Content uploaded by Arben Pitarka

Author content

All content in this area was uploaded by Arben Pitarka on Jan 31, 2015

Content may be subject to copyright.

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.