Bayesian tsunami fragility modeling considering input data ...users.ntua.gr/divamva/docs/42RHUW/DeRisi_42RHUW_Hydra.pdf · • Empirical Tsunami Fragility modelling requires numerous

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]