Bayesian tsunami fragility modeling considering input data uncertainty

Transcription

1 Bayesian tsunami fragility modeling considering input data uncertainty Raffaele De Risi Research Associate University of Bristol United Kingdom De Risi, R., Goda, K., Mori, N., & Yasuda, T. (2016). Bayesian tsunami fragility modeling considering input data uncertainty. Stochastic Environmental Research and Risk Assessment, doi: /s x

2 Motivation Empirical Tsunami Fragility modelling requires numerous pairs of Tsunami Damage Observations and Explanatory Variable related to both Hazard and Exposure. Tsunami Inundation Depth is the typical intensity measure adopted in developing empirical fragility.

3 Motivation Empirical Tsunami Fragility modelling requires numerous pairs of Tsunami Damage Observations and Explanatory Variable related to both Hazard and Exposure. Tsunami Inundation Depth is the typical intensity measure adopted in developing empirical fragility. Tsunami Inundation Height Tsunami Inundation Depth

4 Motivation Empirical Tsunami Fragility modelling requires numerous pairs of Tsunami Damage Observations and Explanatory Variable related to both Hazard and Exposure. Tsunami Inundation Depth is the typical intensity measure adopted in developing empirical fragility. Tsunami Inundation Depth are subject to errors due to: survey (i) techniques, (ii) equipment, and (iii) conditions. A further source of potential error is the operation of Interpolation when direct measurement are not available. Observations Other locations

5 Motivation Empirical Tsunami Fragility modelling requires numerous pairs of Tsunami Damage Observations and Explanatory Variable related to both Hazard and Exposure. Tsunami Inundation Depth is the typical intensity measure adopted in developing empirical fragility. Tsunami Inundation Depth are subject to errors due to: survey (i) techniques, (ii) equipment, and (iii) conditions. A further source of potential error is the operation of Interpolation when direct measurement are not available. Uncertainty associated with input hazard data can result in potential overestimation of model uncertainty associated with developed Fragilities In Tsunami fragility modelling, incorporation of input data errors and uncertainty has not been explored rigorously.

6 Scientific Questions (1) How to quantify the Uncertainty of input hazard parameters? (2) How to propagate such Uncertainty on tsunami fragility function? To respond these questions, we studied the M W TOHOKU event, for which a large amount of data is available

7 First Available Database: MLIT database Uncertainty Quantification MLIT (Ministry of Land, Infrastructure, and Transportation of Japanese Government) Description Condition Non Structural Structural Observation: Location, h, Material, Damage State, Number of stories, etc.

8 Uncertainty Quantification First Available Database: MLIT database Timber Structures

9 Uncertainty Quantification MLIT database Accuracy Two sources of uncertainty associated to the Intensity Measures: 1. Error due to interpolation/smoothing: recordings are based on MLIT 100-m data; 2. Elevation data at each building sites are not available; therefore there is not a straightforward correlation between tsunami height and tsunami depth. It is not straightforward to estimate the MLIT data accuracy Second Available Database: TTJS database (Tohoku Tsunami Joint Survey) 1. More reliable than MLIT database (vertical accuracy within few centimeters, as DEM the GSI data are used) but less populated; 2. Heights of watermarks on buildings, trees, and walls were measured using a laser range finder, a level survey, a real-time kinematic global positioning system (RTK-GPS) receiver with a cellular transmitter, and total stations.

10 Uncertainty Quantification Procedure for Uncertainty Quantification

11 Uncertainty Quantification Procedure for Uncertainty Quantification Normal Log-Normal

12 Uncertainty Quantification Procedure for Uncertainty Quantification Normal Log-Normal Lognormal, η equal to 1 and logarithmic standard deviation equal to 0.25

13 Uncertainty Propagation First Step: Typical Tsunami Empirical Fragility models (1) Log-Normal Method Binning Change of variables Linear fitting ( ) 1 ln h = ln η+ β Φ P DS ds h + εr Change of variables ln h lnη P ( DS ds h) =Φ β Two Parameters for each damage state: η and β

14 Uncertainty Propagation First Step: Typical Tsunami Empirical Fragility models (2) Binomial Logistic Method Probability of occurrence n i= 1 1 y ( 1 ) 1 i πi π i yi π i may assume different forms Logit ( b1+ b2 ln hi ) ( b b h ) exp π i = 1 + exp + ln 1 2 i y i Two Parameters for each damage state: b 1 and b 2

15 Uncertainty Propagation First Step: Typical Tsunami Empirical Fragility models (3) Multinomial Logistic Method Binning Probability of occurrence k m j= 1 i! y ij! k j= 1 π i may assume different forms π y ij ( b1, j b2, j ln hi) ( b1, j b2, j hi) exp + j 1 1 πil + + l= 1 πij = 1 exp ln ij Two Parameters for each damage state: b 1i and b 2i

16 Uncertainty Propagation Second Step: Bayesian procedure 1 ( θ D) = ( D θ) ( θ ) ( ) ( ) f c L f D θ θ θ L( D θ) f ( Di θ) c= L f d n = The likelihood function depend by the adopted typology of regression. The parameters maximizing the posteriors represent the solution of the Bayesian regression (i.e. the Bayesian maximum likelihood). f (θ ) i= 1 θ i

17 Uncertainty Propagation Second Step: Bayesian procedure 1 ( θ D) = ( D θ) ( θ ) ( ) ( ) f c L f D θ θ θ L( D θ) f ( Di θ) c= L f d n = The likelihood function depend by the adopted typology of regression. The parameters maximizing the posteriors represent the solution of the Bayesian regression (i.e. the Bayesian maximum likelihood). How to implement the uncertainty on the intensity measure? i= 1 + ( D θ) ( D θ) ( ) f i = f i ε, fi ε dε n + ( D θ) ( D θ) ( ) L = f ε, f ε dε i= 1 i i

18 Uncertainty Propagation Second Step: Bayesian procedure (1) Log-Normal Method exp ln h + ε ln η β Φ 2 ( P ( DS ds h )) f ( ε ) dε 2π σ 2 σ R R 1 i ln h i ln h ln h ( ( ε )) ( ε ) ( ) (2) Binomial Logistic Method ( ( ε )) ( ε ) yi 1 yi + 1 exp b1 + b2 ln hi + ln h exp b1 + b2 ln hi + ln h 1 f ( εln h) yi 1+ exp b1 + b2 ln hi + ln h 1+ exp b1 + b2 ln hi + ln h ( ) exp (3) Multinomial Logistic Method ( b1, j b2, j ( ln h ε ln h) ) b b ( h ε ) k j πil f ( εln h ) dε ln h j= 1 1+ exp 1, j + 2, j ln + ln h l= 1 ( ) dε ln h

19 Numerical Results: Log-Normal Method Uncertainty Propagation

20 Numerical Results: Log-Normal Method Uncertainty Propagation

21 Uncertainty Propagation Numerical Results: Binomial Logistic Method Without With

22 Uncertainty Propagation Numerical Results: Binomial Logistic Method

23 Uncertainty Propagation Numerical Results: Multinomial Logistic Method

24 Uncertainty Propagation Numerical Results: Multinomial Logistic Method

25 Uncertainty Propagation Effects on the Risk Assessment k [ ] = j ( j) ( j 1) E L R P DS ds P DS ds + j= 1 µ R = 1600 $/m 2 σ R = 32% &7 0% 5% 20% 40% 60% 100% 1000 simulations

26 Future Developments Multivariate Empirical Tsunami Fragility, i.e. consider not only tsunami depth but also tsunami velocity. Identification of a methodology for the quantification of the input data uncertainty for the velocity. Propagate the entire distribution of the parameters for a robust regression. Potential extension to experimental database to remove from the capacity models the measurement error or other typologies of error that can be quantified.

27 Thank you for your attention! Raffaele De Risi Research Associate University of Bristol United Kingdom