A. Betâmio de Almeida Assessing Modelling Uncertainty A. Betâmio de Almeida Instituto Superior Técnico November 2004 Zaragoza, Spain 4th IMPACT Workshop

A. Betâmio de Almeida


Assessing Modelling Uncertainty

A. Betâmio de AlmeidaInstituto Superior Técnico

November 2004

Zaragoza, Spain

4th IMPACT Workshop



Research Domain : Uncertainty characterization related to risk assessment methods in civil engineering

Specific Topic : Dam-break flooding and risk assessment – uncertainty analysis

Phase I

(2002- )

: Rock and earthfill dam breach modelling and uncertainty analysis

Research Tools : Physical model

Mathematical and computational models

Monte Carlo - Latin Hypercube Sampling

Research Team

(IST Lisbon)

: A. Betâmio de Almeida, Mário Franca, Joana Brito



Risk Assessment Level

• Level 0 → Identification hazard

• Level 1 → “Worst-case” approach

• Level 2 → “Quasi-worst-case” – plausible upper bounds

• Level 3 → “Best estimate”, central value

• Level 4 → Probabilistic risk assessment

Probabilistic uncertainty management

• Level 5 → Separation of different types of uncertainty

Single risk distribution

Uncertainty management



Phase 1

Tools •Experimental studies and bibliography (IST Laboratory – Franca, 2002)

•Computational model – RoDaB model (Franca and Almeida, 2003)

•Uncertainty propagation method of analysis- Monte Carlo and Latin Hypercube Sampling (Brito and Almeida, 2004)

Objectives •To consider the model output precision → model input and model parameter uncertainty → Aleatory Uncertainty

•To consider the model output accuracy (2005) → model structure uncertainty → Epistemic Uncertainty

•To improve the risk management decisions

•To improve the model management → Sensivity analysis



Reference System for Uncertainty Management



Monte Carlo Method of Simulation (L.H.S.)

Uncertainty propagation scheme



Monte Carlo Method of Simulation (L.H.S.)

1 - Generation of random number [0,1] – two tipes of sets– Type 1→ for generation of samples size N for each input / parameter of the model (susbsystem)

kk

numbers random variables kNk

XX jn kj 1

– Type 2→ one set for L.H.S. Special procedure

numbers random 1 variables kNk

YY jn 11 kj



Latin Hypercube Sampling (L.H.S.)2 – Latin Hypercube Sampling (L.H.S.)

Justification – It is a refinement of the classical (standard) Monte Carlo Sampling. In general, it produces substantial variance reductions over standard Monte Carlo in Risk Analysis applications

• Each (input/parameter) probability distribuition is divided into N intervals of equal probability (N ≡ sample size). Each strata is identified (1≤n≤N)

• Each random number of set 1 [X] is renormalized according to each strata number of order → transformed matrix [X’]

• Input samples of size N are generated based on [X’] and the inverse transform of each input/parameter distribution

N

n

N

XX jn

jn1

kj 1

Nn 1



Latin Hypercube Sampling (L.H.S.)



Monte Carlo simulation procedure



Example: RoDaB Model (Franca and Almeida 2004)

• 1)

• 2)

• with

• 3)

• 4)

• 5)

• 6)

dt

dVQQQ R

CBi

5,1' 2 BRCCC NNgLCQ

B

NNC CR

C

132,0333,0

tBCC WLL

5,1BRBBB NNWCQ

01

1,

bs

B qpdt

dN

bsBbsbs Uq ,

,,

bs

bs

B

Bbs

B

A

QC

dt

dN,

,

,

ms

ms

B

Bms

B

A

QC

dt

dW,

,

,

(Exner Equation)

Initial conditions and model parameters 7 input/parameter for uncertainty analysis



Example LHS (shuffling)• Size of each sample: N=1000 N≡number of strata• Number of variables: k=7

Sample matrix

1000,71000,21000,1

500,7500,2500,1

1,71,21,1

xxx

xxx

xxx

The vectors are correlated

In order to break this correlation, we use the random number matrix [Y]

k-1 samples will be randomly shuffled

jnx

1000,71000,31000,2

500,7500,3500,2

1,71,31,2

yyy

yyy

yyy

1000,2

1,2

x

x

1000,2

1,2

y

y

sort

Indu

ced

sort



Parameter Analysis

Output

Input Output Sensivity Analysis

Comparative analysis of all parameters

0.00

0.01

0.02

0.03

1.5E-03 1.8E-03 2.1E-03 2.4E-03 2.7E-03

Erosion Coef. (-)

Fre

qu

ency

30

35

40

45

50

55

60

1.5E-03 1.7E-03 1.9E-03 2.1E-03 2.3E-03 2.5E-03 2.7E-03 2.9E-03

Erosion Coef. (-)

Tim

e (m

in)

4000

4200

4400

4600

4800

5000

5200

Flo

w (

m3/s

)

Time to peak Peak Flow

0.0

0.2

0.4

0.6

0.8

1.0

3332 3555 3779 4002 4225 4448 4671 4894 5117 5340

Peak Flow (m3/s)

Fre

qu

en

cy

Breach Discharge Coef. Erosion Coef. Correlation FactorBreach Final Height Breach Final Width Breach Initial Width

0.00

0.01

0.02

0.03

0.04

4227 4374 4521 4668 4815 4962 5110

Peak Flow (m3/s)

Fre

qu

ency



Integrated Monte Carlo Analysis

Empiric FormulasPeak Flow

(m3/s)

Froehlich (1995) 3237

Taher-Shamsi et al. (2003) 4466

Monte Carlo SimulationPeak Flow

(m3/s)

Average 4468

Standard deviation 630

0.00

0.01

0.02

0.03

0.04

0.05

0.06

2784 3175 3566 3957 4348 4738 5129 5520 5911 6302

Peak Flow (m3/s)

Fre

qu

ency



Upper and Lower Bounds of the Outflow Hydrographs obtained through

Monte Carlo Simulation

0

1000

2000

3000

4000

5000

6000

7000

8000

0 60 120 180 240 300 360 420

Time (minutes )

Flo

w (

m3/

s)

Average value of Monte Carlo Simulation Upper bound of Monte Carlo Simulation Lower bound of Monte Carlo Simulation Deterministic value on best estimates



Example of hydrographs obtained from the Monte Carlo Iterations

0

1000

2000

3000

4000

5000

6000

7000

0 60 120 180 240 300 360

Time (minutes)

Flo

w (

m3 /s

)



25

35

45

55

65

75

2783 3283 3783 4283 4783 5283 5783 6283

Peak Flow (m3/s)

Tim

e to

Pe

ak (

min

)

Documents

A. Betâmio de Almeida Assessing Modelling Uncertainty A. Betâmio de Almeida Instituto Superior Técnico November 2004 Zaragoza, Spain 4th IMPACT Workshop