceleration (not shown) is very similar to that of the velocity. The peak values of the FP and
FN components of displacement also differ by a factor of about two, as for the velocity
components. In contrast with the velocity, however, both the FP and FN components of
displacement show a strong forward directivity effect on the free surface. Interestingly, the
decrease in the velocity with distance from the fault plane is quite rapid, compared with
attenuation of the displacement. The important implications that these differences in ground
motion have for the response of long- and short-period structures are examined in the
companion paper (Fenves et al., 2004).
To explore further the effects of directivity on the ground motion, the synthetic
seismoscope records in Figure 5 show the displacement path at a number of locations on a
regular grid on the ground surface. Solid circles indicate the position of each point at the end
of the earthquake. These records show that the direction and amplitude of motion vary
widely over the region. Due to symmetry, points directly north of the fault experience
motion only in the FN direction. This motion reaches its maximum in the forward directivity
direction, some distance away from the edge of the dislocation. Further north and south of
the fault the motion is predominantly in the FP direction, while at other locations the ground
experiences FP and FN motion of comparable amplitude.
In addition to the differences in amplitude and orientation of motion throughout the
region, the ground motion also varies in duration and frequency content, depending on
location. Figure 6 shows seismograms of ground velocity and displacement in two
orthogonal directions at selected locations. These have been low-pass filtered up to 5 Hz.
Open circles indicate the locations of the observation points, and the direction of particle
motion at each location is perpendicular to the time axis of the corresponding seismogram.
Thus, for example, the point at station S3 experiences a peak velocity of 1.92 m/s in the north
(FN) direction and a peak value of 0.03 m/s in the west (FP) direction, while the peak FP
velocity at station S5 is 0.93 m/s, and the FN component is 0.09 m/s. It is noteworthy that
whereas at stations in the forward directivity direction (e.g., S3, S4, S7, S8) the FN
component of velocity that exhibits a strong pulse-like ground motion, at stations located
directly north of the epicenter (e.g., S5) it is the FP component of velocity that exhibits a
strong pulse-like behavior. This behavior is evident also for the other stations located at short
distances north (and south) of the fault (e.g., S17, S6). Farther away, the seismograms
become more complex in shape and their duration becomes longer; also, the dominant
periods appear to become longer with distance from the fault (e.g., S15). These effects are
mainly due to wave dispersion and contributions from the surface waves. In the direction of
forward directivity only the FN component is significant, whereas at S6, north and middle of
the fault, the amplitude of both the FP and FN components is significant. The same is true at
the location S17 at the back of the fault. As expected, the displacement seismograms are
Bielak -- 9
smoother than those for the velocity. In addition, displacement seismograms exhibit
permanent offsets. This fling is related to the permanent deformation at the site.
To examine differences in frequency content between the seismograms it is useful to use
response spectra, as these also provide insight into the spatial distribution of the dynamic
response of elastic SDF systems to the ground motion. Figures 7 and 8 show the response
spectra for five percent critical damping in the FN and FP directions, respectively, calculated
from the corresponding free-field synthetic seismograms. The spectra are plotted in the usual
tripartite logarithmic representation for the pseudo-velocity, pseudo-acceleration and
displacement to study the period ranges for which the spectra are most sensitive to the ground
velocity, ground acceleration, and ground displacement. All the spectra are drawn for
periods, T, in the range of 0.5 sec 8.0 sec. Also shown in the spectra plots are the peak
ground acceleration (PGA), peak ground velocity (PGV), and peak ground displacement
(PGD) at each location, drawn as piecewise straight lines.
Displacement
Envelope
Fault Normal
Fault Parallel
Velocity
Figure 4. Strike-slip earthquake. Spatial distribution of the absolute values of the fault parallel and
fault normal components of the free surface peak velocity and displacement, and the maximum
amplitudes of the corresponding resultant velocities and displacements.
Bielak -- 10
Figure 5. Strike-slip earthquake. Synthetic seismoscope records of the horizontal displacement path
at a number of locations on a regular grid on the free surface.
In the FN direction (Figure 7), the largest pseudo-velocity occurs, not surprisingly, at
stations where the effects of forward directivity are strongest (e.g., S3, S4, S7, S8). The
largest values of pseudo-velocity occur at periods where the lines representing the PGA and
PGD intersect (around a period of 0.6 sec). For long periods, the response spectrum is close
to the constant PGD line and for the shorter periods it approaches the constant PGA line. The
PGV line lies above the point of intersection of the constant PGA and PGD lines, and is,
therefore, not shown. Immediately adjacent to the fault in the backward direction, the shape
of the response spectra is quite similar to that in the forward direction, though the amplitude
is smaller. Farther north from the fault, the spectral shape begins to change, acquiring a clear
velocity-sensitive band as the PGV line becomes sufficiently small to intersect the PGA and
PGD lines below their own intersection. Consistent with the increase of the dominant
periods with increased distance to the fault observed in the seismograms, the period at which
the pseudo-velocity attains a peak value also increases with distance from the fault, e.g., the
peak for station S15 occurs at 2.2 sec.
Bielak -- 11
(a) Velocity
.
(b) Displacement
Figure 6. Strike-slip earthquake. Seismograms of (a) velocity (m/sec) and (b) displacement in two
orthogonal directions at selected locations on the free surface. Open circles indicate the locations of
the observation points, and the direction of motion is denoted by the orientation of the seismograms.
The direction of particle motion at each location is perpendicular to the time axes of the
corresponding seismogram. <<<Have Jaesung move the legend S7 in (a) a little to the left>>>
Bielak -- 12
Figure 7. Strike-slip earthquake. Response spectra for 5 percent critical damping for ground motion
in the fault normal direction, calculated from the corresponding free-field accelerograms. The spectra
are plotted in the usual tripartite logarithmic representation, with pseudo-velocity on the vertical axis
and period on the horizontal axis. The period range is 0.5 sec to 8.0 sec.
Figure 8. Strike-slip earthquake. Response spectra in fault parallel direction, as described in Figure 7.
Bielak -- 13
The response spectra for the FP component of the ground motion exhibit trends that are
the reverse of the FN component. By comparing the spectra at stations S12, S5, and S6, with
those at S3, S4, S7, and S8 in Figures 7 and 8, the FP component of the former look like the
FN components of the latter; and, vice versa, the FN components S12, S5, and S6 look like
the FP components of S3, S4, S7, and S8. The peak spectral ordinate of the FP response at
S5 is almost the same, around 2 m/s, and the shape of the spectrum is quite similar, to that at
S4 in the FN direction. Moving away from the fault, the FP and FN components begin to
look more similar. One important distinction is that the spectra for the FN components differ
significantly from the corresponding peak ground motion parameters because they exhibit a
dynamic effect. In contrast, the spectra for the FP components are similar to the
corresponding peak values of the ground motion, denoting little effective dynamic action of
the structural response. This is not true for stations S9, S10, and S11, which exhibit
amplification within the velocity-sensitive period band.
THRUST FAULT
Similarly to the idealized strike-slip event, the synthetics for the thrust fault scenario exhibit a
strong spatial and temporal dependence with the location of the observer with respect to the
fault. Figure 9 shows the absolute values of the EW and NS components of the peak
displacements and velocities at the ground surface throughout the region. The white circle
denotes the epicenter and the white rectangle is the projection of the causative fault plane
onto the free surface. Slip occurs in the direction of the hanging wall relative to the foot wall.
The rake of the slip is such that the thrust of the hanging wall on the free surface points in the
west direction. Thus, for this inclined fault we denote the particle motion in the east-west
direction as fault parallel (FP) since it is normal to the extended intersection of the fault plane
and free surface; the north-south direction is fault normal (FN). The simulations show a clear
forward directivity effect. The most salient feature for the ground displacement is the
concentration of motion around the upper corners for the FP component and along the entire
edge in the FN direction, directly above locations where there are stress concentrations. In
contrast with the strike-slip event, however, the displacement in the FP direction is greater
than that in the FN direction, by a factor of about two for the thrust fault. Both components
show significant displacement beyond the intersection of the extended fault with the free
surface. Backward directivity, although present, is small compared with the forward
directivity. The distribution of the peak velocity is similar to that of the displacement, except
that the decay is faster with distance from the fault. In addition, the FP component of the
peak velocity is significantly smaller than that of the corresponding FN component.
The synthetic seismoscope records depicted in Figure 10 show that, in addition to its
amplitude, the direction of particle motion varies considerably depending on station, and that
for this shallow thrust fault there is a noticeable permanent offset within large portions of the
epicentral region. The seismograms for velocity and displacement are shown in Figure 11.
They, too, show large velocity pulses in the direction of forward directivity, and a decay and
increased complexity with distance from the fault. Consistent with the distribution of peak
values in Figure 9, and contrary to the results for the strike-slip fault, the stronger pulses
occur in the FP direction. As observed already, the displacements show some permanent
offset, but much smaller than for the strike-slip fault.
Bielak -- 14
Displacement
Envelope
Fault Normal
Fault Parallel
Velocity
Figure 9. Thrust fault earthquake. Spatial distribution of the absolute values of the fault parallel and
fault normal components of the peak velocity and displacement at free surface. White dot denotes the
location of the epicenter and white rectangle the projection of the causative fault on the horizontal
plane.
The response spectra for five percent critical damping are shown in Figures 12 and 13 for
the same stations (S1-S15) as in the preceding figures for the thrust fault scenario. The
shapes of the response spectra do not show the same degree of variability with location with
respect to the fault as those for the strike-slip scenario. The main features are: (a) the FP
components exhibit distinct acceleration-sensitive, velocity-sensitive, and displacementsensitive bands, except for stations in the forward directivity direction that lie outside the
fault region (S1, S2, S3); and for these, the velocity-sensitive band is very narrow or almost
nonexistent; (b) the velocity-sensitive band for the FN components is generally narrower than
for the FP region; (c) the displacement spectral ordinates at long periods, both for FP and FN
components, are generally smaller than the PGD.
Bielak -- 15
Figure 10. Thrust fault earthquake. Synthetic seismoscope records of the horizontal displacement
path at a number of locations on a regular grid on the free surface.
CONCLUSIONS
The spatial and temporal distribution of ground motion near the causative faults for two
scenario earthquake events have been examined using detailed computational simulation.
Although the models of the crustal structure, causative fault, and the slip are idealized, the
synthetic ground motion exhibits many of the significant characteristics of ground motion
recorded during actual earthquakes. Moreover, the high spatial resolution that can be
achieved through simulation provides information that cannot be gleaned from recorded
ground motion data alone.
In particular, the results of this study show that for the strike-slip fault event:





The fault normal (FN) component of velocity exhibits a very strong forward directivity
effect, but the dynamic effect in the fault parallel (FP) direction occurs in the
neighborhood of the epicenter.
The peak value of the FN components of velocity and displacement are twice those of the
corresponding FP components.
In contrast with the velocity, both the FP and FN components of displacement show a
strong forward directivity effect. Displacement seismograms exhibit permanent offsets,
or fling, as a consequence of the permanent tectonic deformation at the site.
The decay of the velocity with distance from the fault plane is much faster than that of the
displacement.
The ground motion also varies in duration and frequency content depending on location.
The FN component of velocity exhibits a strong pulse-like behavior at stations located in
the forward directivity direction. On the other hand, at stations located directly north of
the epicenter, it is the FP component of the velocity that has a strong pulse-like behavior.
Bielak -- 16
(a) Velocity
(b) Displacement
Figure 11. Thrust fault earthquake. Seismograms of (a) velocity (m/s) and displacement (m) in two
orthogonal directions at selected locations on the free surface. Open circles indicate the locations of
the observation points, and the direction of motion is denoted by the orientation of the seismograms.
The direction of particle motion at each location is perpendicular to the time axes of the
corresponding seismogram
Bielak -- 17
Figure 12. Thrust fault earthquake. Response spectra for 5 percent critical damping for ground
motion in the SN direction, calculated from the corresponding free-field accelerograms. The spectra
are plotted in the usual tripartite logarithmic representation, with pseudo-velocity on the vertical axis
and period on the horizontal axis. The periods considered range from 0.5s to 8.0s.
Figure 13. Thrust fault earthquake. Same as Figure 12 for ground motion in the fault parallel
direction.
Bielak -- 18

The seismograms become more complex in shape and their duration becomes longer
farther away from the fault; also, the dominant periods become longer with distance from
the fault.
As with the strike-slip fault event, the simulations of the thrust fault event show that the
ground motion has strong spatial and temporal dependence on the location of the observer
with respect to the fault. The most salient features for the thrust fault event are:

The greatest concentration of ground displacement occurs near the corners of the fault
opposite the hypocenter, in the rake direction.
 In contrast with the strike-slip fault, the displacement in the fault parallel direction (also
in the direction of the slip) is greater than that in the fault normal direction (normal to the
slip) by a factor of about two. The displacement of both components beyond the
intersection of the extended fault with the ground surface is significant.
 The peak velocity decays faster than the peak displacement with distance from the fault,
as for the strike-slip fault.
The important implications that this behavior of the ground motion has for the response
of long- and short-vibration period elastoplastic models structures is examined in detail in the
companion paper (Fenves et al., 2004)
ACKNOWLEDGMENTS
The research described in this paper was supported by the National Science Foundation
under grant number 0121989 to Mississippi State University. The authors appreciate the
encouragement of Dr. Lynn Preston and Dr. Joy Pauschke of the NSF Division of
Engineering Education and Centers. Drs. Michael Stokes and Donald Trotter of MSU were
instrumental in establishing the coordinated research program on seismic performance of
urban regions, for which the reported research is one component. The use of the Pittsburgh
Supercomputer Center for the computations is greatly appreciated. The strike-slip earthquake
simulation is based on one conducted as part of a PEER/SCEC project on the validation of
numerical methods for ground motion modeling in large basins.
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