Macro-Spatial Correlation Model of
Seismic Ground Motion
Min WANG (Presenter)
Tsuyoshi TAKADA
The University of Tokyo
0. Contents
‹
Introduction
‹
Stochastic Model of Seismic Ground Motion Intensity
‹
Data Analysis of Ground Motion
‹
Macro-Spatial Correlation Model
‹
Discussions
‹
Conclusions
2
1. Introduction: Deterministic and Probabilistic
Evaluation of Seismic Ground Motion
Ground Motion Evaluation:
Ai = A(M, Xi)･εi
‹
For deterministic evaluation:
Source A
(scenario-based evaluation)
Not taking account of uncertainty,
that is, εi is deterministic.
‹
For probabilistic evaluation:
(probability-based evaluation)
Ai
Source B
Aj
Not taking account of the spatial
correlation of seismic ground
motion, that is εi and εj is
dependent.
3
1. Introduction: Seismic Design of Infrastructure and
Portfolio Analysis
‹
When seismic design is applied to the
spatially-spread infrastructures, the
reliability index of the system which is
associated with the joint exceedance
probability with correlation should be
reasonably set, such as the case of the
traffic line.
Source
Bridge 3
Bridge 1
Bridge 2
Traffic Line
‹
In the portfolio analysis of the building
assets, the simultaneous damages of
buildings located in different sites are
concerned.
Building 1
Building 3
Source
Building 2
4
1. Introduction: Past Studies
‹
Either perfect correlation or independency of the uncertainty
of GM is assumed in the past researches on the risk
management.
‹
So far, only Takada et al. study (Takada and Shimomura,
2003) on macro-spatial correlation based on the Chi-Chi
earthquake is available.
‹
Contributed to the K-NET and KiK-NET developed recently
in Japan, the macro-spatial correlation analysis for ground
motion intensity can be done.
5
2. Stochastic Model of Seismic GM Ai = Ti･εi
2.1 Mean Attenuation Characteristics of GM
‹
‹
1. the Annaka attenuation relation (1997)
0.653 M j
log10 T = a1M j + a2 H − a3 log10 ( D + 0.334e
) − a4
(1)
2. the Midorikawa-Ohtake attenuation relation (2002)
log10 T = c − log( D + m1e 0.5 M w ) − m2 D
( H ≤ 30 km )
(2a)
) − m2 D ( H > 30km)
(2b)
log10 T = c + 0.6log(1.7 H + m1e0.5 M w )
− 1.6log( D + m1e
Where:
‹
0.5 M w
c = m3 M w + m4 H + ∑ di si − m5
3. Amplification factor (ARV)
(3)
Variable of
fault type
log10 ARV = 1.83 − 0.66log10 (V 30)
(4)
•The uncertainty associated in Eq. (4) is suggested as 0.37.
6
2.2 Uncertainty of Ground Motion Intensity
The uncertainty of seismic ground motion intensity can be
decomposed into inter-event and intra-event uncertainties.
Inter-event uncertainty
Perfect correlation
Observed
A = T ( M , x) × ε × ε
S
INTRA
A
‹
εS2
εS1
(x)
ε2(xk)
ε1(xj)
Mean Trend
Intra-event uncertainty
Auto-covariance function CLL
ε2(xl)
ε1(xi)
Uncertainties of the Attenuation Relations in Natural Logarithm
Uncertainty
Annaka
Midorikawa-Ohtake
PGA
PGV
PGA
PGV
Inter-event εS
0.37
0.37
0.37
0.37
Intra-event εINTRA
0.51
‹
Mean Trend
T(M,X)
X
0.51
0.62
0.55
This study concerns about the intra-event uncertainty,
and proposed is its spatial correlation model.
7
2.3 Auto-correlation Function
L(x) ≡ ln( A(x) / T (x)) = ln ε (x)
‹
1.5
Logarithmic deviation:
Auto-covariance function:
C LL ( h ) = E [ L ( x1 ) − µ L )( L ( x 2 − µ L ) ]
where h = | x1 - x 2 | , µ L = E [ L(x)]
L(x)
‹
1
logarithmic deviatoic component, L(x)
moving average of L(x), (ML)
ML±σ
0.5
0
-0.5
-1
PGA: Mean Component, Midorikawa-Ohtake Attenuation Relation(2002)
Assumption:
50
100
150
200
Logarithmic deviation L(x) constitutes a homogeneous
2-D stochastic field so that the auto-covariance function
CLL(h) is only a function of separation distance h.
-1.5
250
300
X
Ž For simplicity, the moving average curve is used to examine
homogeneity of the logarithmic deviation, and the distancedependency of the logarithmic deviation can be easily illustrated.
8
3. Data Analysis of GM
3.1 Database of the Seismic Ground Motion
Profile of Earthquakes
No.
Earthquakes
Date
Mj
Mw
1
Tottori-ken
Seibu
2000/10/06
7.3
6.8
2
Geiyo
2001/03/24
6.7
6.7
3
Miyagi-kenoki
2003/05/26
7.0
7.0
4
Miyagi-ken
Hokubu
2003/07/26
6.2
6.2
5
Tokachi-oki
2003/09/26
8.0
8.0
6
Mid Niigataprefecture
2004/10/23
6.8
6.5
9
3.2 Mean Attenuation characteristic of GM
3
10
2
10
1
¾
PGA
10
3
10
2
10
1
PGA (gal)
PGA (gal)
10
the Tottori-ken Seibu Earthquake
0
10 0
10
the Mid Niigata-prefecture Earthquake
PGA observed
the Annaka Relation
the Midorikawa-Ohtake Relation
0
1
10 0
10
2
10
10
Closest Distance to the Fault Plane (km)
Earthquake
Annaka
PGA observed
the Annaka Relation
the Midorikawa-Ohtake Relation
1
2
10
10
Closest Distance to the Fault Plane (km)
Midorikawa-Ohtake
μL
σL
μL
σL
Tottori-ken Seibu
0.08
0.64
0.07
0.59
Geiyo
0.59
0.75
-0.13
0.60
Miyagi-ken-oki
0.83
0.90
0.03
0.82
Miyagi-ken Hokubu
0.68
0.84
0.15
0.83
Tokachi-oki
-0.16
0.85
-0.45
0.74
Mid Niigata-prefecture
0.03
0.76
-0.19
0.74
Inter-event uncertainty
0.41
0.22
μL: mean of the L(x)
σL: standard deviation of L(x)
Statistical values of
logarithmic deviation
L(x) of PGA
10
10
2
10
1
10
0
¾
PGV
PGV on Stiff Ground (kine)
PGV on Stiff Ground (kine)
3.2 Mean Attenuation characteristic of GM
the Miyagi-ken-oki Earthquake
10
-1
PGV observed
the Annaka Relationship
the Midorikawa-Ohtake Relationship
10
1
2
10
1
10
0
the Tokachi-oki Earthquake
10
2
10
Closest Distance to the Fault Plane (km)
Earthquake
10
Annaka
-1
PGV observed
the Annaka Relationship
the Midorikawa-Ohtake Relationship
10
1
2
10
Closest Distance to the Fault Plane (km)
Midorikawa-Ohtake
μL
σL
μL
σL
Tottori-ken Seibu
0.46
0.51
0.18
0.53
Geiyo
-0.11
0.56
-0.07
0.49
Miyagi-ken-oki
0.32
0.58
0.21
0.51
Miyagi-ken Hokubu
0.14
0.58
0.06
0.57
Tokachi-oki
-0.48
0.64
-0.14
0.57
Mid Niigata-prefecture
-0.48
0.58
-0.12
0.58
Inter-event uncertainty
0.37
0.34
Statistical values of
logarithmic deviation
L(x) of PGV
11
4. Proposal of Macro-Spatial Correlation Model
‹
Auto-covariance function:
900
the Mid Niigata-prefecture Earthquake
800
1 N (h)
CLL ( h ) =
∑ ( L(xai ) − µL )( L(xbi ) − µL )
N (h) i =1
600
N all
N (h )
1
µL =
N all
700
∑ L( x )
i
i =1
500
400
300
200
N(h)： the number of pairs of sites (xa, xb)
that meet the condition：
h – ∆h/2 < |xa – xb| ≤ h + ∆h/2;
‹
0
0
20
40
60
80
100
h (km)
Normalized auto-covariance function
RLL ( h ) =
‹
100
CLL (h)
σ L2
Macro-spatial correlation model:
(
RLL (h) = exp − h
b
)
b： correlation length,
RLL(h=b) = 1/e
RLL(0) = 1, and RLL (∞) = 0.
12
----Results of Correlation Length for PGA and PGV
Correlation Lengths b (km)
Earthquake
Annake
Midorikawa-Ohtake
PGA
PGV
PGA
PGV
Tottori-ken Seibu
23.1
21.0
12.9
28.6
Geiyo
41.7
47.8
11.7
35.8
Miyagi-ken-oki
45.2
39.7
38.4
21.6
Miyagi-ken Hokubu
31.0
27.7
31.0
24.0
Tokachi-oki
59.1
44.5
45.2
22.4
Mid Niigata-prefecture
43.5
21.6
39.7
22.0
1
1
The Miyagi-ken Hokubu Earthquake, PGA
Midorikawa-Ohtake Attenuation Relation(2002)
0.9
0.8
0.7
y = exp(-h/31.0)
0.6
RLL(h)
RLL(h)
0.7
0.5
0.4
0.5
0.4
0.3
0.2
0.2
0.1
0.1
20
40
60
h (km)
80
100
y = exp(-h/21.6)
0.6
0.3
0
0
PGV: 21~36 km
The Miyagi-ken-oki Earthquake, PGV
Midorikawa-Ohtake Attenuation Relation(2002)
0.9
0.8
PGA: 11~45 km
0
0
20
40
60
h (km)
80
100
13
5. Discussions
‹
‹
The model is fully dependent on the homogeneity of the logarithmic
deviation.
The factors, such as source characteristic, wave propagation and site
effect affecting the uncertainty of attenuation relation, may have
effects on this model.
Ž The effect of source characteristic
1
1
The Mid Niigata-prefecture Earthquake, PGA
Annaka Attenuation Relation(1997)
0.9
0.8
0.8
0.7
y = exp(-h/43.5)
0.6
RLL(h)
RLL(h )
0.7
0.5
0.4
0.5
0.4
0.3
0.2
0.2
0.1
0.1
20
40
60
h (km)
ρL=-0.42
80
100
y = exp(-h/39.7)
0.6
0.3
0
0
The Mid Niigata-prefecture Earthquake, PGA
Midorikawa-Ohtake Attenuation Relation(2002)
0.9
0
0
20
40
60
80
100
h (km)
ρL=-0. 34
14
5. Discussions
Ž The effect of wave propagation
Ž Site effect, Inter-event uncertainty
1
1
The Tokachi-oki Earthquake, PGV
Annaka Attenuation Relation(1997)
0.9
0.8
0.8
0.7
y = exp(-h/44.5)
0.6
RLL(h)
RLL(h )
0.7
0.5
0.4
0.5
0.4
0.3
0.2
0.2
0.1
0.1
20
40
60
h (km)
ρL=-0.50
80
100
y = exp(-h/22.4)
0.6
0.3
0
0
The Tokachi-oki Earthquake, PGV
Midorikawa-Ohtake Attenuation Relation(2002)
0.9
0
0
20
40
60
80
100
h (km)
ρL=-0.01
15
6. Conclusions
‹
In this study, focusing on the residual value between the
observed value and the predicted value by empirical mean
attenuation relations, the macro-spatial correlation of the
logarithmic deviation is proposed.
1. The assumption that the logarithmic deviation L(x) constitutes the
homogeneous 2-D stochastic field can be approximately satisfied.
2. The macro-spatial correlation length for PGA falls in 11-45 km. For
PGV, it falls in 21-36km.
3. The effects on the macro-spatial correlation model are discussed.
‹
The proposed correlation model can be effectively utilized
such as in the real-time prediction of ground motion
intensities, evaluation of joint exceedance probability and
evaluation of seismic risk for portfolio, etc.
16
END
THANKS FOR YOUR ATTENTIONS