4th International Conference on Earthquake Engineering Taipei, Taiwan October 12-13, 2006

Site-specific Prediction of Seismic Ground Motion with Bayesian Updating Framework

Site-specific Prediction of Seismic Ground Motion with Bayesian Updating Framework

Min Wang, and Tsuyoshi Takada

The University of Tokyo

Min Wang, and Tsuyoshi Takada

The University of Tokyo

2006/10/12 2

Takada Lab. UT

Introduction

Prediction of ground motion Important step of PSHA (Probabilistic Seismic Hazard Analysis) By the (past empirical) attenuation relation

Multi-event&

Multi-site

Status quo

Hazard / Risk@

specific-site

Needs

Past attenuation relation Site-specific attenuation relation

2006/10/12 3

Takada Lab. UT

Midorikawa & Ohtake 2003

Prediction : biased Uncertainty: average characteristic

Problems in the past attenuation relation

TKCH07 IBR005 TKY011 TCG009 TKY010-3

-2

-1

0

1

2

3

4 Number of data n : 42+54+61+63+73 = 293

Station Code

Log

arit

hmic

Dev

iati

on

Number of data n :Number of data n :Number of data n :Number of data n :

P=0P

Statistical uncertainty: not considered

TKCH07 IBR005 TKY011 TCG009 TKY010-3

-2

-1

0

1

2

3

4 Number of data n :

42

Station Code

Log

arit

hmic

Dev

iati

on

Number of data n :

54

Number of data n :

61

Number of data n :

63

Number of data n :

73

ˆˆ( , ) Py g x β

2006/10/12 4

Takada Lab. UT

Site-specific attenuation relation

Model

Mean value of ground motion y :

Variance of ground motion y : Var(y) = y2 =

2 + 2

Specific Only applied to the specific site

Local soil condition, topographic effects…(any local geologic conditions)

( , , )ˆˆ ( , )y g m r θx β

ˆ( )E y g

: median of the past attenuation relation(m,r,) : correction term = 0+ Mm+ Rr =(0,M, R ), random variables: random term, ~N(0,

2)

g

y2 =

2

2006/10/12 5

Takada Lab. UT

Bayesian updating framework

Bayesian theorem

ˆˆ( , ,..., ) ( , , ) ,y g m r m r β θ

Model A: (m,r,) = 0+ Mm+ Rr

Model B: (0) = 0

ΘyΘyΘ pcLf ||

= (, 2), = (0, M, R)

y : Observed data

p() : Prior distribution

L(|y) : Likelihood function

f() : Posterior distribution

),0(~ 2 N

-- Knowledge about before making observations

-- Information contained in the set of observations

-- Updated-state knowledge about

p()L(|y)

f()

2006/10/12 6

Takada Lab. UT

Bayesian estimation

Prior distribution Noinformative, independent about and

2 (Jeffrey’s rule, 1961)

p(, 2) 1/

2

Likelihood function

Marginal posterior distribution

2 2 / 22

( ) '( )( , ) ( ) exp ,

2nL

γ xθ γ xθθ

ˆ γ y g

3( , , )nx 1 m r

2

2

ˆ ˆ( ) ' ' ( )( ) 1

n

fvs

θ θ x x θ θθ

2

( / 2 1)2 22

( ) exp2

v vsf

1ˆ ( ' ) 'θ x x x y2 ˆ ˆ(1/ )( ) '( )s v γ γ γ γ

ˆˆ γ xθ

3v n

x = (1, m, r)

2ˆ ˆ, θ

2006/10/12 7

Takada Lab. UT

Evaluation of site-specific attenuation relation

Sites K-NET, KiK-NET, etc.

Data 1997~2005, Mw ≥ 5.0, R ≤ 250km,

PGA ≥ 10gal

Past attenuation relation (PGA) Si-Midorikawa (1999)

After S. Midorikawa (2005)

2006/10/12 8

Takada Lab. UT

Results

Site HKD100 & EKO.ERI

2006/10/12 9

Takada Lab. UT

Results

Site HKD100 & EKO.ERI

Site HKD10047 earthquakes

Site EKO.ERI20 earthquakes

4.5 5 5.5 6 6.5 7 7.5 8 8.50

50

100

150

200

250

Moment Magnitude m

Hyp

ocen

ter D

ista

nce

r (k

m)

Site Code : EKO.ERI Number of Observations : 20

4.5 5 5.5 6 6.5 7 7.5 8 8.50

50

100

150

200

250

Moment Magnitude m

Hyp

ocen

ter D

ista

nce

r (k

m)

Site Code : HKD100Number of Observations : 47

EKO.ERI HKD100-2

-1

0

1

2

3

4

Site Code

Loga

rithm

ic D

evia

tion

Number of data n =

20 47

ˆˆ ˆ( , )y g x β

^

2006/10/12 10

Takada Lab. UT

Parameter estimation

- 6 - 4 - 2 0 2 40

0.2

0.4

0.6

0.8

0

f ( 0)

- 0.5 0 0.5 10

1

2

3

4

m

f ( m

)

- 0.01 0 0.01 0.020

50

100

150

200

250

r

f ( r )

0 0.2 0.4 0.60

2

4

6

8

2

f ( 2 )

Parameters Estimator Standard deviation

0 -1.543 0.716

m 0.290 0.129

r 0.004 0.002

2 0.262 0.061

• Model A: HKD100

2006/10/12 11

Takada Lab. UT

Parameter estimation

• Model B

Site EKO.ERI (n=20) Site HKD100 (n=47)

Parameters EstimatorStandarddeviation

EstimatorStandarddeviation

0 0.497 0.286 0.597 0.087

2 1.459 0.596 0.338 0.077

PDF

- 2 0 20

0.5

1

1.5

0

f ( 0)

0 2 40

0.2

0.4

0.6

0.8

1

2

f ( 2 )

0 0.5 1 1.50

1

2

3

4

5

0

f ( 0)

0 0.5 10

2

4

6

2

f ( 2 )

2006/10/12 12

Takada Lab. UT

Prediction of ground motion

Predictive PDF of ground motion y

* * * *( | ) ( | , ) ( )f y L y f dΘx x Θ Θ Θ

Expectation over = (, 2)

100

101

10210

0

101

102

103

104

Mw

= 8.0

R (km)

PG

A (

gal)

this studySi-Midorikawa

Mw

= 5.0

6.0

7.0

Focal depth : 50 km Interplate earthquake

2ˆ ˆ( , )θ

* * *ˆ ˆ( , ) ( , )y g x β x θ

2006/10/12 13

Takada Lab. UT

Prediction of ground motion

Site EKO.ERI, Model B

0 50 100 150 200 250 3000

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

PGA (gal)

f PG

A(p

ga)

Site Code : EKO.ERI (35.72N, 139.76E)CENTRAL CHIBA PREF 2005/ 07/ 23

Observed: 113.10 (gal)Si- Midorikawa: 55.93 (gal), = 0.70This study: 90.98 (gal),

all = 1.27

This study

Median

Si- Midorikawa

Observation Mw = 6.0

R = 82 kmDepth = 73 kmInterplate

0 100 200 300 400 500 600 700 800 9000

1

2

3

4

5

6

7

8x 10

- 3

PGA (gal)f P

GA(p

ga)

Site Code : HKD100 (42.28N, 143.32E)SE OFF ERIMOMISAKI 2003/ 09/ 26

Observed: 425.91 (gal)Si- Midorikawa: 139.49 (gal), = 0.70This study: 305.24 (gal),

all = 0.58

This Study

Si- Midorikawa

Observation

Median

Mw = 7.3

R = 74 kmDepth = 21 kmInterplate

Site HKD100, Model A

2006/10/12 14

Takada Lab. UT

Discussions

Site-specific attenuation relation

Past attenuation relation

Method Bayesian approach Classical regression

Estimator PDF of parameter Point estimator

Statistical uncertainty

Possible Impossible

Uncertainty Specific site Common

2006/10/12 15

Takada Lab. UT

Conclusions

The site-specific attenuation are developed based on the past attenuation relation and observations with Bayesian framework.

It shows more flexibility that the correction term can expressed in a linear model and its reduced models according to the observations.

Although the statistical uncertainty will decrease, the inherent variability and model uncertainty remain unchanged no matter how much data increase.

The site-specific attenuation relation is suggested to be incorporated into PSHA because its median component and uncertainty component can represent those at the specific site.

2006/10/12 16

Takada Lab. UT

Thank you for your attention!

2006/10/12 17

Takada Lab. UT

Uncertainty of Ground Motion

Inherent Variability: temporal variability or spatial variability or both.

Model Uncertainty: missing variables and simplifying the function form in the prediction model (attenuation relation).

Statistical Uncertainty: limited data.

---- aleatory uncertainty

---- epistemic uncertainty

---- epistemic uncertainty

What is P2 of the past attenuation relation ? ˆ , ˆ

Py f x β

Answers to

---- represent inherent variability and model uncertainty.

---- represent the average character of uncertainty for all sites.

2006/10/12 18

Takada Lab. UT

Ground Motion

Modeling the ground motion

Effects of ground Motion： ---- Source, path, site

( , , ) aGM f Source Path Site

a: inherent variability, aleatory uncertainty

when f represents the real world of ground motion

m , s: epistemic uncertainty

Mathematical modeling

x: variables, : parameters ,ˆamGM f x θ

m: model uncertainty, when replaces f .f

Buildings

Engineering bedrock

Seismic bedrock

Source

Path

Site

s: statistical uncertainty, when is estimated with limited number of data.

Parameter estimate

ˆ , ˆsm aGM f θx

2006/10/12 19

Takada Lab. UT

Past attenuation relation

Model of ground motion

Mathematical modeling

Py

0.5100.5 0.0043 0.61 log ( 0.0055 10 ) 0.003wM

w i i Py M D d E R R

e.g. Si-Midorikawa(1999)

Mathematical modeling

ˆˆ( , ) Py g x β y: ground motion in natural logarithmx: variables, such as Mw, R, D, …, etc. : regression coefficientsP: random term ~N(0, P

2)β

P :

• inherent variability a , aleatory uncertainty

• model uncertainty m , epistemic uncertainty

( , , ) aGM f Source Path Site a: ~ N(0, a2)

a: inherent variability

2006/10/12 20

Takada Lab. UT

Uncertainty considering statistical uncertainty

* * *

* 1 *

ˆ ˆˆ( , )( ) ~

ˆ 1 ( ) '( ' )v

y gt

x β x θy

x x x x

x : observations of magnitude m and distance r.x*: new value of magnitude m and distance r.y*: new prediction of ground motion given x*.

5 5.5 6 6.5 7 7.5 8

50

100

150

200

250

Mw

R (

km)

0.54

0.56

0.58

0.6

0.62

0.64

Contour of y

2006/10/12 21

Takada Lab. UT

5 5.5 6 6.5 7 7.5 80.45

0.5

0.55

0.6

0.65

0.7

Mw

y

R = 10 (km)R = 50 (km)R = 100 (km)Observations

0 50 100 150 200 2500.45

0.5

0.55

0.6

0.65

0.7

R (km)

y

Mw = 6.0

Mw = 7.0

Mw = 8.0

Observations

512.0ˆ 512.0ˆ

2006/10/12 22

Takada Lab. UT

Soil-specific attenuation relation

Attenuation relation on a baseline condition

Amplification factor

Engineering bedrock

Surface

Amplification factor e.g. f(Vs)

Attenuation relation on Engineering bedrock

2006/10/12 23

Takada Lab. UT

Prediction for unobserved site --Macro-spatial Correlation Model

Conditional PDF of GMs at Unobserved Site:

Assuming GM is a log-normal field,

Conditional PDF can be given:

x1

x2

x3

y = ?

Unobserved Site

Observed Site

),,(

),,,()(

1

1,|

n

nYY xxf

xxyfyf

X

XxX

2|

| 2||

ˆ( )1( ) exp

22Y x

Y x

Y xY x

y yf y

XX

XX

2| 1 1

| 2

1 2|

( )2

/ 1 /

Y t tY Y c c Y

tY Y c Y

C C

C

X xX x X X

X x X X

x x

( ) expXYhh b 0 20 40 60 80 100

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y = exp(- h/ 22.4)

h (km)

RLL

(h)

( ) exp 22.4XYhh

Ref.: Wang, M. and Takada, T. (2005): Macrospatial correlation model of seismic ground motion, Earthquake Spectra, Vol. 21, No. 4, 1137-1156.

2006/10/12 24

Takada Lab. UT

Conclusions

Give a new thinking on the prediction of ground motion. Change from common to specific Mean component is unbiased. Uncertainty represents that of specific site.

Reclassify the uncertainty of the prediction of ground motion. Inherent variability, model uncertainty, statistical uncertainty. Can deal with uncertainty due to data. Answer to how much degree the future earthquake is like the past.