Stochastic Simulation
of Ground Motion Components
for a Specified Design Scenario
Sanaz Rezaeian
Armen Der Kiureghian (PI)
University of California, Berkeley
Sponsor:
State of California through Transportation Systems Research
Program of Pacific Earthquake Engineering Research (PEER)
Outline:

Motivation

Ground motion model

Extend to simulate multiple components
o
Principal axes of ground motions
o
High correlations between model parameters

Example

Conclusion
Motivation:
In seismic hazard analysis, development of design ground motions is a crucial step.

High levels of intensity
Expected structural behavior: Nonlinear
Approach: Response-history dynamic analysis
Requires: Ground motion time-series
Motivation:
In seismic hazard analysis, development of design ground motions is a crucial step.

High levels of intensity
Expected structural behavior: Nonlinear
Approach: Response-history dynamic analysis
Requires: Ground motion time-series
Difficulties come from scarcity of previously recorded motions.
Controversies come from methods of selecting and modifying real records.
Alternative: Use simulated time-series in conjunction or in the place of real records.
Motivation:
In seismic hazard analysis, development of design ground motions is a crucial step.

High levels of intensity
Expected structural behavior: Nonlinear
Approach: Response-history dynamic analysis
Requires: Ground motion time-series
Our Goal: Earthquake and site characteristics  Suite of simulated time-series
(F, M, Rrup, Vs30)
Site
Controlling Fault
F: Faulting mechanism
M: Moment magnitude
…
VS30: Shear wave velocity of top 30m
R: Closest distance to ruptured area
Motivation:
In seismic hazard analysis, development of design ground motions is a crucial step.

High levels of intensity
Expected structural behavior: Nonlinear
Approach: Response-history dynamic analysis
Requires: Ground motion time-series
Our Goal: Earthquake and site characteristics  Suite of simulated time-series
(F, M, Rrup, Vs30)
Site
Controlling Fault
F: Faulting mechanism
M: Moment magnitude
…
VS30: Shear wave velocity of top 30m
R: Closest distance to ruptured area
For 2D/3D structural analysis, need ground motion components.
Ground Motion Model:
[Rezaeian and Der Kiureghian, 2008]
 1

t
x(t )  q(t, α)
 h[t   , λ ( )]w( )d 
  f (t ) 

White-noise
w(t )
Linear filter
with
time-varying
parameters
Filtered
white-noise
Normalization
by
standard
deviation
Fully non-stationary process
x(t )
Unit-variance process with
spectral non-stationarity
Time
modulating
filter
Simulated ground acceleration
z(t )
High-pass
filter
Ground Motion Model:
[Rezaeian and Der Kiureghian, 2008]
 1

t
x(t )  q(t, α)
 h[t   , λ ( )]w( )d 
  f (t ) 

White-noise
w(t )
Linear filter
with
time-varying
parameters
Filtered
white-noise
Normalization
by
standard
deviation
Fully non-stationary process
x(t )
Unit-variance process with
spectral non-stationarity
Time
modulating
filter
Simulated ground acceleration
z(t )
High-pass
filter
Acceleration time-series
Ground Motion Model:
[Rezaeian and Der Kiureghian, 2008]
 1

t
x(t )  q(t, α)
 h[t   , λ ( )]w( )d 
  f (t ) 

White-noise
w(t )
Linear filter
with
time-varying
parameters
Filtered
white-noise
Normalization
by
standard
deviation
Fully non-stationary process
x(t )
Unit-variance process with
spectral non-stationarity
Time
modulating
filter
Simulated ground acceleration
z(t )
High-pass
filter
Temporal non-stationarity: Variation of intensity in time
Spectral non-stationarity: Variation of frequency content in time
Ground Motion Model:
[Rezaeian and Der Kiureghian, 2008]
 1

t
x(t )  q(t, α)
 h[t   , λ ( )]w( )d 
  f (t ) 

White-noise
w(t )
Linear filter
with
time-varying
parameters
Filtered
white-noise
Normalization
by
standard
deviation
Fully non-stationary process
x(t )
Unit-variance process with
spectral non-stationarity
Time
modulating
filter
Simulated ground acceleration
z(t )
High-pass
filter
Source of stochasticity
Ground Motion Model:
[Rezaeian and Der Kiureghian, 2008]
 1

t
x(t )  q(t, α)
 h[t   , λ ( )]w( )d 
  f (t ) 

White-noise
w(t )
Linear filter
with
time-varying
parameters
Filtered
white-noise
Normalization
by
standard
deviation
Fully non-stationary process
x(t )
Unit-variance process with
spectral non-stationarity
Time
modulating
filter
Simulated ground acceleration
z(t )
High-pass
filter
Impulse response function (IRF)
corresponding to pseudo-acceleration response of a SDOF linear oscillator
Ground Motion Model:
[Rezaeian and Der Kiureghian, 2008]
 1

t
x(t )  q(t, α)
 h[t   , λ ( )]w( )d 
  f (t ) 

White-noise
w(t )
Linear filter
with
time-varying
parameters
Filtered
white-noise
Normalization
by
standard
deviation
Fully non-stationary process
x(t )
Unit-variance process with
spectral non-stationarity
Time
modulating
filter
Simulated ground acceleration
z(t )
High-pass
filter
Duhamel’s integral
(superposition of filter responses to a sequence of statistically independent
pulses with the time of application τ)
Ground Motion Model:
[Rezaeian and Der Kiureghian, 2008]
 1

t
x(t )  q(t, α)
 h[t   , λ ( )]w( )d 
  f (t ) 

White-noise
w(t )
Linear filter
with
time-varying
parameters
Filtered
white-noise
Normalization
by
standard
deviation
Fully non-stationary process
x(t )
Unit-variance process with
spectral non-stationarity
Time
modulating
filter
Simulated ground acceleration
z(t )
High-pass
filter
Ground Motion Model:
[Rezaeian and Der Kiureghian, 2008]
 1

t
x(t )  q(t, α)
 h[t   , λ ( )]w( )d 
  f (t ) 

White-noise
w(t )
Linear filter
with
time-varying
parameters
Filtered
white-noise
Normalization
by
standard
deviation
Fully non-stationary process
x(t )
Unit-variance process with
spectral non-stationarity
Time
modulating
filter
Simulated ground acceleration
z(t )
High-pass
filter
Ground Motion Model:
[Rezaeian and Der Kiureghian, 2008]
 1

t
x(t )  q(t, α)
 h[t   , λ ( )]w( )d 
  f (t ) 

White-noise
w(t )
Linear filter
with
time-varying
parameters
Filtered
white-noise
Normalization
by
standard
deviation
Fully non-stationary process
x(t )
Unit-variance process with
spectral non-stationarity
Time
modulating
filter
Simulated ground acceleration
z(t )
High-pass
filter
0
time
tn
Ground Motion Model:
[Rezaeian and Der Kiureghian, 2008]
 1

t
x(t )  q(t, α)
 h[t   , λ ( )]w( )d 
  f (t ) 

White-noise
w(t )
Linear filter
with
time-varying
parameters
Filtered
white-noise
Unit-variance process with
spectral non-stationarity
Normalization
by
standard
deviation
Fully non-stationary process
x(t )
Time
modulating
filter
Simulated ground acceleration
z(t )
High-pass
filter
Non-zero residuals!
Over estimates response spectrum at long periods!
Ground Motion Model:
[Rezaeian and Der Kiureghian, 2008]
 1

t
x(t )  q(t, α)
 h[t   , λ ( )]w( )d 
  f (t ) 

White-noise
w(t )
Linear filter
with
time-varying
parameters
Filtered
white-noise
Normalization
by
standard
deviation
Fully non-stationary process
x(t )
Unit-variance process with
spectral non-stationarity
Time
modulating
filter
Simulated ground acceleration
z(t )
High-pass
filter
Critically damped oscillator
z  2 c z  c2 z  x(t)


Ground Motion Model:
[Rezaeian and Der Kiureghian, 2008]
 1

t
x(t )  q(t, α)
 h[t   , λ ( )]w( )d 
  f (t ) 

White-noise
w(t )
Linear filter
with
time-varying
parameters
Filtered
white-noise
Normalization
by
standard
deviation
Fully non-stationary process
x(t )
Unit-variance process with
spectral non-stationarity
Time
modulating
filter
Simulated ground acceleration
z(t )
High-pass
filter
Model Parameters:
 1

t
x(t )  q(t, α)
 h[t   , λ ( )]w( )d 
  f (t ) 

Model Parameters:
 1

t
x(t )  q(t, α)
 h[t   , λ ( )]w( )d 
  f (t ) 

Modulating function parameters:
Ia 

2g
t
2
0n (accel (t )) dt
: Arias intensity
D595 : Effective duration, between 5% to 95% Ia
tmid
: Time at the middle of strong shaking, at 45% Ia
Filter parameters:
ωmid : Frequency at the middle of strong shaking
ω'
: Rate of change of frequency over time

: Damping ratio
Model Parameters:
 1

t
x(t )  q(t, α)
 h[t   , λ ( )]w( )d 
  f (t ) 

Modulating function parameters:
Ia 

2g
t
2
0n (accel (t )) dt
: Arias intensity
D595 : Effective duration, between 5% to 95% Ia
tmid
: Time at the middle of strong shaking, at 45% Ia
Filter parameters:
ωmid : Frequency at the middle of strong shaking
ω'
: Rate of change of frequency over time

: Damping ratio
Model Parameters:
 1

t
x(t )  q(t, α)
 h[t   , λ ( )]w( )d 
  f (t ) 

Modulating function parameters:
Ia 

2g
t
2
0n (accel (t )) dt
Filter parameters:
ωmid : Frequency at the middle of strong shaking
: Arias intensity
D595 : Effective duration, between 5% to 95% Ia
: Time at the middle of strong shaking, at 45% Ia
: Rate of change of frequency over time

: Damping ratio
Model parameters are identified for many recorded motions
to develop predictive equations in terms of F, M, R, VS30
Acceleration, g
tmid
ω'
0.15
0
Recorded
-0.25
0
Time, sec
40
Match
statistical
characteristics
Representing:
• Intensity
• Frequency
• Bandwidth
Identify
model parameters
Ia, tmid, D5-95
ωmid, ω’ , ζ
Two Horizontal Components:
Component 1:
 1 t

x1 (t )  q(t, α1 ) 
h
[
t

τ,
λ
(

)]
w
(
τ
)
dτ


1
1
σ
(
t
)


 h

Component 2:
 1 t

x 2 (t )  q(t, α 2 ) 
 h[t  τ,λ 2 ( )]w2 ( τ )dτ 
σ
(
t
)
 h 

Two Horizontal Components:
Component 1:
Component 2:

 1 t

x1 (t )  q(t, α1 ) 
h
[
t

τ,
λ
(

)]
w
(
τ
)
dτ


1
1
σ
(
t
)


 h

 1 t

x 2 (t )  q(t, α 2 ) 
 h[t  τ,λ 2 ( )]w2 ( τ )dτ 
σ
(
t
)
 h 

w1(τ) and w2(τ) are statistically independent if along the principal axes.
source of
stochasticity
Two Horizontal Components:
 1 t

x1 (t )  q(t, α1 ) 
h
[
t

τ,
λ
(

)]
w
(
τ
)
dτ


1
1
σ
(
t
)


 h

Component 1:
 1 t

x 2 (t )  q(t, α 2 ) 
 h[t  τ,λ 2 ( )]w2 ( τ )dτ 
σ
(
t
)
 h 

Component 2:

source of
stochasticity
w1(τ) and w2(τ) are statistically independent if along the principal axes.
Principal Axes of Ground Motion:
A set of orthogonal axes along which the components are uncorrelated.
Minor
Expected Epicenter
Site
Major
Horizontal Plane
Intermediate
Penzien and Watabe (1975)
Two Horizontal Components:
 1 t

x1 (t )  q(t, α1 ) 
h
[
t

τ,
λ
(

)]
w
(
τ
)
dτ


1
1
σ
(
t
)


 h

Component 1:
 1 t

x 2 (t )  q(t, α 2 ) 
 h[t  τ,λ 2 ( )]w2 ( τ )dτ 
σ
(
t
)
 h 

Component 2:

source of
stochasticity
w1(τ) and w2(τ) are statistically independent if along the principal axes.
Principal Axes of Ground Motion:
A set of orthogonal axes along which the components are uncorrelated.
Rotate recorded motions in the database.
ρ a ,a ≠0
ρ a ,a =0
1
1,θ
a2,θ
θ
2
2,θ
Site
a1,θ
a1
a2
Horizontal Plane
Rotating Recorded Motions:
Correlation Coefficient
Between The Two
Components
Northridge earthquake recorded at Mt. Wilson Station
0.5
(55,0)
0
(0,-0.42)
-0.5
0
10
20
30
40
50
60
70
80
90
Rotation Angle, degrees
0.3
0.2
As-Recorded
Component 1
0.1
Acceleration, g
0.3
0.2
0
-0.1
-0.2
-0.3
0
-0.1
-0.2
-0.3
0.3
0.2
0.3
0.2
As-Recorded
Component 2
0.1
0
-0.1
-0.2
-0.3
5
10
15
20
Time, s
25
30
35
Principal
Component 2
0.1
0
-0.1
-0.2
-0.3
0
Principal
Component 1
0.1
40
0
5
10
15
20
Time, s
25
30
35
40
Two Horizontal Components:
Component 1:
 1 t

x1 (t )  q(t, α1 ) 
h
[
t

τ,
λ
(

)]
w
(
τ
)
dτ


1
1
σ
(
t
)


 h

Component 2:
 1 t

x 2 (t )  q(t, α 2 ) 
 h[t  τ,λ 2 ( )]w2 ( τ )dτ 
σ
(
t
)
 h 

Two Horizontal Components:

Component 1:
 1 t

x1 (t )  q(t, α1 ) 
h
[
t

τ,
λ
(

)]
w
(
τ
)
dτ


1
1
σ
(
t
)


 h

Component 2:
 1 t

x 2 (t )  q(t, α 2 ) 
 h[t  τ,λ 2 ( )]w2 ( τ )dτ 
σ
(
t
)
 h 

Predictive equations:

R rup 

 M 
  β 4  ln Vs30
  β3  ln
 -1 [ F p ( p)]  β0  β1 (F)  β 2 
 750 m/s
 25 km 
 7.0 



 M 
  β3
 1[ F p ( p)]  β0  β1 (F)  β 2 
7
.
0


 R rup 

  β4
 25 km 


 Vs30

 750 m/s

  η  ε

if
p  I a,maj , I a,int

  η  ε

if
p  D5-95 , tmid , ωmid , ω' , ζ f
Two Horizontal Components:

Component 1:
 1 t

x1 (t )  q(t, α1 ) 
h
[
t

τ,
λ
(

)]
w
(
τ
)
dτ


1
1
σ
(
t
)


 h

Component 2:
 1 t

x 2 (t )  q(t, α 2 ) 
 h[t  τ,λ 2 ( )]w2 ( τ )dτ 
σ
(
t
)
 h 

Predictive equations:

R rup 

 M 
  β 4  ln Vs30
  β3  ln
 -1 [ F p ( p)]  β0  β1 (F)  β 2 
 750 m/s
 25 km 
 7.0 



 M 
  β3
 1[ F p ( p)]  β0  β1 (F)  β 2 
7
.
0


Model parameter p
transformed to the
standard normal space
 R rup 

  β4
 25 km 


 Vs30

 750 m/s

  η  ε

if
p  I a,maj , I a,int

  η  ε

if
p  D5-95 , tmid , ωmid , ω' , ζ f
Two Horizontal Components:

Component 1:
 1 t

x1 (t )  q(t, α1 ) 
h
[
t

τ,
λ
(

)]
w
(
τ
)
dτ


1
1
σ
(
t
)


 h

Component 2:
 1 t

x 2 (t )  q(t, α 2 ) 
 h[t  τ,λ 2 ( )]w2 ( τ )dτ 
σ
(
t
)
 h 

Predictive equations:

R rup 

 M 
  β 4  ln Vs30
  β3  ln
 -1 [ F p ( p)]  β0  β1 (F)  β 2 
 750 m/s
 25 km 
 7.0 



 M 
  β3
 1[ F p ( p)]  β0  β1 (F)  β 2 
7
.
0


 R rup 

  β4
 25 km 


 Vs30

 750 m/s
Predicted mean
conditioned on
earthquake and site characteristics

  η  ε

if
p  I a,maj , I a,int

  η  ε

if
p  D5-95 , tmid , ωmid , ω' , ζ f
Independent
normally-distributed
errors
Two Horizontal Components:
Component 1:
 1 t

x1 (t )  q(t, α1 ) 
h
[
t

τ,
λ
(

)]
w
(
τ
)
dτ


1
1
σ
(
t
)


 h

Component 2:
 1 t

x 2 (t )  q(t, α 2 ) 
 h[t  τ,λ 2 ( )]w2 ( τ )dτ 
σ
(
t
)
 h 


Predictive equations.

High correlations expected between parameters of the two components.
Two Horizontal Components:
Correlation Matrix:
Major Component (larger Arias intensity)
I a, tf
D595, tf
tmid , tf
ωmid , tf
−0.38
−0.04
−0.21
−0.25
−0.06
+0.92
−0.30
1
+0.68
−0.07
−0.21
−0.26
−0.31
1
−0.24
−0.22
−0.26
1
−0.19
1
ω'tf
ω'tf
ζ tf
−0.03
−0.13
+0.09
+0.02
+0.89
+0.68
−0.17
−0.11
−0.17
+0.04
+0.65
+0.96
−0.30
−0.24
−0.21
+0.28
−0.13
−0.15
−0.29
+0.94
−0.10
+0.29
I a, tf
D595, tf
tmid , tf
ωmid , tf
−0.06
+0.19
−0.21
−0.22
−0.10
+0.52
−0.13
ω'tf
1
−0.01
−0.23
−0.29
+0.32
−0.02
+0.75
ζ tf
1
−0.31
+0.01
−0.08
+0.07
−0.005
1
+0.69
−0.20
−0.18
−0.17
1
−0.34
−0.24
−0.22
1
−0.19
+0.28
I a, tf
D595, tf
tmid , tf
ωmid , tf
1
−0.05
ω'tf
I a, tf
D595, tf tmid , tf
1
ζ tf
Intermediate Component
Symmetric
ωmid , tf
ζ tf
Major Component
1
Intermediate Component (smaller Arias intensity)
Two Horizontal Components:
Correlation Matrix:
Major Component (larger Arias intensity)
I a, tf
D595, tf
tmid , tf
ωmid , tf
−0.38
−0.04
−0.21
−0.25
−0.06
+0.92
−0.30
1
+0.68
−0.07
−0.21
−0.26
−0.31
1
−0.24
−0.22
−0.26
1
−0.19
1
ω'tf
ω'tf
ζ tf
−0.03
−0.13
+0.09
+0.02
+0.89
+0.68
−0.17
−0.11
−0.17
+0.04
+0.65
+0.96
−0.30
−0.24
−0.21
+0.28
−0.13
−0.15
−0.29
+0.94
−0.10
+0.29
I a, tf
D595, tf
tmid , tf
ωmid , tf
−0.06
+0.19
−0.21
−0.22
−0.10
+0.52
−0.13
ω'tf
1
−0.01
−0.23
−0.29
+0.32
−0.02
+0.75
ζ tf
1
−0.31
+0.01
−0.08
+0.07
−0.005
1
+0.69
−0.20
−0.18
−0.17
1
−0.34
−0.24
−0.22
1
−0.19
+0.28
I a, tf
D595, tf
tmid , tf
ωmid , tf
1
−0.05
ω'tf
I a, tf
D595, tf tmid , tf
1
ζ tf
Intermediate Component
Symmetric
ωmid , tf
ζ tf
Major Component
1
Intermediate Component (smaller Arias intensity)
Two Horizontal Components:
Correlation Matrix:
Major Component (larger Arias intensity)
I a, tf
D595, tf
tmid , tf
ωmid , tf
−0.38
−0.04
−0.21
−0.25
−0.06
+0.92
−0.30
1
+0.68
−0.07
−0.21
−0.26
−0.31
1
−0.24
−0.22
−0.26
1
−0.19
1
ω'tf
ω'tf
ζ tf
−0.03
−0.13
+0.09
+0.02
+0.89
+0.68
−0.17
−0.11
−0.17
+0.04
+0.65
+0.96
−0.30
−0.24
−0.21
+0.28
−0.13
−0.15
−0.29
+0.94
−0.10
+0.29
I a, tf
D595, tf
tmid , tf
ωmid , tf
−0.06
+0.19
−0.21
−0.22
−0.10
+0.52
−0.13
ω'tf
1
−0.01
−0.23
−0.29
+0.32
−0.02
+0.75
ζ tf
1
−0.31
+0.01
−0.08
+0.07
−0.005
1
+0.69
−0.20
−0.18
−0.17
1
−0.34
−0.24
−0.22
1
−0.19
+0.28
I a, tf
D595, tf
tmid , tf
ωmid , tf
1
−0.05
ω'tf
I a, tf
D595, tf tmid , tf
1
ζ tf
Intermediate Component
Symmetric
ωmid , tf
ζ tf
Major Component
1
Intermediate Component (smaller Arias intensity)
Two Horizontal Components:
Correlation Matrix:
Major Component (larger Arias intensity)
I a, tf
D595, tf
tmid , tf
ωmid , tf
−0.38
−0.04
−0.21
−0.25
−0.06
+0.92
−0.30
1
+0.68
−0.07
−0.21
−0.26
−0.31
1
−0.24
−0.22
−0.26
1
−0.19
1
ω'tf
ω'tf
ζ tf
−0.03
−0.13
+0.09
+0.02
+0.89
+0.68
−0.17
−0.11
−0.17
+0.04
+0.65
+0.96
−0.30
−0.24
−0.21
+0.28
−0.13
−0.15
−0.29
+0.94
−0.10
+0.29
I a, tf
D595, tf
tmid , tf
ωmid , tf
−0.06
+0.19
−0.21
−0.22
−0.10
+0.52
−0.13
ω'tf
1
−0.01
−0.23
−0.29
+0.32
−0.02
+0.75
ζ tf
1
−0.31
+0.01
−0.08
+0.07
−0.005
1
+0.69
−0.20
−0.18
−0.17
1
−0.34
−0.24
−0.22
1
−0.19
+0.28
I a, tf
D595, tf
tmid , tf
ωmid , tf
1
−0.05
ω'tf
I a, tf
D595, tf tmid , tf
1
ζ tf
Intermediate Component
Symmetric
ωmid , tf
ζ tf
Major Component
1
Intermediate Component (smaller Arias intensity)
Example:
Design scenario: F = 1 (Reverse) , M = 7.35 , R =14 km , VS30 = 660 m/s
Example:
Design scenario: F = 1 (Reverse) , M = 7.35 , R =14 km , VS30 = 660 m/s
Major Component
Recorded
Intermediate Component
Ia
D5-95
tmid
ωmid/(2π)
ω’/(2π)
s.g
s
s
Hz
Hz/s
16.7
17.3
27.2
18.3
10.1
17.1
3.9
8.1
–0.08
–0.12
–0.03
0.0165
0.0147
Simulated
0.0099
3.2
ζf
0.12
0.42
0.20
Ia
D5-95
tmid
ωmid/(2π)
ω’/(2π)
s.g
s
s
Hz
Hz/s
0.0135
0.0047
0.0034
17.0
21.0
24.8
17.8
10.7
16.9
4.1
8.6
3.7
–0.02
–0.18
–0.13
ζf
0.11
0.50
0.35
Example:
Design scenario: F = 1 (Reverse) , M = 7.35 , R =14 km , VS30 = 660 m/s
Major Component
Recorded
Intermediate Component
Ia
D5-95
tmid
ωmid/(2π)
ω’/(2π)
s.g
s
s
Hz
Hz/s
16.7
17.3
27.2
18.3
10.1
17.1
3.9
8.1
–0.08
–0.12
–0.03
0.0165
0.0147
Simulated
0.0099
3.2
Ia
D5-95
tmid
ωmid/(2π)
ω’/(2π)
s.g
s
s
Hz
Hz/s
0.0135
0.0047
0.0034
17.0
21.0
24.8
17.8
10.7
16.9
4.1
8.6
3.7
–0.02
–0.18
–0.13
ζf
0.12
0.42
0.20
0.1
Recorded
Recorded
Simulated
Simulated
Simulated
Simulated
ζf
0.11
0.50
0.35
0
Acceleration, g
-0.1
0.1
0
-0.1
0.05
0
-0.05
0
20
40
Time, s
60
80
0
20
40
60
Time, s
80
Example:
Design scenario: F = 1 (Reverse) , M = 7.35 , R =14 km , VS30 = 660 m/s
Major Component
Recorded
Intermediate Component
Ia
D5-95
tmid
ωmid/(2π)
ω’/(2π)
s.g
s
s
Hz
Hz/s
16.7
17.3
27.2
18.3
10.1
17.1
3.9
8.1
–0.08
–0.12
–0.03
0.0165
0.0147
Simulated
0.0099
3.2
0.01
Ia
D5-95
tmid
ωmid/(2π)
ω’/(2π)
s.g
s
s
Hz
Hz/s
0.0135
0.0047
0.0034
17.0
21.0
24.8
17.8
10.7
16.9
4.1
8.6
3.7
–0.02
–0.18
–0.13
ζf
0.12
0.42
0.20
Recorded
Recorded
Simulated
Simulated
Simulated
Simulated
ζf
0.11
0.50
0.35
0
Velocity, m/s
-0.01
0.05
0
-0.05
0.05
0
-0.05
0
20
40
Time, s
60
80
0
20
40
60
Time, s
80
Example:
Design scenario: F = 1 (Reverse) , M = 7.35 , R =14 km , VS30 = 660 m/s
Major Component
Recorded
Intermediate Component
Ia
D5-95
tmid
ωmid/(2π)
ω’/(2π)
s.g
s
s
Hz
Hz/s
16.7
17.3
27.2
18.3
10.1
17.1
3.9
8.1
–0.08
–0.12
–0.03
0.0165
0.0147
Simulated
0.0099
3.2
0.02
Ia
D5-95
tmid
ωmid/(2π)
ω’/(2π)
s.g
s
s
Hz
Hz/s
0.0135
0.0047
0.0034
17.0
21.0
24.8
17.8
10.7
16.9
4.1
8.6
3.7
–0.02
–0.18
–0.13
ζf
0.12
0.42
0.20
Recorded
Recorded
Simulated
Simulated
Simulated
Simulated
ζf
0.11
0.50
0.35
Displacement, m
0
-0.02
0.05
0
-0.05
0.05
0
-0.05
0
20
40
Time, s
60
80
0
20
40
60
Time, s
80
Conclusion:

Developed a stochastic model for earthquake ground motion components

Created a database of principal ground motion components

Identified model parameters for the records in the database
 predictive equations for model parameters in terms of F , M , R , VS30

Identified correlation coefficients between model parameters of the components

For given F , M , R , VS30 , correlated model parameters are randomly simulated and
used along with statistically independent white-noise processes to generate a pair of
horizontal ground motion components in the directions of principal axes.
The proposed methods can be easily extended to simulate the vertical component.
Related Publications:
1. Rezaeian S, Der Kiureghian A. "A stochastic ground motion model with separable temporal and spectral
nonstationarities”, Earthquake Engineering and Structural Dynamics, 2008, Vol. 37, pp. 1565-1584.
2. Rezaeian S, Der Kiureghian A. "Simulation of synthetic ground motions for specified earthquake and site
characteristics”, Earthquake Engineering and Structural Dynamics, 2010, Vol. 39, pp. 1155-1180.
3. Rezaeian S, Der Kiureghian A. "Simulation of orthogonal horizontal ground motion components for specified
earthquake and site characteristics”, Submitted to Earthquake Engineering and Structural Dynamics.
Thank You
This project was made possible with support from:
State of California through Transportation Systems
Research Program of Pacific Earthquake Engineering
Research Center (PEER TSRP).
44
Ground Motion Model: Advantages
Extra

Small number of parameters that have physical meaning and can be easily
identified by matching with features of a given accelerogram

Completely separable temporal and spectral nonstationary characteristics,
which facilitates identification and interpretation of the parameters

No need for sophisticated processing of the target accelerogram, e.g. Fourier
analysis or estimation of evolutionary PSD

Simple simulation of sample functions, requiring little more than generation
of standard normal random variables
k
xˆ (t )  q(t ) si (t )  ui
i 1

for
where
where
tk  t  tk 1
ti  i  t
i  0,...,n
ui ~ N(0,1)
Form of the model facilitates nonlinear random vibration analysis (e.g., by
using TELM).