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

2. A. Betâmio de Almeida 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

3. A. Betâmio de Almeida 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

4. A. Betâmio de Almeida 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

5. A. Betâmio de Almeida Reference System for Uncertainty Management

6. A. Betâmio de Almeida Monte Carlo Method of Simulation (L.H.S.) Uncertainty propagation scheme

7. A. Betâmio de Almeida 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) –Type 2→ one set for L.H.S. Special procedure

8. A. Betâmio de Almeida 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

9. A. Betâmio de Almeida Latin Hypercube Sampling (L.H.S.)

10. A. Betâmio de Almeida Monte Carlo simulation procedure

11. A. Betâmio de Almeida Example: RoDaB Model (Franca and Almeida 2004) 1) 2) with 3) 4) 5) 6) (Exner Equation) Initial conditions and model parameters 7 input/parameter for uncertainty analysis

12. A. Betâmio de Almeida Example LHS (shuffling) Size of each sample: N=1000N≡number of strata Number of variables: k=7 Sample matrix The vectors are correlated In order to break this correlation, we use the random number matrix [Y] k-1 samples will be randomly shuffled sort Induced sort

13. A. Betâmio de Almeida Parameter Analysis Output InputOutput Sensivity Analysis Comparative analysis of all parameters

14. A. Betâmio de Almeida Integrated Monte Carlo Analysis Empiric Formulas Peak Flow (m 3 /s) Froehlich (1995) 3237 Taher-Shamsi et al. (2003) 4466 Monte Carlo Simulation Peak Flow (m 3 /s) Average 4468 Standard deviation 630

15. A. Betâmio de Almeida Upper and Lower Bounds of the Outflow Hydrographs obtained through Monte Carlo Simulation

16. A. Betâmio de Almeida Example of hydrographs obtained from the Monte Carlo Iterations

17. A. Betâmio de Almeida