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Bayesian tsunami fragility modeling considering input data uncertainty
Raffaele De RisiResearch AssociateUniversity of Bristol
United Kingdom
De Risi, R., Goda, K., Mori, N., & Yasuda, T. (2016). Bayesian tsunami fragility modelingconsidering input data uncertainty. Stochastic Environmental Research and Risk Assessment,doi:10.1007/s00477-016-1230-x
Motivation
• Empirical Tsunami Fragility modelling requires numerous pairs ofTsunami Damage Observations and Explanatory Variable related toboth Hazard and Exposure.
• Tsunami Inundation Depth is the typical intensity measure adoptedin developing empirical fragility.
Motivation
• Empirical Tsunami Fragility modelling requires numerous pairs ofTsunami Damage Observations and Explanatory Variable related toboth Hazard and Exposure.
• Tsunami Inundation Depth is the typical intensity measure adoptedin developing empirical fragility.
Tsunami Inundation Depth
Tsunami Inundation Height
Motivation
• Empirical Tsunami Fragility modelling requires numerous pairs ofTsunami Damage Observations and Explanatory Variable related toboth Hazard and Exposure.
• Tsunami Inundation Depth is the typical intensity measure adoptedin 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 Interpolationwhen direct measurement are not available.
Observations
Other locations
Motivation
• Empirical Tsunami Fragility modelling requires numerous pairs ofTsunami Damage Observations and Explanatory Variable related toboth Hazard and Exposure.
• Tsunami Inundation Depth is the typical intensity measure adoptedin 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 Interpolationwhen direct measurement are not available.
• In Tsunami fragility modelling, incorporation of input data errors anduncertainty has not been explored rigorously.
Uncertainty associated with input hazard data can result in potential overestimation of model uncertainty associated with developed Fragilities
Scientific Questions
• (2) How to propagate such Uncertainty on tsunami fragility function?
• (1) How to quantify the Uncertainty of input hazard parameters ?
To respond these questions, we studied the MW9 2011 TOHOKU event, for which a large amount of data is available
Uncertainty Quantification
First Available Database: MLIT databaseMLIT (Ministry of Land, Infrastructure, and Transportation of Japanese Government)
2
3456
7
Description Condition
1 2 3 4 5 6 7
Non Structural Structural
∀ Observation: Location, h, Material,Damage State, Number of stories, etc.
Uncertainty Quantification
First Available Database: MLIT database
Timber Structures
Uncertainty QuantificationMLIT database AccuracyTwo sources of uncertainty associated to the Intensity Measures:
1. Error due to interpolation/smoothing: recordings are based onMLIT 100-m data;
2. Elevation data at each building sites are not available; thereforethere is not a straightforward correlation between tsunami heightand 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 fewcentimeters, as DEM the GSI data are used) but less populated;
2. Heights of watermarks on buildings, trees, and walls weremeasured using a laser range finder, a level survey, a real-timekinematic global positioning system (RTK-GPS) receiver with acellular transmitter, and total stations.
Uncertainty QuantificationProcedure for Uncertainty Quantification
Uncertainty QuantificationProcedure for Uncertainty Quantification
Normal Log-Normal
Uncertainty QuantificationProcedure for Uncertainty Quantification
Normal Log-Normal
Lognormal, η equal to 1 and logarithmic standard deviation equal to 0.25
Uncertainty Propagation
First Step: Typical Tsunami Empirical Fragility models
( )1ln ln | Rh P DS ds hη β ε−= + ⋅Φ ≥ +
( ) ln ln| hP DS ds h ηβ
−≥ = Φ
(1) Log-Normal Method
• Binning
• Change of variables
• Linear fitting
• Change of variables
Two Parameters for each damage state: η and β
Uncertainty Propagation
First Step: Typical Tsunami Empirical Fragility models
(2) Binomial Logistic Method
( )1
1
11 ii
nyy
i ii iy
π π −
=
⋅ ⋅ −
∏
• Probability of occurrence
( )( )1 2
1 2
exp ln1 exp ln
ii
i
b b hb b h
π+ ⋅
=+ + ⋅
• πi may assume different forms
• Logit
Two Parameters for each damage state: b1 and b2
Uncertainty Propagation
First Step: Typical Tsunami Empirical Fragility models
(3) Multinomial Logistic Method
• Binning• Probability of occurrence
1
1
!
!
ijk
yiijk
jij
j
m
yπ
=
=
∏∏
• πi may assume different forms
( )( )
11, 2,
11, 2,
exp ln1
1 exp ln
jj j i
ij illj j i
b b h
b b hπ π
−
=
+ ⋅ = ⋅ − + + ⋅
∑
Two Parameters for each damage state: b1i and b2i
Uncertainty Propagation
Second Step: Bayesian procedure
( ) ( ) ( )1 |f c L f−| = ⋅ ⋅D Dθ θ θ ( ) ( )|c L f d= ⋅ ⋅∫ D θ θ θ ( ) ( )1
| |n
i
L f=
= ∏D iDθ θ
The likelihood function depend by the adopted typology of regression.
The parameters maximizing the posteriors represent the solution of theBayesian regression (i.e. the Bayesian maximum likelihood).
θi
f (θ )
Uncertainty Propagation
Second Step: Bayesian procedure
( ) ( ) ( )1 |f c L f−| = ⋅ ⋅D Dθ θ θ ( ) ( )|c L f d= ⋅ ⋅∫ D θ θ θ ( ) ( )1
| |n
i
L f=
= ∏D iDθ θ
The likelihood function depend by the adopted typology of regression.
The parameters maximizing the posteriors represent the solution of theBayesian regression (i.e. the Bayesian maximum likelihood).
How to implement the uncertainty on the intensity measure?
( ) ( ) ( )| | if f f dε ε ε+∞
−∞= , ⋅ ⋅∫i iD Dθ θ
( ) ( ) ( )1
| |n
ii
L f f dε ε ε+∞
−∞=
= , ⋅ ⋅∏∫D iDθ θ
Uncertainty Propagation
Second Step: Bayesian procedure
(1) Log-Normal Method
( )( ) ( )21
ln ln ln2
1 1exp ln ln |22 i h i h h
RR
h P DS ds h f dε η β ε εσπ σ
+∞ −
−∞
⋅ − ⋅ + − − ⋅Φ ≥ ⋅ ⋅ ⋅⋅
∫
(2) Binomial Logistic Method
( )( )( )( )
( )( )( )( ) ( )
1
1 2 ln 1 2 lnln ln
1 2 ln 1 2 ln
exp ln exp ln11
1 exp ln 1 exp ln
i iy y
i h i hh h
i i h i h
b b h b b hf d
y b b h b b hε ε
ε εε ε
−+∞
−∞
+ ⋅ + + ⋅ + ⋅ ⋅ − ⋅ ⋅ + + ⋅ + + + ⋅ +
∫
(3) Multinomial Logistic Method
( )( )( )( ) ( )
11, 2, ln
ln ln11 1, 2, ln
exp ln1
1 exp ln
jkj j h
il h hlj j j h
b b hf d
b b h
επ ε ε
ε
−+∞
−∞==
+ ⋅ + ⋅ − ⋅ ⋅ + + ⋅ +
∑∏∫
Uncertainty Propagation
Numerical Results: Log-Normal Method
Uncertainty Propagation
Numerical Results: Log-Normal Method
Uncertainty Propagation
Numerical Results: Binomial Logistic Method
Without With
Uncertainty Propagation
Numerical Results: Binomial Logistic Method
Uncertainty Propagation
Numerical Results: Multinomial Logistic Method
Uncertainty Propagation
Numerical Results: Multinomial Logistic Method
Uncertainty Propagation
Effects on the Risk Assessment
[ ] ( ) ( )11
k
j j jj
E L R P DS ds P DS ds +=
= ⋅ ≥ − ≥ ∑
µR = 1600 $/m2
σR = 32%
1 2 3 4 5
0% 5% 20% 40% 60% 100%
6&7 1000 simulations
Future Developments
• Multivariate Empirical Tsunami Fragility, i.e. consider not onlytsunami depth but also tsunami velocity.
• Identification of a methodology for the quantification of the inputdata uncertainty for the velocity.
• Propagate the entire distribution of the parameters for a robustregression.
• Potential extension to experimental database to remove from thecapacity models the measurement error or other typologies of errorthat can be quantified.
Thank you for your attention!
Raffaele De RisiResearch AssociateUniversity of Bristol
United [email protected]