1. 4th International Conference on Earthquake Engineering Taipei, Taiwan October 12-13, 2006 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

2. Takada Lab. UT 2006/10/12 2 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 relationSite-specific attenuation relation

3. Takada Lab. UT 2006/10/12 3 Midorikawa & Ohtake 2003 Prediction : biased Uncertainty: average characteristic Problems in the past attenuation relation   P =0   P Statistical uncertainty: not considered

4. Takada Lab. UT 2006/10/12 4 Site-specific attenuation relation Model Mean value of ground motion y : Variance of ground motion y : Var(y) =  y 2 =   2 +   2 Specific Only applied to the specific site Local soil condition, topographic effects…(any local geologic conditions) : median of the past attenuation relation  (m,r,  ) : correction term =  0 +  M m+  R r  =(  0,  M,  R ), random variables  : random term, ~N(0,   2 )  y 2 =   2

5. Takada Lab. UT 2006/10/12 5 Bayesian updating framework Bayesian theorem Model A:  (m,r,  ) =  0 +  M m+  R r Model B:  (  0 ) =  0  = ( ,   2 ),  = (  0,  M,  R ) y : Observed data p(  ) : Prior distribution L(  |y) : Likelihood function f(  ) : Posterior distribution -- Knowledge about  before making observations -- Information contained in the set of observations -- Updated-state knowledge about  p()p() L(|y)L(|y) f()f()

6. Takada Lab. UT 2006/10/12 6 Bayesian estimation Prior distribution Noinformative, independent about  and   2 (Jeffrey’s rule, 1961) p( ,   2 )  1/   2 Likelihood function Marginal posterior distribution x = (1, m, r)

7. Takada Lab. UT 2006/10/12 7 Evaluation of site-specific attenuation relation Sites K-NET, KiK-NET, etc. Data 1997~2005, M w ≥ 5.0, R ≤ 250km, PGA ≥ 10gal Past attenuation relation (PGA) Si-Midorikawa (1999) After S. Midorikawa (2005)

8. Takada Lab. UT 2006/10/12 8 Results Site HKD100 & EKO.ERI

9. Takada Lab. UT 2006/10/12 9 Results Site HKD100 & EKO.ERI Site HKD100 47 earthquakes Site EKO.ERI 20 earthquakes ^

10. Takada Lab. UT 2006/10/12 10 Parameter estimation ParametersEstimatorStandard deviation 00 -1.5430.716 mm 0.2900.129 rr 0.0040.002 22 0.2620.061 Model A: HKD100

11. Takada Lab. UT 2006/10/12 11 Parameter estimation Model B Site EKO.ERI (n=20)Site HKD100 (n=47) ParametersEstimator Standard deviation Estimator Standard deviation 00 0.4970.2860.5970.087 22 1.4590.5960.3380.077 PDF

12. Takada Lab. UT 2006/10/12 12 Prediction of ground motion Predictive PDF of ground motion y Expectation over  = ( ,   2 )

13. Takada Lab. UT 2006/10/12 13 Prediction of ground motion Site EKO.ERI, Model BSite HKD100, Model A

14. Takada Lab. UT 2006/10/12 14 Discussions Site-specific attenuation relation Past attenuation relation MethodBayesian approachClassical regression EstimatorPDF of parameterPoint estimator Statistical uncertainty PossibleImpossible UncertaintySpecific siteCommon

15. Takada Lab. UT 2006/10/12 15 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.

16. Takada Lab. UT 2006/10/12 16 Thank you for your attention!

17. Takada Lab. UT 2006/10/12 17 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 What is  P 2 of the past attenuation relation ? Answers to ---- represent inherent variability and model uncertainty. ---- represent the average character of uncertainty for all sites.

18. Takada Lab. UT 2006/10/12 18 Ground Motion Modeling the ground motion Effects of ground Motion ： ---- 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  m : model uncertainty, when replaces f. Buildings Engineering bedrock Seismic bedrock Source Path Site  s : statistical uncertainty, when  is estimated with limited number of data. Parameter estimate

19. Takada Lab. UT 2006/10/12 19 Past attenuation relation Model of ground motion Mathematical modeling e.g. Si-Midorikawa(1999) Mathematical modeling y : ground motion in natural logarithm x: variables, such as M w, R, D, …, etc. : regression coefficients  P : random term ~N(0,   P 2 )   P : inherent variability  a, aleatory uncertainty model uncertainty  m, epistemic uncertainty  a : ~ N(0,   a 2 )   a : inherent variability

20. Takada Lab. UT 2006/10/12 20 Uncertainty considering statistical uncertainty 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 *. Contour of  y

21. Takada Lab. UT 2006/10/12 21

22. Takada Lab. UT 2006/10/12 22 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

23. Takada Lab. UT 2006/10/12 23 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 : x1x1 x2x2 x3x3 y = ? Unobserved Site Observed Site Ref.: Wang, M. and Takada, T. (2005): Macrospatial correlation model of seismic ground motion, Earthquake Spectra, Vol. 21, No. 4, 1137-1156.

24. Takada Lab. UT 2006/10/12 24 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.