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Sanaz Rezaeian (Doctoral Candidate) Armen Der Kiureghian (PI)
University of California, Berkeley
Simulation of Synthetic Ground Motions for
Specified Earthquake and Site Characteristics
Sponsor: State of California through Transportation Systems Research Program of Pacific Earthquake Engineering Research (PEER) Center
Our Goal: Earthquake and site characteristics Suite of simulated design time-histories (F, M, Rrup, Vs30 ,…)
Objective:
F: Faulting mechanism M: Moment magnitude
…
Rrup: Closest distance to ruptured area
Vs30: Shear wave velocity of top 30m Controlling Fault
Site
What we have done so far: Developed a stochastic site-based model for far-field strong ground motions. Developed empirical predictive equations for the model parameters. Compared elastic response spectra (median and variability) to NGA relations.
Ongoing activity and what we plan to accomplish by May 2010: Simulate orthogonal horizontal ground motion components. Extend the model to near-field ground motions. Scrutinize the simulated motions for inelastic structural responses.
Ground Motion Model:
0 5 10 15 20
0
Acceleration
High-pass Filtering
0 5 10 15 20
0
Time, sec
Unit-variance process
Controls spectral nonstationarity
0 5 10 15 20
0
Time, sec
Time modulating function
Controls temporal nonstationarity
0 5 10 15 20
0
Time, sec
Ground Motion Model Parameters:
t0 tn
Let:
0 tn
0 tn
: Arias intensity
: Time at the middle of strong shaking
: Effective duration
Ground Motion Model Parameters:
t0 tn
Let:
0 tn
0 tn
: Time at the middle of strong shaking
: Effective duration
If the model parameters are given, time-histories can be simulated.
: Arias intensity
Simulate a suite of ground motions for a given design scenario:
Simulate a given accelerogram:
Applications in Practice:
Given Earthquake/Site characteristics
(design scenario)
Generate several possible sets of
model parameters
Simulations … 0 5 10 15 20 25 30 35 40 45 50 -0.1
0
0.1
-0.1 0
0.1
0.1
-0.1
0
…
model formulation
F, M, Rrup, Vs30
predictive equations
Ia, tmid, D5-95 ωmid, ω’ , ζ
Match statistical
characteristics Representing:
• Intensity • Frequency • Bandwidth
Identify model parameters
ωmid, ω’ , ζ Ia, tmid, D5-95 Recorded
0 40 -0.25
0
0.15
Time, sec Acc
eler
atio
n, g
model formulation
Simulations …
…
-0.25
0
0.15
0 -0.25
0
0.15
-0.25
0
0.15
40
Simulate a suite of ground motions for a given design scenario:
Simulate a given accelerogram:
Applications in Practice:
Given Earthquake/Site characteristics
(design scenario)
Generate several possible sets of
model parameters
Simulations … 0 5 10 15 20 25 30 35 40 45 50 -0.1
0
0.1
-0.1 0
0.1
0.1
-0.1
0
…
model formulation
F, M, Rrup, Vs30
predictive equations
Ia, tmid, D5-95 Regression
Predictor variables
Response variables
Done for many records to get observational data for predictor and response variables
ωmid, ω’ , ζ
Match statistical
characteristics Representing:
• Intensity • Frequency • Bandwidth
Identify model parameters
ωmid, ω’ , ζ Ia, tmid, D5-95 Recorded
0 40 -0.25
0
0.15
Time, sec Acc
eler
atio
n, g
model formulation
Simulations …
…
-0.25
0
0.15
0 -0.25
0
0.15
-0.25
0
0.15
40
Ground Motion Database (far-field):
Earthquake #ofrecords1 ImperialValley 2
2 Victoria,Mexico 2
3 Morganhill 10
4 Landers 4
5 BigBear 10
6 Kobe,Japan 4
7 Kocaeli,Turkey 4
8 Duzce,Turkey 2
9 Sitka,Alaska 2
10 Manjil,Iran 2
11 HectorMine 16
12 Denali,Alaska 4
13 SanFernando 14
14 Tabas,Iran 2
15 Coalinga 2
16 NPalmSprings 12
17 LomaPrieta 28
18 Northridge 38
19 ChiChi,Taiwan 48
Strike-slip
Reverse Mom
ent M
agni
tude
Rrup , km 10 20 30 40 50 60 70 80 90 100
8.0
7.5
7.0
6.5
6.0
Strike-slip Reverse
Vs30 > 600 m/sec
Two horizontal components
Shallow crustal earthquakes in tectonically active regions
Total: 206 Accelerograms
Predictive Equations (Regression):
Independent Normally-distributed
errors
Observed Data Fitted PDF
Norm
alize
d Fr
eque
ncy
(Tot
al:2
06)
5 10 15 20 25 30 35 40 45 0
0.02
0.04
0.06
-2 -1.5 -1 -0.5 0 0.5 0
1
2
3
4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0
1
2
3
4
0 5 10 15 20 25 0
0.04
0.08
0.12
0.16
0 5 10 15 20 25 30 35 40 0
0.02
0.04
0.06
0.08
ln(Ia, sec.g) D5-95, sec tmid, sec
ω'/(2π), Hz/sec ζ ωmid/(2π), Hz
-7.5 -5.5 -3.5 -1.5 0 0
0.1
0.2
0.3
0.4 Normal Beta Beta
Gamma Two-Sided Exponential
Beta
Distributions assigned to the model parameters:
Regression model (for jth earthquake and kth record of that earthquake):
Predicted mean conditioned on
earthquake and site characteristics
Model parameter θ transformed to the
standard normal space
Standard deviation of
−1.844 −0.071 2.944 −1.356 −0.265 0.27 0.59 0.65
−6.195 −0.703 6.792 0.219 −0.523 0.46 0.57 0.73
−5.011 −0.345 4.638 0.348 −0.185 0.51 0.41 0.66
2.253 −0.081 −1.810 −0.211 0.012 0.69 0.72 1.00
−2.489 0.044 2.408 0.065 −0.081 0.13 0.95 0.96
−0.258 −0.477 0.905 −0.289 0.316 0.68 0.76 1.02
Regression Results (Predictive Equations):
if
if
Maximum Likelihood Estimation:
Formulation:
Regression Results (Correlations):
1 −0.36 0.01 −0.15 0.13 −0.01
1 0.67 −0.13 −0.16 −0.20
1 −0.28 −0.20 −0.22
1 −0.20 0.28
1 −0.01
1
Transformed model parameters:
Symmetric
(given the earthquake and site characteristics)
F = 1 (Reverse) M = 7.35 Rrup =14 km VS30 = 660 m/sec
Example 1 : Acceleration
4 simulated motions and 1 real recording for the given design scenario:
Acce
lera
tion,
g
Time, sec
0 5 10 15 20 25 30 35
-0.2 0
0.2
0 5 10 15 20 25 30 35 -0.2
0
0.2
0 5 10 15 20 25 30 35 -0.1
0
0.1
0 5 10 15 20 25 30 35 -0.1
0
0.1
0 5 10 15 20 25 30 35 -0.1
0 0.1
Simulated
Recorded
Simulated
Simulated
Simulated
RealizaVonsofmodelparameters:
0.01217.236.276.88‐0.010.14
0.14512.306.785.900.120.26
0.05514.227.224.48‐0.160.38
0.01414.076.3110.75‐0.240.26
0.03614.878.324.36‐0.150.03
Ia sec.g
D5-95 sec
tmid sec
ω’/(2π) Hz/sec
ζ ωmid /(2π) Hz
(1978 Tabas at Dayhook)
F = 1 (Reverse) M = 7.35 Rrup =14 km VS30 = 660 m/sec
4 simulated motions and 1 real recording for the given design scenario:
Velo
city,
m/s
ec
Time, sec
0 5 10 15 20 25 30 35
-0.2
0
0.2
0 5 10 15 20 25 30 35 -0.2
0
0.2
0 5 10 15 20 25 30 35 -0.05
0
0.05
0 5 10 15 20 25 30 35 -0.1
0
0.1
0 5 10 15 20 25 30 35 -0.1
0
0.1
Example 1 : Velocity
Simulated
Recorded
Simulated
Simulated
Simulated
RealizaVonsofmodelparameters:
0.01217.236.276.88‐0.010.14
0.14512.306.785.900.120.26
0.05514.227.224.48‐0.160.38
0.01414.076.3110.75‐0.240.26
0.03614.878.324.36‐0.150.03
Ia sec.g
D5-95 sec
tmid sec
ω’/(2π) Hz/sec
ζ ωmid /(2π) Hz
F = 1 (Reverse) M = 7.35 Rrup =14 km VS30 = 660 m/sec
4 simulated motions and 1 real recording for the given design scenario:
Disp
lace
men
t, m
Time, sec
0 5 10 15 20 25 30 35
-0.1
0
0.1
0 5 10 15 20 25 30 35
-0.1
0
0.1
0 5 10 15 20 25 30 35 -0.05
0
0.05
0 5 10 15 20 25 30 35 -0.05
0
0.05
0 5 10 15 20 25 30 35
-0.05 0
0.05
Example 1 : Displacement
Simulated
Recorded
Simulated
Simulated
Simulated
RealizaVonsofmodelparameters:
0.01217.236.276.88‐0.010.14
0.14512.306.785.900.120.26
0.05514.227.224.48‐0.160.38
0.01414.076.3110.75‐0.240.26
0.03614.878.324.36‐0.150.03
Ia sec.g
D5-95 sec
tmid sec
ω’/(2π) Hz/sec
ζ ωmid /(2π) Hz
RealizaVonsofmodelparameters:
0.14512.306.785.900120.26
0.14512.799.867.48‐0.520.13
0.14522.1116.248.05‐0.090.12
0.1458.145.317.34‐0.020.30
0.14511.0110.304.430.120.29
Example 2 :
F = 1 (Reverse) M = 7.35 Rrup =14 km VS30 = 660 m/sec
If desired, a fixed value may be assigned to one or more of the model parameters:
Acce
lera
tion,
g
Time, sec
0 5 10 15 20 25 30 35 40 -0.5
0
0.5
0 5 10 15 20 25 30 35 40 -0.5
0
0.5
0 5 10 15 20 25 30 35 40 -0.5
0
0.5
-0.5
0
0.5
0 5 10 15 20 25 30 35 40 -0.5
0
0.5 0 5 10 15 20 25 30 35 40
Recorded
Simulated
Simulated
Simulated
Simulated
Ia sec.g
D5-95 sec
tmid sec
ω’/(2π) Hz/sec
ζ ωmid /(2π) Hz
Example 3 : Response Spectrum (5% damped) 2 horizontal components of a recorded motion (1994 Northridge at LA Wonderland Ave)
Vs. 50 simulated motions
Corresponding to earthquake and site characteristics:
Period, sec
Def
orm
atio
n R
espo
nse
Spec
trum
, m
10 -1 10 0 10 -4
10 -3
10 -2
10 -1
10 0
5×10 0 10 -1 10 0 10 -3
10 -2
10 -1
10 0
10 1
5×10 0 Period, sec
Pseu
do-A
ccel
erat
ion
Res
pons
e Sp
ectru
m, g
Recorded Simulated
F = 1 (Reverse) M = 6.69 Rrup = 20.3 km VS30 = 1223 m/sec
Comparison with NGA Models:
10 -3
10 -2
10 -1
10 0
M=6.0, R=20km
Campbell-Bozorgnia (z2.5 = 1km) Abrahamson-Silva (z1.0 = 34m) Chiou-Youngs (z1.0 = 24m) Boore-Atkinson
5% D
ampe
d Ps
eudo
-Acc
eler
atio
n R
espo
nse
Spec
trum
, g
10 -3
10 -2
10 -1
10 0
M=7.0, R=20km M=7.0, R=20km
M=7.0, R=10km
Period, sec 0.1 1.0
M=7.0, R=40km
5.0
F = 0 (Strike-Slip) Vs30 = 760 m/sec
Avg NGA 500 Simulations
Median Median −1 logarithmic stdv.
Median +1 logarithmic stdv.
NGA Parameters: Rupture width = 20km Rupture depth = 1km
Selected NGA Models:
Note: Models based on different subsets of NGA database.
Observe: Except for M=6.0 (lower bound of database), deviations are much smaller than the variability present in the NGA prediction equations. Synthetics are in close agreement with NGA.
Period, sec
10 -3
10 -2
10 -1
10 0
M=8.0, R=20km
0.1 1.0 5.0
Current & Future Developments: Simulating correlated orthogonal horizontal ground motion components.
Component 1:
Component 2:
Motions in the database are rotated to the principal axes so that w1(τ) and w2(τ) are statistically independent.
Model is fitted to the rotated database to estimate correlations: ρα1, α2 and ρλ1, λ2
Current & Future Developments: Simulating near-field ground motions.
Separately model and superimpose:
1) The directivity pulse Long period pulse in the velocity time-series of the fault-normal component. Develop prediction equations for characteristics of the pulse in terms of earthquake/site parameters. Collaboration with Jack Baker: Using wavelet analysis, directivity pulse extracted from a database of near-field motions, this database will be used to develop prediction equations.
2) The fling step Permanent displacement may exist in the fault-parallel component. Incorporate the available seismological models (e.g., Somerville 1998, Abrahamson 2001).
3) The residue motion The total motion minus the directivity pulse and the fling step. Model by a modified version of the far-field stochastic ground motion process.
Current & Future Developments: Scrutinize the simulated motions for inelastic structural responses.
Compare inelastic response spectra (for given ductility ratios) of synthetic motions with real recordings and existing prediction equations (e.g., Bozorgnia et. al., 2010).
Case Study: Compare inelastic response of a multi-degree-of-freedom structure to simulated and recorded motions.
Related Publications:
Rezaeian, S. and A. Der Kiureghian, "A stochastic ground motion model with separable temporal and spectral nonstationarities," Earthquake Engineering and Structural Dynamics, July 2008, Vol. 37, pp. 1565-1584.
Rezaeian, S. and A. Der Kiureghian, "Simulation of synthetic ground motions for specified earthquake and site characteristics," Earthquake Engineering and Structural Dynamics, 2009. Submitted.
MATLAB software to be made available Current abilities:
Fitting the stochastic model to a target accelerogram. Simulating far-field strong motions on firm-ground for specified F, M, Rrup, Vs30.
Will be added by May 2010: Two component simulation. Near-field simulation.
Thank You
Cumulative energy
Cumulative number of zero-level up crossings – a measure of dominant frequency
Cumulative number of positive minima and negative maxima – a measure of bandwidth
Features of target accelerogram
0 5 10 15 20 25 30 35 40 0
0.005
0.01
0.015
0.02
0.025
0.03
Cum
ulat
ive
ener
gy
0 5 10 15 20 25 30 35 40 0
20
40
60
80
100
120
140 160
Cum
ulat
ive
num
ber o
f zer
o - level
up - cro
ssin
gs
Time, sec
0 5 10 15 20 25 30 35 40 0
20
40
60
80
100
120
140
Cum
ulat
ive n
umbe
r of n
egat
ive m
axim
a a
nd p
ositiv
e m
inim
a
Response spectrum
Discretized white noise (input)
Unit-variance process with spectral nonstationarity
Linear time-varying filter
Fully non-stationarity process
Time modulation
High-pass filter
Simulated ground motion (output)
Post-processing is needed for long-period range. A critically damped oscillator is used as a high-pass filter.
corner frequency
10 10 0 10 1 10 -3
10
10
10
10
T (sec)
A(g)
-1
-1
-2
0
1
After high-pass filtering
10 10 0 10 1
T (sec) -1
10 -3
10
10
10
10
A(g)
-1
-2
0
1
Acce
lera
tion,
g
Recorded motion
0 5 10 15 20 25 30 35 40 -0.2
-0.1
0
0.1
0.2
0 5 10 15 20 25 30 35 40 -0.2
0
0.2
0 5 10 15 20 25 30 35 40 -0.2
-0.1
0
0.1
0.2
Time (sec)
Simulation
Simulation
Northridge earthquake
Acce
lera
tion,
g
Recorded motion
Time (sec)
Simulation
Simulation
Kobe earthquake