Broadband time history simulation using a hybrid approach

Abstract

We present a methodology for generating broadband (0 -10 Hz) ground motion time histories for moderate and larger crustal earthquakes. Our hybrid technique combines a stochastic approach at high frequencies with a deterministic approach at low frequencies. The broadband response is obtained by summing the separate responses in the time domain using matched filters centered at 1 Hz. We use a kinematic description of fault rupture, incorporating spatial heterogeneity in slip, rupture velocity and rise time by discretizing an extended finite-fault into a number of smaller subfaults. The stochastic approach sums the response for each subfault assuming a random phase, an omega-squared source spectrum and generic ray-path Green's functions. Gross impedance effects are incorporated using quarter wavelength theory to bring the response to a reference baserock velocity level. The deterministic approach sums the response for many point sources distributed across each subfault. Wave propagation at frequencies below 1 Hz is modeled using a 3D viscoelastic finite difference algorithm with the minimum shear wave velocity set between 600 and 1000 m/s, depending on the scope and complexity of the velocity structure. To account for site-specific geologic conditions, short-and mid-period empirical amplification factors provided by Borcherdt [1] are used to develop frequency-dependent non-linear site response functions. The amplification functions are applied to the stochastic and deterministic responses separately since these may have different (computational) reference site velocities. We note that although the amplification factors are strictly defined for response spectra, we have applied them to the Fourier amplitude spectra of our simulated time histories. This process appears to be justified because the amplification functions vary slowly with frequency and the method produces favorable comparisons with observations. We demonstrate the applicability of the technique by modeling the broadband strong ground motion recordings from the 1989 Loma Prieta and 1994 Northridge earthquakes.

Full text

13th World Conference on Earthquake Engineering
Vancouver, B.C., Canada
August 1-6, 2004
Paper No. 1098
BROADBAND TIME HISTORY SIMULATION USING A HYBRID
APPROACH
Robert GRAVES1 and Arben PITARKA2
SUMMARY
We present a methodology for generating broadband (0 - 10 Hz) ground motion time histories for
moderate and larger crustal earthquakes. Our hybrid technique combines a stochastic approach at high
frequencies with a deterministic approach at low frequencies. The broadband response is obtained by
summing the separate responses in the time domain using matched filters centered at 1 Hz. We use a
kinematic description of fault rupture, incorporating spatial heterogeneity in slip, rupture velocity and rise
time by discretizing an extended finite-fault into a number of smaller subfaults. The stochastic approach
sums the response for each subfault assuming a random phase, an omega-squared source spectrum and
generic ray-path Green's functions. Gross impedance effects are incorporated using quarter wavelength
theory to bring the response to a reference baserock velocity level. The deterministic approach sums the
response for many point sources distributed across each subfault. Wave propagation at frequencies below
1 Hz is modeled using a 3D viscoelastic finite difference algorithm with the minimum shear wave velocity
set between 600 and 1000 m/s, depending on the scope and complexity of the velocity structure. To
account for site-specific geologic conditions, short- and mid-period empirical amplification factors
provided by Borcherdt [1] are used to develop frequency-dependent non-linear site response functions.
The amplification functions are applied to the stochastic and deterministic responses separately since
these may have different (computational) reference site velocities. We note that although the
amplification factors are strictly defined for response spectra, we have applied them to the Fourier
amplitude spectra of our simulated time histories. This process appears to be justified because the
amplification functions vary slowly with frequency and the method produces favorable comparisons with
observations. We demonstrate the applicability of the technique by modeling the broadband strong
ground motion recordings from the 1989 Loma Prieta and 1994 Northridge earthquakes.
INTRODUCTION
Our primary motivation in developing an enhanced broadband simulation methodology is to provide more
robust estimates of the ground shaking expected in future earthquakes. The most comprehensive manner
of ground shaking characterization is through the use of full waveform time histories. These ground
motion time histories can have many potential applications, including engineering design studies
1 Principal Scientist, URS Corporation, Pasadena, CA, USA. Email: robert_graves@urscorp.com
2 Senior Project Scientist, URS Corporation, Pasadena, CA, USA. Email: arben_pitarka@urscorp.com

incorporating non-linear structural analysis, seismic hazard characterization, disaster mitigation planning
and high-resolution real time and post earthquake (e.g., ShakeMap) ground motion estimation.
Traditionally, ground motion recordings from past earthquakes have been used as surrogates to represent
the motions expected during future earthquakes. Unfortunately, the library of existing recordings only
samples a small subset of possible earthquake scenarios. Thus, the ground motion records typically must
be scaled or modified in order to fit the magnitude, mechanism, distance and site characteristics of the
target earthquake. As an alternative, advances in the understanding of fault rupture processes, wave
propagation phenomena and site response characterization, coupled with the tremendous growth in
computational power and efficiency, has made the prospect of large-scale ground motion time history
synthesis for future earthquakes much more feasible.
The idea of simulating broadband strong ground motion time histories is not new, and dates back at least
to the pioneering work of Hartzell [2] and Irikura [3]. These early studies proposed a method of summing
recordings of small earthquakes (empirical Green’s functions) to estimate the response of a larger
earthquake. Since then, the simulation techniques have been extended to include stochastic representation
of source and path effects (e.g., Boore [4]), theoretical full waveform Green’s functions (e.g., Zeng, [5]),
or various combinations of these approaches (e.g., Hartzell [6]). Over the years, a large number of
investigators have made significant contributions and refinements to these methodologies. Hartzell [7]
provides a detailed and comprehensive review of many of these existing simulation methodologies.
In our approach, the broadband ground motion simulation procedure is a hybrid technique that computes
the low frequency and high frequency ranges separately and then combines the two to produce a single
time history (Hartzell [7]). At frequencies below 1 Hz, the methodology is deterministic and contains a
theoretically rigorous representation of fault rupture and wave propagation effects, and attempts to
reproduce recorded ground motion waveforms and amplitudes. At frequencies above 1 Hz, it uses a
stochastic representation of source radiation, which is combined with a simplified theoretical
representation of wave propagation and scattering effects. The use of different simulation approaches for
the different frequency bands results from the seismological observation that source radiation and wave
propagation effects tend to become stochastic at frequencies of about 1 Hz and higher.
Our methodology offers a significant enhancement over previous broadband simulation techniques
through the use of frequency-dependent non-linear site amplification factors. These factors are
incorporated by first restricting the computational velocity model in both the deterministic and stochastic
bandwidths to have an average near-surface shear wave velocity between 600 and 1000 m/s. We then
apply site-specific amplification factors, which are derived using the empirical relations of Borcherdt [1].
This approach significantly reduces the numerical computational burden, particularly for the deterministic
domain, and also provides an efficient mechanism for including detailed site-specific geologic information
in the ground motion estimates.
In the sections that follow, we first provide detailed descriptions of the deterministic and stochastic
simulation methodologies. Next, we discuss the derivation and implementation of the non-linear site
amplification factors. The final sections present validation studies of the simulation methodology using
ground motions recorded during the 1989 Loma Prieta and 1994 Northridge earthquakes.
SIMULATION METHODOLOGY
Determinstic Methodology (f < 1 Hz)
The low frequency simulation methodology uses a deterministic representation of source and wave
propagation effects and is based on the approach described by Hartzell [8]. The basic calculation is
carried out using a 3D viscoelastic finite-difference algorithm, which incorporates both complex source

rupture as well as wave propagation effects within arbitrarily heterogeneous 3D geologic structure. The
details of the finite-difference methodology are described by Graves [9] and Pitarka [10]. Anealsticity is
incorporated using the coarse-grain approach of Day [11].
The earthquake source is specified by a kinematic description of fault rupture, incorporating spatial
heterogeneity in slip, rupture velocity and rise time. Following Hartzell [8], the fault is divided into a
number of subfaults. The slip and rise time are constant across each individual subfault, although these
parameters are allowed to vary from subfault to subfault. We use a slip velocity function that is
constructed using two triangles as shown in Figure 1. This functional form is based on results of dynamic
rupture simulations (e.g., Guaterri [12]). We
constrain the parameters of this function as
follows:
A
h
TT
MT
rp
r
⋅=
⋅=
⋅×= −
−
2.0
2.0
1083.1 3/1
0
9
(1)
where M0 is the seismic moment, Tr is the rise time
and A is normalized to give the desired final slip.
The expression for Tr comes from the empirical
analysis of Somerville [13]. In general, Tr may
vary across the fault; however, in practice we only
allow a depth dependent scaling such that Tr
increases by a factor of 2 if the rupture is between
0 and 5 km depth. This is consistent with observations of low slip velocity on shallow fault ruptures
(Kagawa [14]).
The rupture initiation time (Ti) is determined using the expression
sr
ri
VV
tVRT
⋅=
−=
8.0
/
δ
(2)
where R is the rupture path length from the hypocenter to a given point on the fault surface, Vr is the
rupture velocity and is set at 80% of the local shear wave velocity (Vs), and
δ
t is a timing perturbation that
scales linearly with slip amplitude such that 0
tt
δ
δ
= where the slip is at its maximum and 0=t
δ
where
the slip is at the average slip value. We typically set .sec5.0
0=t
δ
This scaling results in faster rupture
across portions of the fault having large slip as suggested by source inversions of past earthquakes (Hisada
[15]).
For scenario earthquakes, the slip distribution can be specified using randomized spatial fields,
constrained to fit certain wave number properties (e.g., Somerville [13]; Mai [16]). In the simulation of
past earthquakes, we use smooth representations of the static slip distribution determined from finite-fault
source inversions. Typically, these inversions will also include detailed information on the spatial
variation of rupture initiation time and slip velocity function, either by solving for these parameters
directly or by using multiple time windows. However, we do not include these in our simulations, but
rather rely on equations (1) and (2) to provide them. Our philosophy is that the level of detailed resolution
of these parameters provided by the source inversions will generally not be available a priori for future
earthquakes. Furthermore, since the inversions determine these parameters by optimally fitting the
Figure 1. Slip velocity function used in the deterministic
simulations [see equation (1)].

selected observations, there are no guarantees that they will produce an optimal waveform fit at sites not
used in the inversion. Hopefully, an improved understanding of dynamic rupture processes will help to
provide better constraints on these parameters in the future.
Stochastic Methodology (f > 1 Hz)
The high frequency simulation methodology is a stochastic approach that sums the response for each
subfault assuming a random phase, an omega-squared source spectrum and simplified Green’s functions.
The methodology follows from the procedure that was first presented by Boore [4] with the extension to
finite-faults given by Beresnev [17]. We have incorporated several modifications of the original finite-
fault methodology of Beresnev [17], which are described below.
In our approach, each subfault is allowed to rupture with a subfault moment weighting that is proportional
to the final static slip amount given by the prescribed rupture model. The final summed moment release is
then scaled to the prescribed target mainshock moment. This alleviates the problem of requiring that each
of the subfaults scale to an integer multiple of 3
dl
p⋅
σ
(where p
σ
is the stress parameter and dl is the
subfault dimension), which tends to make many of the subfaults have zero moment release. The subfault
dimensions are determined using the scaling relation of Beresnev [18].
Beresnev [19] define a radiation-strength factor (s), which is used as a free parameter in the specification
of the subfault corner frequency (fc)
dl
V
zsf r
c⋅
⋅=
π
(3)
where z is a scaling factor relating fc to the rise time of the subfault source. In our approach, instead of
allowing this to be a free parameter, we set 6.1=z and let
ff ADs ⋅=







<+
≥≥
−
−
+
>
=







≥+
≤≤
−
−
+
<
=
01
01
01
1
1
1
10
10
01
0
0
0
if1
if1
if1
if1
if1
if1
δδ
δδδ
δδ
δδ
δδ
c
cA
hhc
hhh
hh
hh
c
hh
D
f
f
(4)
where Df is a depth scaling factor, 5
0=hkm, 10
1=hkm and h is the depth of the subfault center in km,
and Af is a dip scaling factor, o
054=
δ
, o
160=
δ
and δ is the subfault dip. The constants c0 and c1 are set
at 0.4 and 0.35, respectively, based on calibration experiments. This parameterization follows from the
observation in crustal earthquakes that slip velocity is relatively low for shallow near-vertical ruptures and
increases with increasing rupture depth and decreasing fault dip (Kagawa [14]). Since corner frequency is
proportional to slip velocity, this formulation replicates the trend of the observations. We note that
although this formulation reduces the number of free parameters, it certainly is not unique and probably
has trade-offs with other parameters in the stochastic model. In particular, allowing the subfault stress
parameter (σp) to be variable across the fault would accommodate a similar type of slip velocity scaling.
Instead, we fix 50=
p
σ
in our simulations.

Our formulation also allows the specification of a plane layered velocity model from which we calculate
simplified Green’s functions (GFs) and impedance effects. The GFs are comprised of the direct and
Moho-reflected rays, which are traced through the specified velocity structure. Following Ou [20], each
ray is attenuated by 1/Rp where Rp is the path length traveled by the particular ray. For each ray and each
subfault, we calculate a radiation pattern coefficient by averaging over a range of slip mechanisms and
take-off angles, varying o
45± about their theoretical values. Anelasticity is incorporated using a travel-
time weighted average of the Q values for each of the velocity layers and using a kappa operator set at
05.0=
κ
. Finally, gross impedance effects are included using quarter wavelength theory (Boore [21]) to
derive amplification functions that are consistent with the specified velocity structure.
Site Specific Amplification Factors
Borcherdt [1] derived empirically based amplification functions for use in converting response spectra
from one site condition to a different site condition. The general form of these functions is given by
x
m
refsitexVVF )/(= (5)
where Vsite is the 30 m travel-time averaged shear wave velocity (Vs
30) at the site of interest, Vref is the
corresponding velocity measure at a reference site where the ground response is known, and mx is an
empirically determined factor. Borcherdt [1] specified one set of factors at short periods (centered around
0.3 s) and one set at mid-periods (centered around 1.0 s). Furthermore, non-linear effects are included
since the mx decrease as the reference ground response PGA increases. The decrease in the mx is sharper
for the short period factors than the mid-period ones, reflecting the observed increase of non-linear effects
at shorter periods.
In our simulation methodology, we restrict the
computational velocity models in both the
deterministic and stochastic calculations to have Vs
30
values between 600 and 1000 m/s. This is our Vref. To
obtain an amplification function for a given site
velocity, we first determine the short- and mid-period
factors from equation (5) using the tabulated mx from
Borcherdt [1] given the reference PGA from the
stochastic response. Next, we construct a smoothly
varying function in the frequency domain by applying
a simple taper to interpolate the factors between the
short- and mid-period bands. The function tapers back
to unity at very short and very long periods. An
example set of these functions is shown in Figure 2
In practice, we apply these amplification functions to the amplitude spectra of the Fourier transformed
simulated time histories. This process is done to the deterministic and stochastic results separately since
these may have different computational reference velocities. Although the amplification factors are
strictly defined for response spectra, the application in the Fourier domain appears to be justified since the
functions vary slowly with frequency. Finally, the individual responses are combined into broadband
response using a set of matched butterworth filters. The filters are 4th-order and zero-phase with a
lowpass corner at 1 Hz for the deterministic response and a highpass corner at 1 Hz for the stochastic
response. The important properties of the matched filters are 1) they do not alter the phase of the response
and 2) they sum to unity for all frequencies. After applying the filters to the individual responses, they are
summed together to produce a single broadband time history.
Figure 2. Frequency dependent amplification
functions with an input PGA of 20% g,
V
ref
= 620 m/s
and various site velocities.

VALIDATION STUDIES
In order to test the adequacy of our simulation methodology, we compare our computed synthetic strong
motion time histories with those recorded during past earthquakes. The only earthquake specific
parameters we use are seismic moment, overall fault dimensions and geometry, hypocenter location, and a
generalized model of the final slip distribution. For future earthquakes, these are the parameters that we
feel can either be reliably estimated (e.g., seismic moment, fault dimensions) or parametrically assessed
using multiple realizations (e.g., hypocenter location, slip distribution). All other source parameters are
determined using the scaling relations described in the previous section. Since we have not optimized the
rupture models for these exercises, we cannot hope to match all the details of the recorded waveforms.
However, our goal is to reproduce the overall characteristics of the observed motions over a broad
frequency range throughout the region surrounding the fault. This includes matching the trends and levels
of common ground motion parameters such as PGA, PGV, SA and duration of shaking, adequately
capturing near-fault phenomena such as rupture directivity and footwall / hanging wall effects, and
reproducing region or site specific effects such as basin response and site amplification.
1994 Northridge Earthquake
Our model region for the Northridge earthquake
covers the area within about 40 km of the rupture
surface, which includes 69 strong ground motion
recording sites (Figure 3). Site types range from Vs
30
categories BC to D (Wills [22]). We adopt the fault
geometry of Hartzell [23] for our simulations. The
fault is 20 km long and extends from a depth of 5 km
to 21 km giving a down-dip width of 25 km. The
strike is 122o, dip is 40o and the rake is nearly pure
thrust. We use a moment of cmdyne 1014.1 26 ⋅× ,
resulting in a moment magnitude (Mw) of 6.7.
Figure 4 shows the final static slip distribution
obtained from Hartzell [23]. There are 3-4 regions of
large slip (asperities) located updip and northwest of
the hypocenter. The contour lines in Figure 4 indicate
the propagation of the rupture front at 1 sec intervals.
We have determined the rupture times using equation
(2). Note that the rupture is advanced in large slip regions and is delayed in low slip regions.
The subsurface velocity structure used for the deterministic simulations is taken from Version 2 of the
SCEC 3D Seismic Velocity Model (Magistrale [24]). We set the lowest shear wave velocity to be 620 m/s
in our simulations. With a minimum finite-difference grid spacing of 120 m in the lowest velocity regions
of the model, we obtain reliable results for frequencies of about 1 Hz and less. These lowest velocity
regions occur over the deep sediments of the Los Angeles and San Fernando basins. Surrounding these
basin structures are more consolidated sedimentary and crystalline rocks with generally higher near
surface velocities. Most of the non-basin regions of the model have a computational near-surface shear
wave velocity of 1000 m/s.
Figure 3. Map of the near source region of the
Northridge earthquake. Strong motion stations are
indicated by red triangles. Dashed lines show the
surface projection of the fault plane with a star at the
epicenter.

The lateral complexity in the velocity structure extends far
beneath the near surface layers. Figure 5 shows a vertical
cross section of the shear wave velocity structure along a
profile extending from NW of the San Fernando basin into the
middle of the Los Angeles basin (A-A’ in Figure 4). There is a
clear expression of the low velocity sediments of the San
Fernando and Los Angeles basins down to several km in depth.
The lateral contrast between the various rock types can have a
profound effect on the propagation of seismic energy,
particularly at frequencies less than about 1 Hz. For this
reason, we have used this complex representation of the
subsurface velocity structure in our deterministic simulations.
From the 3D velocity structure, we extract 1D velocity profiles
to use in the stochastic simulations. We select one profile for
rock sites and another profile for basin sites (Figure 5). Even
though both 1D profiles are constrained to have Vs
30 of 760
m/s, the basin profile has significantly lower velocities than the
rock profile in the upper 5 km. This will have two primary
effects on the stochastic simulations: 1) the GF travel times
will more closely match the phasing of the deterministic
results, which are calculated with the fully 3D model and 2)
the impedance amplification determined from the quarter-
wavelength approach will have a broader frequency response for the basin sites compared to the rock
sites.
For each of the 69 strong motion sites, site category and Vs
30 values are obtained from Wills [22]. Using
equation (5), we then construct frequency-dependent amplification functions that are applied to the results
of the deterministic and stochastic simulations. The final simulated broadband time histories are
computed using the match-filter and summation procedure described earlier. Figure 6 compares the
observed and simulated three-component ground velocities at 18 selected sites. These 18 sites include
near-fault locations (e.g., rrs), rock sites (e.g., ssus) and deep basin sites (e.g. pdrc). In general, the
waveform character, amplitude and duration of the observed data are matched reasonable well by the
simulation. The simulation reproduces key phenomena such as the pulse-like motions at forward
directivity sites (jeng, sylm, pard, rrs), the relatively brief duration and higher frequency motions at rock
sites (ssus, scrs), and the relatively long duration and lower frequency motions at deep basin sites (verm,
Figure 4. Slip distribution of the Northridge
earthquake from Hartzell [23]). Contours
show rupture front at 1 sec intervals
determined from equation (2).
Figure 5. (Left) Shear wave velocity cross-section along profile A-A’. Minimum Vs is set at 620 m/s. (Right) 1D velocity
profiles used for the stochastic simulations. Solid lines are for rock sites and dashed lines are for basin sites. Both rock
and basin 1D profiles are constrained to have Vs
30 of 760 m/s.

pdrc, bald). The simulation does not match exactly the phasing of the observed waveforms, as we expect
since we have used a smooth representation of the rupture process. In addition, we do not predict several
of the large-amplitude high-frequency pulses seen in the data (e.g., scrs, uhsp) due to the use of random
phasing in the stochastic calculations.
Figure 7 compares the observed and
simulated PGA and PGV for all 69 sites.
These values are plotted as a function of
closest distance to the rupture plane. The
agreement between the observed and
simulated values is good, both in terms of
amplitude level and distance dependence.
The simulation also reproduces several
important features seen in the observations.
These include: 1) the flattening of PGA
attenuation between 10 km and 20 km
distance, which is likely due to hanging
wall effects and amplification within the
San Fernando basin, 2) the large PGV at
close distance, which results from strong
forward directivity effects, and 3) the
elevation in PGA and PGV around 25 km
distance, which is probably due to
amplification effects along the northern
margin of the Los Angeles basin.
We also compare the data and simulations
using goodness-of-fit measures for 5%
damped spectral acceleration calculated
from the broadband time histories (e.g.,
Schneider [25]). For an individual station,
the residual r(Ti) at each period Ti is given
by
[]
)(/)(ln)( iSiOi TsaTsaTr =, where
saO(Ti) and saS(Ti) are the observed and
simulated spectral acceleration values,
respectively. The model bias is obtained by
averaging the residuals for all stations and
both horizontal components at each period. A model bias of zero indicates the simulation, on average,
matches the observed ground motion level. A negative model bias indicates over-prediction and a positive
model bias indicates under-prediction of the observations. The top panel of Figure 8 plots the model bias
and standard error for the Northridge simulation. The simulation result has no significant bias over the
period range 0.1 to 10 seconds, indicating that the simulation model adequately captures the main
characteristics of the ground motion response. In addition, the standard error is about 0.4 to 0.5 (natural
log) over this period band. The bottom panel plots the model bias for the simulation when the site-
specific amplification factors are not included in the response. Without the site-specific amplification
factors, the simulation under-predicts the observed response by about 20 to 30%, with the largest
difference at the longer periods. More significantly, the standard error increases to about 0.6 to 0.7. This
indicates that including the site-specific factors significantly reduces the uncertainty of the ground motion
estimates.
Figure 6. Comparison of observed (black traces) and simulated
(red traces) three-component ground velocities at 18 selected
sites for the Northridge earthquake. For each station and
component, the traces are scaled to the maximum amplitude of
the observed or simulated time history. The maximum value
(cm/s) is indicated above each pair of traces.

We also examine the spectral acceleration residuals as a function of site type and location. Figure 9 plots
the residuals as a function of distance to the rupture plane at periods of 0.3, 1.0 and 3.0 sec. The sites are
grouped into three categories using the classification of Wills [22], B-BC, C-CD, and D. These plots
indicate that there is little systematic trend in the residuals as a function of site type or distance for these
periods. Figure 10 displays the residuals in map view. These plots indicate some slight trends in the
simulations such as a tendency to under-predict the response along the Santa Monica Mountains and
northern Los Angeles basin, and a tendency to over predict the response in the San Fernando basin and
downtown Los Angeles regions, particularly at the longer periods.
1989 Loma Prieta earthquake
Our model region for the Loma Prieta earthquake covers the area within about 35 km of the rupture
surface, which includes 32 strong ground motion recording sites (Figure 11). Site types range from Vs
30
categories BC to D (Wills [22]). We adopt the fault geometry of Wald [26] for our simulations. The fault
is 40 km long and has a down-dip width of 20 km. The strike is 130o, dip is 70o and the rake averages
about 150o. We use a moment of cmdyne 1053.2 26 ⋅× , giving a moment magnitude of 6.9.
Figure 8. (Top) Spectral acceleration goodness-of-fit
computed for the average of both horizontal components
for the Northridge earthquake. Red line plots mean
model bias averaged over 69 sites. Gray shading
denotes 90% confidence interval of the mean and green
shading denotes interval of one standard deviation.
(Bottom) Same as top except simulation does not
incorporate site-specific amplification factors.
Figure 7. Observed (red crosses) and simulated (green
circles) peak ground acceleration (top) and peak ground
velocity (bottom) plotted as a function of closest
distance to fault rupture at 69 sites for the Northridge
earthquake. PGA and PGV values are measured from the
observed and simulated time histories.

Figure 12 shows the final static slip distribution obtained from Wald [26]. The hypocenter is in the
middle of the fault a depth of about 18 km. There are 2 main asperities; one located northwest and the
other southeast of the hypocenter. The contour lines in Figure 12 indicate the propagation of the rupture
front at 1 sec intervals. We have determined the rupture times using equation (2). Again, the rupture is
advanced in large slip regions and is delayed in low slip regions.
The subsurface velocity structure used for the deterministic simulations is constructed using two 1D
velocity profiles, one for the region west of the San Andreas fault and the other for the region east of the
fault (Figure 13). Above 17 km depth, the velocities on the west side of the fault are about 5% higher than
Figure 10. Maps of spectral acceleration residuals at 0.3
s (top), 1.0 s (middle and 3.0 s (bottom). Symbols are
plotted at station locations. Crosses indicate over-
prediction and circles indicate under-prediction.
Figure 9. Residuals between observed and simulated
spectral acceleration at periods of 0.3 s (top), 1.0 s
(middle) and 3.0 s (bottom) for all sites plotted as a
function of closest distance to fault plane. Sym bols
denote site type.

the velocities on the east side of the fault. These
profiles are taken from Stidham [27], who found
that the lateral velocity contrast across the San
Andreas fault had a strong influence on wave
propagation effects during the Loma Prieta
earthquake. In particular, energy propagating
along the fault is laterally refracted toward the
eastern side due to the velocity contrast. This
tends to increase the amplitudes of waves
traveling northward into the Santa Clara Valley
and southeastward into the Gilroy and Hollister
areas. We set the lowest shear wave velocity to
be 1000 m/s in our simulations. With a minimum
finite-difference grid spacing of 200 m in the
lowest velocity regions of the model, we obtain
reliable results for frequencies of about 1 Hz and
less.
Several studies have proposed 3D basin velocity
models for this region, including Brocher [28]
and Stidham [27]. In our current simulations we
have not included these more detailed structural
representations, mainly because there are some
notable differences between the proposed models, and we did not want the uncertainty in the 3D velocity
structure to have a strong influence on the uncertainty in our ground motion estimates. We fully expect
that future refinement of the 3D velocity structure will also improve the simulation results.
For the stochastic simulations, we have used the
same basic 1D profiles that are used for the
deterministic calculations. The only
modification is that both models are tapered in
the near-surface to have a Vs
30 of 760 m/s.
Again, impedance amplification effects at high
frequencies are modeled using the quarter-
wavelength approach.
For each of the 32 strong motion sites, site
category and Vs
30 values are obtained from Wills
[22]. Using equation (5), we then construct
frequency-dependent amplification functions that
are applied to the results of the deterministic and
stochastic simulations. The final simulated
broadband time histories are computed using the
match-filter and summation procedure described earlier. Figure 14 compares the observed and simulated
three-component ground velocities at 18 selected sites. These 18 sites include near-fault locations (e.g.,
lgpc), rock sites (e.g., lex1) and Santa Clara Valley sites (e.g. sjin). In general, the waveform character,
amplitude and duration of the observed data are matched reasonable well by the simulation. The
simulation reproduces key phenomena such as the pulse-like motions at forward directivity sites (lgpc,
lex1, srtg), the longer duration and non-pulse-like motions at neutral directivity near-fault sites (bran, cor),
and the relatively long duration and lower frequency motions at the more distant sites (sjin, agnw, hall).
Figure 11. Map of the near source region of the Loma Prieta
earthquake. Strong motion stations are indicated by red
triangles. The surface projection of the fault plane is shown
by dashed lines with a star at the epicenter.
Figure 12. Slip distribution of the Lom a Prieta earthquake
from Wald [26]. Contours show rupture propagation at 1
sec intervals determined from equation (2).

As with the Northridge simulation, we do not match exactly the
phasing of the observed waveforms, which is to be expected since we
use a smooth representation of the rupture process.
Figure 7 compares the observed and simulated PGA and PGV for all 32
sites. These values are plotted as a function of closest distance to the
rupture plane. The agreement between the observed and simulated
values is good, both in terms of amplitude level and distance
dependence. The simulation also reproduces several important features
seen in the observations. These include: 1) the flattening of PGA
attenuation between 10 km and 20 km distance, which may be related
to site response (most of these sites are soil), 2) the large PGV at close
distance, which results from strong forward directivity effects, and 3)
the elevation in PGA and PGV around 22 - 30 km distance, which is
probably due to amplification effects in the Santa Clara Valley and the
Gilroy / Hollister area.
We also compare the data and simulations
using the same goodness-of-fit measures
for 5% damped spectral acceleration
calculated from the broadband time
histories that we described earlier for the
Northridge simulation. Figure 16 plots the
model bias and standard error for the
Loma Prieta simulation. On average, the
simulation result tends to slightly over-
predict the recorded motions (on the order
of 5 to 10%), which could be accounted
for by adjusting the moment used in the
simulation. Aside from this feature, there
is no significant bias over the period range
0.1 to 10 seconds, indicating that the
simulation model adequately captures the
main characteristics of the ground motion
response. In addition, the standard error is
about 0.4 (natural log) over this period
band.
As was done in the Northridge
comparison, we also examine the spectral
acceleration residuals as a function of site
type and location. Figure 17 plots the
residuals as a function of distance to the
rupture plane at periods of 0.3, 1.0 and 3.0
sec. The sites are grouped into three
categories using the classification of Wills
[22], B-BC, C-CD, and D. These plots
indicate that there is little systematic trend
in the residuals as a function of site type or
distance for these periods. Figure 18
Figure 13. 1D Loma Prieta
velocity profiles. Solid lines are
for west of the San Andreas fault
and dashed lines are for east of
the fault.
Figure 14. Comparison of observed (black traces) and simulated
(red traces) three-component ground velocities at 18 selected sites
for the Loma Prieta earthquake. For each station and component,
the traces are scaled to the maximum amplitude of the observed or
simulated time history. The maxim um value (cm/s) is indicated
above each pair of traces.

displays the residuals in map view. These plots
suggest some systematic behavior of the residuals,
such as over-prediction in the Santa Clara Valley
and under-prediction in Gilroy at 0.3 sec period;
under-prediction in the near-fault region at 1
second period, and under-prediction in the Santa Clara Valley at 3 sec period. However, in general, these
trends are of relatively small magnitude, and we suspect that increased knowledge of the 3D sub-surface
geology will improve the simulation response.
DISCUSSION AND CONCLUSIONS
The broadband simulation methodology presented here provides a general framework for synthesizing
ground motion time histories for future scenario earthquakes. One of the main enhancements of our
approach over previous techniques is the use of frequency-dependent non-linear site amplification factors.
Our methodology produces quite favorable results when compared against the strong ground motions
recorded during the 1989 Loma Prieta and 1994 Northridge earthquakes.
In developing this methodology, we have tried to incorporate as much detail as possible in describing the
source, path and site effects in order to adequately capture the main characteristics of the expected ground
motions. For the path and site effects, this stresses the importance of developing detailed 3D seismic
velocity models for earthquake prone regions. However, we recognize that extremely detailed descriptions
of the earthquake rupture process will generally not be available a priori for future events. Thus, our
methodology uses simple, yet flexible, rules to parameterize the slip, slip velocity function and rupture
velocity. Our hope is that more robust constraints on these parameters can be obtained from detailed
source inversion studies and dynamic rupture analyses.
REFERENCES
1. Borcherdt R. "Estimates of site-dependent response spectra for design (methodology and
justification)." Earthquake Spectra 1994; 10(4): 617-653.
2. Hartzell S. "Earthquake aftershocks as Green's functions." Geophys. Res. Lett. 1978; 5: 1-4.
Figure 15. Observed (red crosses) and simulated (green
circles) PGA (top) and PGV (bottom) plotted as a function
of closest distance to fault rupture at 32 sites for the
Loma Prieta earthquake.
Figure 16. Spectral acceleration goodness-of-fit
computed for the average of both horizontal components
for the Loma Prieta earthquake. Red line is mean model
bias averaged over 32 sites. Gray shading denotes 90%
confidence interval of the mean and green shading
denotes interval of one standard deviation.

3 Irikura K. "Semi-empirical estimation of strong ground motions during large earthquakes." Bull.
Disast. Prev. Res. Inst., Kyoto Univ. 1978; 33: 63-104.
4. Boore D. "Stochastic simulation of high frequency ground motions based on seismological models
of the radiated spectra." Bull. Seism. Soc. Am. 1983; 73: 1865-1894.
5. Zeng Y, Anderson JG, Yu G. "A composite source model for computing syntheticstrong ground
motions." Geophys. Res. Lett. 1994; 21: 725-728.
6. Hartzell S. "Comparison of seismic waveform inversion results for the rupture history of a finite
fault: application to the 1986 North Palm Springs, California, earthquake." J. Geophys. Res. 1989;
94: 7515-7534.
7. Hartzell S, Harmsen S, Frankel A, Larsen S. "Calculation of broadband time histories of ground
motion: comparison of methods and validation using strong ground motion from the 1994
Northridge earthquake." Bull. Seism. Soc. Am. 1999; 89: 1484-1504.
Figure 17. Residuals between observed and simulated
spectral acceleration at periods of 0.3 s (top), 1.0 s
(middle) and 3.0 s (bottom) for all sites plotted as a
function of closest distance to fault plane.
Figure 18. Spectral acceleration residuals at 0.3 s (top),
1.0 s (middle and 3.0 s (bottom). Crosses indicate over-
prediction and circles indicate under-prediction.

8. Hartzell S, Heaton T. "Inversion of strong ground motion and teleseismic waveform data for the
fault rupture history of the 1979 Imperial Valley, California earthquake." Bull. Seism. Soc. Am.
1983; 73: 1553-1583.
9. Graves R. "Simulating seismic wave propagation in 3D elastic media using staggered grid finite
differences." Bull. Seism. Soc. Am. 1996; 86: 1091-1106.
10. Pitarka A. "3D elastic finite difference modeling of seismic wave propagation using staggered grid
with non-uniform spacing." Bull. Seism. Soc. Am. 1998; 88: 54-68.
11. Day S, Bradley C. "Memory efficient simulation of anelastic wave propagation." Bull. Seism. Soc.
Am. 2001; 91: 520-531.
12. Guatteri M, Mai PM, Beroza G, Boatwright J. "Strong ground motion prediction from stochastic-
dynamic source models." Bull. Seism. Soc. Am. 2003; 93: 301-313.
13. Somerville P, Irikura K, Graves R, Sawada S, Wald D, Abrahamson N, Iwasaki Y, Kagawa T, Smith
N, Kowada A. "Characterizing crustal earthquake slip models for the prediction of strong ground
motion." Seism. Res. Lett. 1999; 70: 59-80.
14. Kagawa T, Irikura K, Somerville P. "A study on ground motion and fault rupture due to subsurface
faults." Eos. Trans. AGU 2001; 82(47): Fall Meet. Suppl., Abstract S31C-06.
15. Hisada Y. "A theoretical omega-square model considering spatial variation in slip and rupture
velocity. part 2. case for a two-dimensional source model." Bull. Seism. Soc. Am. 2001; 91: 651-
666.
16. Mai P, Beroza G. "A spatial random field model to characterize complexity in earthquake slip." J.
Geophys. Res. 2002; 107(B11): doi:10.1029/2001JB000588.
17. Beresnev I, Atkinson G. "Modeling finite fault radiation from the ωn spectrum." Bull. Seism. Soc.
Am. 1997; 87: 67-84.
18. Beresnev I, Atkinson G. "Subevent structure of large earthquakes - A ground motion perspective."
Geophys. Res. Lett. 2001; 28(1): 53-56.
19. Beresnev I, Atkinson G. "Stochastic finite fault modeling of ground motions from the 1994
Northridge, California earthquake. I. validation on rock sites." Bull. Seism. Soc. Am. 1998; 88:
1392-1401.
20. Ou GB, Herrmann R. "A statistical model for ground motion produced by earthquakes at local and
regional distances." Bull. Seism. Soc. Am. 1990; 80: 1397-1417.
21. Boore D, Joyner W, "Site amplification for generic rock sites." Bull. Seism. Soc. Am. 1997; 87: 327-
341.
22. Wills C, Petersen M, Bryant W, Reichle M, Saucedo G, Tan S, Taylor G, Treiman J. "A site
conditions map for California based on geology and shear wave velocity." Bull. Seism. Soc. Am.
2000; 90(6B): S187-S208.
23. Hartzell S, Liu P, Mendoza C. "The 1994 Northridge, California earthquake: Investigation of
rupture velocity, risetime, and high frequency radiation." J. Geophys. Res. 1996; 101; 20,091-
20,108.
24. Magistrale H, Day S, Clayton R, Graves R. "The SCEC Southern California reference three-
dimensional seismic velocity model version 2." Bull. Seism. Soc. Am. 2000; 90(6B): S77-S94.
25. Schneider J, Silva W, Stark C. "Ground motion model for the 1989 M6.9 Loma Prieta earthquake
including effects of source, path and site." Earthquake Spectra 1993; 9: 251-287.
26. Wald D, Helmberger D, Heaton T. "Rupture history of the 1989 Loma Prieta, California
earthquake." Bull. Seism. Soc. Am. 1991; 81: 1540-1572.
27. Stidham C, Antolik M, Dreger D, Larsen S, Romanowicz B. "Three-dimensional structure
influences on the strong motion wavefield of the 1989 Loma Prieta earthquake" Bull. Seism. Soc.
Am. 1999; 99: 1184-1202.
28. Brocher T, Brabb R, Catchings R, Fuis G, Fumal T, Jachens R, Jayko A, Kayen R, McLaughlin R,
Parsons T, Rymer M, Stanley R, Wentworth C. "A crustal-scale 3D seismic velocity model for the
San Francisco Bay area, California." Eos 1997; 78; F435-F436.