Site-specific Pre diction of Seismic Ground Motion with Bayesian Updating Framework

1. Site-specific Prediction of Seismic Ground Motion with Bayesian Updating Framework Min Wang, and Tsuyoshi Takada The University of Tokyo

2. Needs Status quo Hazard / Risk @ specific-site Multi-event & Multi-site Introduction • Prediction of ground motion • Important step of PSHA (Probabilistic Seismic Hazard Analysis) • By the (past empirical) attenuation relation Past attenuation relation Site-specific attenuation relation

3. P=0P Midorikawa & Ohtake 2003 Problems in the past attenuation relation • Prediction : biased • Uncertainty: average characteristic • Statistical uncertainty: not considered

4. : median of the past attenuation relation(m,r,) : correction term = 0+ Mm+ Rr =(0,M, R ), random variables : random term, ~N(0, 2) y2 = 2 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)

5. L(|y) p() f() Bayesian updating framework • Bayesian theorem Model A: (m,r,) = 0+ Mm+ Rr 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 

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

7. 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)

8. Results • Site HKD100 & EKO.ERI

9. ^ Site HKD10047 earthquakes Site EKO.ERI20 earthquakes Results • Site HKD100 & EKO.ERI

10. Parameter estimation • Model A: HKD100

11. Parameter estimation • Model B

12. Prediction of ground motion • Predictive PDF of ground motion y Expectation over  = (, 2)

13. Prediction of ground motion Site EKO.ERI, Model B Site HKD100, Model A

14. Discussions

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. Thank you for your attention!

17. ----epistemic uncertainty ----epistemic uncertainty Answers to What is P2 of the past attenuation relation ? 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 ----represent inherentvariabilityandmodel uncertainty. ----represent the average character of uncertainty for all sites.

18. Mathematical modeling x: variables, : parameters m: model uncertainty, when replaces f . Parameter estimate s: statistical uncertainty, when  is estimated with limited number of data. Modeling the ground motion • Effects of ground Motion： ---- Source, path, site Buildings Ground Motion a: inherent variability, aleatory uncertainty when f represents the real world of ground motion Site Engineering bedrock Path Seismic bedrock Source m, s:epistemic uncertainty

19. Past attenuation relation • Model of ground motion • Mathematical modeling a: ~ N(0, a2)a: inherent variability • e.g. Si-Midorikawa(1999) • Mathematical modeling y: ground motion in natural logarithmx: variables, such as Mw, R, D, …, etc.: regression coefficientsP: random term ~N(0, P2) • P : • inherent variability a , aleatory uncertainty • model uncertainty m, epistemic uncertainty

20. 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*. Uncertainty considering statistical uncertainty Contour of y

22. Surface Amplification factor e.g. f(Vs) Soil-specific attenuation relation Attenuation relation on a baseline condition Amplification factor Engineering bedrock Attenuation relation on Engineering bedrock

23. x1 x2 y = ? Observed Site x3 Unobserved Site 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: Ref.: Wang, M. and Takada, T. (2005):Macrospatial correlation model of seismic ground motion, Earthquake Spectra, Vol. 21, No. 4, 1137-1156.

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.