stochastic simulation of ground motion components for a specified design scenario

Sanaz RezaeianArmen Der Kiureghian (PI)University of California, Berkeley

Stochastic Simulation of Ground Motion Components for a Specified Design Scenario

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

In seismic hazard analysis, development of design ground motions is a crucial step.

High levels of intensityExpected structural behavior: NonlinearApproach: Response-history dynamic analysisRequires: Ground motion time-series

Motivation:



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:



Our Goal: Earthquake and site characteristics Suite of simulated time-series(F, M, Rrup, Vs30)

Motivation:

F: Faulting mechanismM: Moment magnitude

…

R: Closest distance to ruptured area

VS30: Shear wave velocity of top 30mControlling Fault

Site



Our Goal: Earthquake and site characteristics Suite of simulated time-series(F, M, Rrup, Vs30)

For 2D/3D structural analysis, need ground motion components.

Motivation:

F: Faulting mechanismM: Moment magnitude

…

R: Closest distance to ruptured area

VS30: Shear wave velocity of top 30mControlling Fault

Site

t

fdwth

tt,qtx

)()](,[

)(1

)()( λα

Ground Motion Model:

Linear filterwith

time-varyingparameters

Timemodulating

filter

High-passfilter

Unit-variance process with spectral non-stationarity

Normalizationby

standarddeviation

White-noise)(tw

Fully non-stationary process)(tx

Simulated ground acceleration)(tz

Filteredwhite-noise

[Rezaeian and Der Kiureghian, 2008]

t

fdwth

tt,qtx

)()](,[

)(1

)()( λα


Linear filterwith


Timemodulating

filter

High-passfilter


Normalizationby

standarddeviation

White-noise)(tw



Filteredwhite-noise


Acceleration time-series


Linear filterwith


Timemodulating

filter

High-passfilter


Normalizationby

standarddeviation

White-noise)(tw



Filteredwhite-noise


Temporal non-stationarity: Variation of intensity in time

Spectral non-stationarity: Variation of frequency content in time

t

fdwth

tt,qtx

)()](,[

)(1

)()( λα


Linear filterwith


Timemodulating

filter

High-passfilter


Normalizationby

standarddeviation

White-noise)(tw



Filteredwhite-noise


Source of stochasticity

t

fdwth

tt,qtx

)()](,[

)(1

)()( λα

t

fdwth

tt,qtx

)()](,[

)(1

)()( λα


Linear filterwith


Timemodulating

filter

High-passfilter


Normalizationby

standarddeviation

White-noise)(tw



Filteredwhite-noise


Impulse response function (IRF)corresponding to pseudo-acceleration response of a SDOF linear oscillator

t

fdwth

tt,qtx

)()](,[

)(1

)()( λα


Linear filterwith


Timemodulating

filter

High-passfilter


Normalizationby

standarddeviation

White-noise)(tw



Filteredwhite-noise


Duhamel’s integral(superposition of filter responses to a sequence of statistically independent

pulses with the time of application τ)

t

fdwth

tt,qtx

)()](,[

)(1

)()( λα


Linear filterwith


Timemodulating

filter

High-passfilter


Normalizationby

standarddeviation

White-noise)(tw



Filteredwhite-noise



Linear filterwith


Timemodulating

filter

High-passfilter


Normalizationby

standarddeviation

White-noise)(tw



Filteredwhite-noise


t

fdwth

tt,qtx

)()](,[

)(1

)()( λα

t

fdwth

tt,qtx

)()](,[

)(1

)()( λα


Linear filterwith


Timemodulating

filter

High-passfilter


Normalizationby

standarddeviation

White-noise)(tw



Filteredwhite-noise


0 time tn


Linear filterwith


Timemodulating

filter

High-passfilter


Normalizationby

standarddeviation

White-noise)(tw



Filteredwhite-noise


Non-zero residuals!

Over estimates response spectrum at long periods!

t

fdwth

tt,qtx

)()](,[

)(1

)()( λα


Linear filterwith


Timemodulating

filter

High-passfilter


Normalizationby

standarddeviation

White-noise)(tw



Filteredwhite-noise


Critically damped oscillator

)(2 2 txzzz cc

t

fdwth

tt,qtx

)()](,[

)(1

)()( λα


Linear filterwith


Timemodulating

filter

High-passfilter


Normalizationby

standarddeviation

White-noise)(tw



Filteredwhite-noise


t

fdwth

tt,qtx

)()](,[

)(1

)()( λα

t

fdwth

tt,qtx

)()](,[

)(1

)()( λα

Model Parameters:

t

fdwth

tt,qtx

)()](,[

)(1

)()( λα

Model Parameters:

midω

'ω

: Frequency at the middle of strong shaking

: Damping ratio

: Rate of change of frequency over time

Modulating function parameters: Filter parameters:

midt

955D

: Time at the middle of strong shaking, at 45% Ia

: Effective duration, between 5% to 95% Ia

nta tt

gI

02 d))( (accel

2

: Arias intensity

t

fdwth

tt,qtx

)()](,[

)(1

)()( λα

Model Parameters:

midω

'ω

: Frequency at the middle of strong shaking

: Damping ratio

: Rate of change of frequency over time

Modulating function parameters: Filter parameters:

midt

955D

: Time at the middle of strong shaking, at 45% Ia

: Effective duration, between 5% to 95% Ia

nta tt

gI

02 d))( (accel

2

: Arias intensity

Model parameters are identified for many recorded motions to develop predictive equations in terms of F, M, R, VS30

Match statistical

characteristicsRepresenting:• Intensity• Frequency• Bandwidth

Identify model parameters

ωmid, ω’ , ζIa, tmid, D5-95

Recorded0 40-0.25

0

0.15

Time, secAcc

eler

atio

n, g

Two Horizontal Components:

Component 1:

Component 2:

t

hdττwτ,th

tσt,qtx )()]([

)(1

)()( 1111 λα

t

hdττwτ,th

tσt,qtx )()]([

)(1

)()( 2222 λα


Component 1:

Component 2:

t

hdττwτ,th

tσt,qtx )()]([

)(1

)()( 1111 λα

t

hdττwτ,th

tσt,qtx )()]([

)(1

)()( 2222 λα

source of stochasticity

w1(τ) and w2(τ) are statistically independent if along the principal axes.


Component 1:

Component 2:

t

hdττwτ,th

tσt,qtx )()]([

)(1

)()( 1111 λα

t

hdττwτ,th

tσt,qtx )()]([

)(1

)()( 2222 λα



Principal Axes of Ground Motion:A set of orthogonal axes along which the components are uncorrelated.

Expected Epicenter

Site

Horizontal Plane

Major

Intermediate

Minor

Penzien and Watabe (1975)


Component 1:

Component 2:

t

hdττwτ,th

tσt,qtx )()]([

)(1

)()( 1111 λα

t

hdττwτ,th

tσt,qtx )()]([

)(1

)()( 2222 λα



Principal Axes of Ground Motion:A set of orthogonal axes along which the components are uncorrelated.

Rotate recorded motions in the database.

SiteHorizontal Plane

θa2,θ

a1,θ

a2

a1

ρ a1 , a2 ≠ 0ρ a1,θ , a2,θ = 0

Rotating Recorded Motions: Northridge earthquake recorded at Mt. Wilson Station

Time, s

Acc

eler

atio

n, g

As-RecordedComponent 1

0 5 10 15 20 25 30 35 40

As-RecordedComponent 2

Time, s

PrincipalComponent 1

PrincipalComponent 2

-0.2-0.1

00.10.20.3

0 5 10 15 20 25 30 35 40

Cor

rela

tion

Coe

ffici

ent

Bet

wee

n Th

e Tw

oC

ompo

nent

s

Rotation Angle, degrees

(0,-0.42)

(55,0)

0 10 20 30 40 50 60 70 80 90

-0.5

0

0.5

-0.3

-0.2-0.1

00.10.20.3

-0.3

-0.2-0.1

00.10.20.3

-0.3

-0.2-0.1

00.10.20.3

-0.3


Component 1:

Component 2:

t

hdττwτ,th

tσt,qtx )()]([

)(1

)()( 1111 λα

t

hdττwτ,th

tσt,qtx )()]([

)(1

)()( 2222 λα


Component 1:

Component 2:

t

hdττwτ,th

tσt,qtx )()]([

)(1

)()( 1111 λα

t

hdττwτ,th

tσt,qtx )()]([

)(1

)()( 2222 λα

Predictive equations:

εηβββββpFp m/s 750

Vln

km 25

Rln

7.0M

)F( )]([ s304

rup3210

-1


V

km 25

R

0.7M

)F( )]([ s304

rup3210

1

fmidmid- ζ,ω',ω,t,Dp 955 if

if ,intmaj, , aa IIp


Component 1:

Component 2:

t

hdττwτ,th

tσt,qtx )()]([

)(1

)()( 1111 λα

t

hdττwτ,th

tσt,qtx )()]([

)(1

)()( 2222 λα



Vln

km 25

Rln

7.0M

)F( )]([ s304

rup3210

-1


V

km 25

R

0.7M

)F( )]([ s304

rup3210

1



Model parameter p transformed to the

standard normal space


Component 1:

Component 2:

t

hdττwτ,th

tσt,qtx )()]([

)(1

)()( 1111 λα

t

hdττwτ,th

tσt,qtx )()]([

)(1

)()( 2222 λα



Vln

km 25

Rln

7.0M

)F( )]([ s304

rup3210

-1


V

km 25

R

0.7M

)F( )]([ s304

rup3210

1



Independent normally-distributed

errors

Predicted meanconditioned on

earthquake and site characteristics


Component 1:

Component 2:

t

hdττwτ,th

tσt,qtx )()]([

)(1

)()( 1111 λα

t

hdττwτ,th

tσt,qtx )()]([

)(1

)()( 2222 λα

Predictive equations.

High correlations expected between parameters of the two components.

1 −0.38 −0.04 −0.21 −0.25 −0.06 +0.92 −0.30 −0.03 −0.13 +0.09 +0.02

1 +0.68 −0.07 −0.21 −0.26 −0.31 +0.89 +0.68 −0.17 −0.11 −0.17

1 −0.24 −0.22 −0.26 +0.04 +0.65 +0.96 −0.30 −0.24 −0.21

1 −0.19 +0.28 −0.13 −0.15 −0.29 +0.94 −0.10 +0.29

1 −0.06 +0.19 −0.21 −0.22 −0.10 +0.52 −0.13

1 −0.01 −0.23 −0.29 +0.32 −0.02 +0.75

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

1 −0.05

1

tf,aI

Symmetric

tf,955D tf,midt tf,midω tfω' tfζ

tf,aI

tf,955D tf,midt

tf,midω

tfω'

tfζ

Correlation Matrix:

tf,aI tf,955D tf,midt tf,midω tfω' tfζ

tf,aI

tf,955D tf,midt

tf,midω

tfω'

tfζ

Major Component (larger Arias intensity) Intermediate Component (smaller Arias intensity)

Intermediate C

omponent

Major C

omponent


F = 1 (Reverse) , M = 7.35 , R =14 km , VS30 = 660 m/s Design scenario:

Example:


Example:

Ia

s.g

D5-95

s

tmid

s

ωmid/(2π)

Hz

ω’/(2π)

Hz/s ζf

Recorded 0.0165 16.7 18.3 3.9 –0.08 0.12

Simulated 0.0147 17.3 10.1 8.1 –0.12 0.420.0099 27.2 17.1 3.2 –0.03 0.20

Ia

s.g

D5-95

s

tmid

s

ωmid/(2π)

Hz

ω’/(2π)

Hz/s ζf

0.0135 17.0 17.8 4.1 –0.02 0.110.0047 21.0 10.7 8.6 –0.18 0.500.0034 24.8 16.9 3.7 –0.13 0.35

Major Component Intermediate Component


Example:

Ia

s.g

D5-95

s

tmid

s

ωmid/(2π)

Hz

ω’/(2π)

Hz/s ζf

Recorded 0.0165 16.7 18.3 3.9 –0.08 0.12

Simulated 0.0147 17.3 10.1 8.1 –0.12 0.420.0099 27.2 17.1 3.2 –0.03 0.20

Ia

s.g

D5-95

s

tmid

s

ωmid/(2π)

Hz

ω’/(2π)

Hz/s ζf

0.0135 17.0 17.8 4.1 –0.02 0.110.0047 21.0 10.7 8.6 –0.18 0.500.0034 24.8 16.9 3.7 –0.13 0.35


-0.1

0

0.1Simulated

-0.05

0

0.05Simulated

-0.1

0

0.1Recorded

Simulated

Simulated

Recorded

Acc

eler

atio

n, g

0 20 40 60 80 0 20 40 60 80Time, s Time, s


Example:

Ia

s.g

D5-95

s

tmid

s

ωmid/(2π)

Hz

ω’/(2π)

Hz/s ζf

Recorded 0.0165 16.7 18.3 3.9 –0.08 0.12

Simulated 0.0147 17.3 10.1 8.1 –0.12 0.420.0099 27.2 17.1 3.2 –0.03 0.20

Ia

s.g

D5-95

s

tmid

s

ωmid/(2π)

Hz

ω’/(2π)

Hz/s ζf

0.0135 17.0 17.8 4.1 –0.02 0.110.0047 21.0 10.7 8.6 –0.18 0.500.0034 24.8 16.9 3.7 –0.13 0.35


-0.01

0

0.01

-0.05

0

0.05

-0.05

0

0.05

Simulated

Simulated

Recorded

Simulated

Simulated

Recorded

Velo

city

, m/s

0 20 40 60 80 0 20 40 60 80Time, s Time, s


Example:

Ia

s.g

D5-95

s

tmid

s

ωmid/(2π)

Hz

ω’/(2π)

Hz/s ζf

Recorded 0.0165 16.7 18.3 3.9 –0.08 0.12

Simulated 0.0147 17.3 10.1 8.1 –0.12 0.420.0099 27.2 17.1 3.2 –0.03 0.20

Ia

s.g

D5-95

s

tmid

s

ωmid/(2π)

Hz

ω’/(2π)

Hz/s ζf

0.0135 17.0 17.8 4.1 –0.02 0.110.0047 21.0 10.7 8.6 –0.18 0.500.0034 24.8 16.9 3.7 –0.13 0.35


-0.02

0

0.02

-0.05

0

0.05

0 20 40 60 80-0.05

0

0.05

0 20 40 60 80

Simulated

Simulated

Recorded

Simulated

Simulated

Recorded

Dis

plac

emen

t, m

Time, s Time, s

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.

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.

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.

Related Publications:

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.

1.

2.

3.

44

Thank YouThis project was made possible with support from:

State of California through Transportation Systems Research Program of Pacific Earthquake Engineering Research Center (PEER TSRP).

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

Ground Motion Model: Advantages

k

iii utstqtx

1

)()()(ˆ

,...,nititwhere i 0 1 kk tttfor

)1,0(~ N iuwhere

Form of the model facilitates nonlinear random vibration analysis (e.g., by using TELM).

Extra

stochastic simulation of ground motion components for a specified design scenario

Documents

motivation ground motion

pulse ground motion

time tn15ground motion

time of application

crucial step

2d3d structural analysis

recorded motions

vs30 motivation