1. Macro-spatial Correlation of Seismic Ground Motion and Network Reliability Tsuyoshi Takada and Min Wang The University of Tokyo, Graduate School of Engineering Japan IFED (International Forum on Engineering Decision Making), Lake Louise, Canada, April 26-29, 2006

2. 2 1.Introduction 2.Macro-spatial correlation of seismic ground motions 3.Application to system reliability problem 4.Application to first excursion problem 5.Conclusions Content of Presentation

3. 3 Introduction & Bachground For deterministic evaluation: (scenario-based evaluation) Not taking account of uncertainty, that is, ε i is deterministic. For probabilistic evaluation: (probability-based evaluation) Not taking account of the spatial correlation of seismic ground motion, that is the correlation betweenε i and ε j. Source A AjAj AiAi A i = A(M, X i ) ･ ε i Source B Ground Motion Evaluation:

4. 4 Accuracy of Past Attenuation Relation PGA in Kobe Earthquake and Predicted Results (after Fukushima)

5. 5 Infrastructures and Portfolio of Buildings For the seismic design of spatially-spread infrastructures, the reliability of the systems which is associated with the partially correlated failure events should be reasonably taken into account. In the portfolio analysis of building assets, simultaneous damages of buildings located in different sites are of major concern. Bridge 2 Bridge 1 Traffic Line Building 1 Building 2 Building 3 Source

6. 6 Past Research Either perfect correlation or no correlation 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, Wang and Takada, 2005) on macro-spatial correlation based on the past earthquakes is available. Using the data of K-NET and KiK-NET (around 2000 stations throughout Japan) developed recently in Japan, the macro- spatial correlation analysis for ground motion intensity can be done.

7. 7 Stochastic Prediction of Seismic GM 1. the Annaka attenuation relation (1997) 2. the Midorikawa-Ohtake attenuation relation (2002) 3. Amplification factor (AVR) Where: The uncertainty is suggested as 0.37. Variable of fault type

8. 8 Uncertainty of Ground Motion Intensity The uncertainty of seismic ground motion intensity can be decomposed into inter-event and intra-event uncertainty. Mean Trend Inter-event uncertainty Perfect correlation Intra-event uncertainty Auto-covariance function C LL Observed This study concerns about the intra-event uncertainty, and proposed is its spatial correlation model. UncertaintyAnnakaMidorikawa- Ohtake PGAPGVPGAPGV Inter-event0.37 Intra-event0.51 0.620.55 Uncertainties of the Attenuation Relations in Natural Logarithm ε S2 ε2(xk)ε2(xk) ε2(xl)ε2(xl) Mean Trend T(M,X) X A ε1(xj)ε1(xj) ε1(xi)ε1(xi) ε S1

9. 9 Modeling of Macro-Spatial Correlation of GM - Seismic Ground motion intensity A(x) at location x is assumed to the product of mean property T(x) and random component R(x) A(x) = T(x) R(x) T(x): mean property predicted from mean empirical attenuation law R(x): homogeneous 2D random field characterized by mean and auto- correlation function This assumption implies that spatial correlation of R(x) can be defined in terms of relative distance between two different locations L(x) = log( A(x) / T(x) ) Note that L(x) implies the logarithmic deviation of observed record from the mean attenuation law

10. 10 Auto-correlation Function & Homogeneity Logarithmic deviation: Auto-covariance function: Assumption: Logarithmic deviation L(x) constitutes a homogeneous 2-D stochastic field so that the auto-covariance function C LL (h) is only a function of separation distance h. X L(x) –For simplicity, the moving average curve and the correlation coefficient ρ are used to examine homogeneity of the logarithmic deviation, and the dependency between the logarithmic deviation and distance.

11. 11 Proposal of Macro-Spatial Correlation Model Auto-covariance function: Normalized auto-covariance function Macro-spatial correlation function: b ： correlation length, R LL (h=b) = 1/e R LL (0) = 1, and R LL (∞) = 0. N(h) ： the number of pairs of sites (x a, x b ) that meet the condition ： h – ∆h/2 < |x a – x b | ≤ h + ∆h/2 ;

12. 12 Mean Attenuation characteristic of GM (PGA) Statistical values of logarithmic deviation L(x) of PGA EarthquakeAnnakaMidorikawa-Ohtake μLμL σLσL μLμL σLσL Tottori-ken Seibu0.080.640.070.59 Geiyo0.590.75-0.130.60 Miyagi-ken-oki0.830.900.030.82 Miyagi-ken Hokubu0.680.840.150.83 Tokachi-oki-0.160.85-0.450.74 Mid Niigata-prefecture0.030.76-0.190.74 Inter-event uncertainty0.410.22

13. 13 Database of Seismic Ground Motions No.EarthquakesDateMjMj MwMw 1Tottori-ken Seibu 2000/10/067.36.8 2Geiyo2001/03/246.7 3Miyagi-ken- oki 2003/05/267.0 4Miyagi-ken Hokubu 2003/07/266.2 5Tokachi-oki2003/09/268.0 6Mid Niigata- prefecture 2004/10/236.86.5 Profile of Earthquakes

14. 14 Mean Attenuation characteristic of GM (PGV) EarthquakeAnnakaMidorikawa-Ohtake μLμL σLσL μLμL σLσL Tottori-ken Seibu0.460.510.180.53 Geiyo-0.110.56-0.070.49 Miyagi-ken-oki0.320.580.210.51 Miyagi-ken Hokubu0.140.580.060.57 Tokachi-oki-0.480.64-0.140.57 Mid Niigata-prefecture-0.480.58-0.120.58 Inter-event uncertainty0.370.34 Statistical values of logarithmic deviation L(x) of PGV

15. 15 Macro-spatial Correlation model Correlation Lengths b (km) EarthquakeAnnakeMidorikawa-Ohtake PGAPGVPGAPGV Tottori-ken Seibu23.121.012.928.6 Geiyo41.747.811.735.8 Miyagi-ken-oki45.239.738.421.6 Miyagi-ken Hokubu31.027.731.024.0 Tokachi-oki59.144.545.222.4 Mid Niigata-prefecture43.521.639.722.0 PGA: 11~45 km PGV: 21~36 km

16. 16 Probability of events E i Application to system reliability issues EiEi EjEj The system reliability can be described in terms of P(Ei or Ej) for a series system. Bi-modal bounding technique proposed by Ditlevsen can be used.

17. 17 Probability of events E i Closed form solution of joint PDF (1) S i is assumed a normal r.v., then Probability of intersection of events E i and E j u1u1 h k u2u2 0 ↓ Ｐ (Ei,Ej)

18. 18 Closed form solution of joint PDF (2) Probability of intersection of events E 1 and E 2 (Matsumoto and Takada, 2004) u1u1 h k u2u2 0 ↓ Ｐ (Ei,Ej)

19. 19 Accuracy of closed form solution (1) Comparison of result from Ditlevsen’s bi-modal bounds Effect of beta (  1=1.5,  =0.75) Effect of  (h=k=2.0) 22 Correlation coefficient  P(E1 and E2) 11 22 11 22

20. 20 Accuracy of closed form solution (2) Combination with Ditlevsen’s bi-modal bounds Correlation coefficient  P(E1 or E2 or E3) Effect of  (  1=1.0,  2=1.5,  3=2.0) 11 22 33

21. 21 Failure event of a line system Application of first excursion problems Utilization of solution of first excursion probability to non- stationary random process since the random process is defined by the mean attenuation relation and its auto- correlation function s source A(x) = T(x) R(x)

22. 22 Conclusions 1) Macro-spatial correlation model is proposed using data of recent Japanese earthquake data, 2) Macro-spatial correlation can be an exponential decaying function with a correlation length (15-40km), 3) This model can be used for reliability computaion for infrastructures spread widely under earthquake condition, 4) System Reliability or first excursion theory can be utilized.