-2 0 2
-2
-1
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QQPlot [cor=0.99769]
quan
tiles
of i
nput
Sam
ple
Standard Normal Quantiles10-1 100
0.5
1correlation of epsilon
T1
ρ Simulations(0.2s)Baker and Jayaram(0.2s)Simulations(1s)Baker and Jayaram(1s)Simulations(3s)Baker and Jayaram(3s)
100 1020
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Rjb(km)
sd o
f log
(Sa)
PGA
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Rjb(km)
sd o
f log
(Sa)
Sa(0.2s)
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Rjb(km)
sd o
f log
(Sa)
Sa(1.0s)
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Rjb(km)
sd o
f log
(Sa)
Sa(3.0s)
100 102
10-2
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Rjb(km)
Sa(
g)
PGA
100 102
10-2
100
Rjb(km)
Sa(
g)
Sa(0.2s)
100 102
10-2
100
Rjb(km)
Sa(
g)
Sa(1.0s)
100 102
10-2
100
Rjb(km)
Sa(
g)
Sa(3.0s)
Stochastic model for earthquake ground motions using wavelet packetsYoshifumi Yamamoto and Jack W. Baker Stanford University, Dept. of Civil & Environmental Engineering
Summary of results to date
• A stochastic ground motion model with time-frequency nonstationarity has been developed. A wavelet packet based model is used to allow easy control of time-frequency nonstationarity
• The model is empirically calibrated and captures important characteristics of observed ground motion recordings. Simulations reasonably match the em-pirical ground motion prediction models for spectral accelerations, Arias In-tensity, duration, mean period and epsilon.
• Simulations have parameters that vary with earthquake magnitude, distance and local site conditions. This differs from some stochastic simulations which reproduce properties of only a recorded seed ground motion.
• The regression and wavelet packet transform calculations are extremely com-putationally inexpensive, so simulating large numbers of ground motions for engineering use is easy. A desktop PC can produce 1000 simulations in one hour.
• The high freqency portion of these stochastic simulations could be combined with low frequency physics-based simulations to produce broadband simu-lated ground motions.
Wavelet packet transform
The wavelet packet transform is an extended version of the discrete wavelet transform. By controlling wavelet packets at varying scales and locations, the time-frequency character-istics of the time series can be determined.
The following �gures show the relationship between a time series its Fourier spectrum, and its wavelet packets.
Abstract
For performance-based design, non-linear dynamic structural analysis for various types of input ground motions is required. Since the number of ground motion recordings is lim-ited for such analysis, synthetic ground motions can be used as additional input. Here a stochastic ground motion model with time-frequency nonstationarity is developed using wavelet packets.
The wavelet packet transform can decompose time-series data into wavelet packets with constant resolution in the time and frequency axes, and it can also reconstruct a time-series data from wavelet packets. This allows time series to be modeled intuitively in the time-frequency domain using the wavelet packets.
Ground motions siumulated using the proposed approach reasonably match ground motion recordings in several respects, including elastic and inelastic spectral acceleration, duration, bandwidth, cumulative energy, and time-frequency nonstationarity. In addition, the median and log standard deviation of spectral accelerations from the simulated ground motions match those of empirical ground motion prediction models. These re-sults suggest that the synthetic ground motions generated by our model can be used as realistic inputs for non-linear dynamic structural analyses.
Time history
Wavelet packets distribution
Regression analysis to predict model parameters
The 12 parameters are predicted by the regression analysis calibrated using 1408 strong ground motion recordings from 25 earthquakes. The predictors of our model are moment magnitude (Mw), rupture distance (Rrup), hypocentral distance (Rhyp) , and Vs30. The fol-lowing equation predicts the Total Energy parameter (Y). Other model parameters have similar equations.
η is the interevent residual and ε is the intraevent residual, and other variables are deter-mined by regression analysis. η and ε values for pairs of model parameters are correlatied, and this correlation is tracked using a correlation matrix that is then used for simulations .
As seen below, characteristics of the simulated ground motions obtained using these re-gression models reasonably match the characteristics of recorded ground motions as pre-dicted by empirical ground motion prediction (”attenuation”) models.
The standard deviation of log spectral accelerations in long period is a little larger than that of the attenuation models. This is because the constant time-frequency resolution of the wavelet packets makes it di�cult to control the long-period energy of the simulations. An appealing solution would be to use this technique to produce high-frequency simula-tions, and combine them with physics-based low-frequency simulations.
Stochastic model for ground motionsIn our model, time series are de�ned by the combination of two types of wavelet packet groups: the major group (which contains the largest coe�cients representing 70% of the total ground motion energy) and the minor group (which contains the rest). To determine the amplitudes of the wavelet packets, the following 12 parameters are required for three independent probability density functions. for both wavelet packet groups, we de�ne the following parameters • E(t) - mean time • S(t) - standard deviation of time (related to duration) • E(f ) - mean frequency (related to dominant frequency) • S(f ) - standard deviation of frequency (related to bandwidth) • R(t,f ) - correlation between time and frequency (related to frequency evolution in time)for the major wavelet packets group, we de�ne the following • E(a) - mean amplitudeFinally, we specify the total energy in the time series • Energy - total energy
0 2 4 6 8 10−0.2
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Am
plitu
de
Fourier spectrum
0 1 2 3 4 50
1
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7x 10−3
Four
ier a
mpl
itude
Frequency(Hz))
Time(s)
Freq
uenc
y(H
z)
0 2 4 6 8 100
2.5
5
Simulated ground motions
Spectral Acceleration (Sa) vs RjbMw=7, VS30=270m/s
Wavelet packets of ground motion recordings from the Northridge Earthquake
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Example near �eld recordingNorthridge - 17645 Saticoy St
Rhyp=18km, Vs30=281m/s
Example far �eld recordingSanta Barbara - UCSB GoletaRhyp=123km, Vs30=339m/s
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Time(s)
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Acc.(G)
Amp.2
SimulationOriginal
Acc.(G)
Freq.(Hz)
Near �eld stationNorthridge - 17645 Saticoy St [Rhyp=18km, Vs30=281m/s]
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Acc.(G)
Amp.2
SimulationOriginal
Acc.(G)
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Far �eld stationSanta Barbara - UCSB Goleta [Rhyp=123km, Vs30=339m/s]
Cumulative Squared Acceleration
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ent(cm)
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udo
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ocity
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/s)
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Pseudo Acceleration(cm/s2 ) h=0.05
Spectral Acceleration
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Spe
ctra
l Acc
eler
atio
n(g)
Rrup=10kmRrup=20kmRrup=50km
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Rupture Distance = 10km
Rupture Distance = 20km
Rupture Distance = 50kmAcc.(G)
Acc.(G)
Acc.(G)
Simulated ground motionsMw=7, Vs30=270m/s
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Periods(s)
Spe
ctra
l Acc
eler
atio
n(g)
sample 1sample 2sample 3
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Acc.(G)sample 1
sample 2
sample 3
Simulated ground motionsMw=7, Rrup=10km, Vs30=270m/s
ijiSWW VdRcMbMbaY εη ++++++= )ln()ln()ln()ln( 30221
22 hRR rup += ruphypdif RRR −=
∑ ∑−
= =
−
⋅=12
0
2
1, )(),()(
p pn
j k
pkj
pf tkjWtf ψ
: Time history
: Wavelet packet coefficients
: Scaling function
),( kjW pf
)(, tpkjψ
)(tf
j
k
: Frequency order (frequency axis)
: Scale factor (time axis)
: Depth (band width of each level j)
: Total number of data
pn2
Major wavelet packets
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Original
Acc.(G)
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Observed ground motion
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Original
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Simulation
Simulated ground motion
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Simulation
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Simulation
parameters parameters
How to simulate ground motions
Selected simulationsMedian of simulationsSigma of simulationsObservationSimulations
Near field
Far field
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lativ
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uare
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c
Near field observationNear field simulationFar field observationFar field simulation
Standard deviation of log(Sa) vs RjbMw=7, VS30=270m/s
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Dur
atio
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ndar
d de
viat
ion
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(dur
)
Abrahamson and Silva(1996)Simulated ground motions
5-95% duration vs RrupMw=7, Rrup=Rhyp=10km, VS30=270m/s
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Rrup (km)
Aria
s In
tens
ity (m
/s)
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Rrup (km)
Sta
ndar
d de
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ion
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(Ia)
Travasarou et al.(2003)Simulated ground motions
Arias Intensity vs RrupMw=7, Rrup=Rhyp=10km, VS30=270m/s
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10-1
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Closest Distance (km)
Mea
n P
erio
d Tm
(s)
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ndar
d de
viat
ion
of ln
(Tm
)
Rathje et al.(2004)Simulated ground motions
mean Period(Tm) vs RrupMw=7, Rrup=Rhyp=10km, VS30=270m/s
characteristics of epsilonMw=7, Rrup=Rhyp=10km, VS30=270m/s
Abrahamson and Silva(2008)Boore and Atkinson(2008)Campbell and Bozorgnia(2008)Chiou and Youngs(2008)simulated ground motions
Abrahamson and Silva(2008)Boore and Atkinson(2008)Campbell and Bozorgnia(2008)Chiou and Youngs(2008)simulated ground motions