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Bayesian analysis of extremes in hydrology A powerful tool for knowledge integration and uncertainties assessment

Bayesian analysis of extremes in hydrology A powerful tool for knowledge integration and uncertainties assessment Renard, B., Garreta, V., Lang, M. and Bois, P. Introduction. Water is both a resource and a risk. High flows risk…. …and low flows risk.

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Bayesian analysis of extremes in hydrology A powerful tool for knowledge integration and uncertainties assessment

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  1. Bayesian analysis of extremes in hydrology A powerful tool for knowledge integration and uncertainties assessment Renard, B., Garreta, V., Lang, M. and Bois, P. Extreme Value Analysis, August 15-19, 2005

  2. Introduction Water is both a resource and a risk High flows risk… …and low flows risk → Hydrologists are interested in the tail of the discharges distribution Extreme Value Analysis, August 15-19, 2005

  3. Introduction General analysis scheme • Extract a sample of extreme values from the discharges series • Choose a convenient extreme value distribution • Estimate parameters • Compute quantities of interest (quantiles) Estimation methods: moments, L-moments, maximum likelihood, Bayesian estimation Extreme Value Analysis, August 15-19, 2005

  4. Probabilistic Model(s) M1: X~p(θ) M2: X~p’(θ’) … Decision p(M1|X), p(M2|X),… Observations X=(x1, …, xn) Likelihood(s) p(X| θ), p’(X| θ’),… Estimation = … Posterior distribution(s) p(θ|X), p’(θ’|X) Bayes Theorem Prior distribution(s) π(θ), π’(θ’), … Frequency analysis p(q(T)) Introduction Bayesian Analysis Extreme Value Analysis, August 15-19, 2005

  5. Introduction Advantages from an hydrological point of view: Prior knowledge introduction: taking advantage of the physical processes creating the flow (rainfall, watershed topography, …) Model choice: computation of models probabilities, and incorporation of model uncertainties by « model averaging » Drawback for new user: MCMC algorithms… We used combinations of Gibbs and Metropolis samplers, with adaptive jumping rules as suggested by Gelman et al. (1995) Extreme Value Analysis, August 15-19, 2005

  6. The Ardeche river at St Martin d’Ardeche 2240 km2 High slopes and granitic rocks on the top of the catchment Very intense precipitations (September-December) Extreme Value Analysis, August 15-19, 2005

  7. The Ardeche river at St Martin d’Ardeche Discharge data Extreme Value Analysis, August 15-19, 2005

  8. The Ardeche river at St Martin d’Ardeche Model: Annual Maxima follow a GEV distribution Likelihood: Extreme Value Analysis, August 15-19, 2005

  9. The Ardeche river at St Martin d’Ardeche Prior specifications Hydrological methods give rough estimates of quantiles: CRUPEDIX method: use watershed surface, daily rainfall quantile and geographical localization (q10) Gradex method: use extreme rainfall distribution and expert’s judgment about response time of the watershed (q200-q10) Record floods analysis: use discharges data on an extended geographical scale (q1000) The prior distribution on quantiles is then transformed in a prior distribution on parameters Extreme Value Analysis, August 15-19, 2005

  10. The Ardeche river at St Martin d’Ardeche Results: uncertainties reduction 1 3 2 Extreme Value Analysis, August 15-19, 2005

  11. The Drome river at Luc-en-Diois Data: 93 flood events between 1907 and 2003 Extreme Value Analysis, August 15-19, 2005

  12. The Drome river at Luc-en-Diois Models: Inter-arrivals duration: M0 : X~Exp(λ) M1 : X~Exp(λ0(1+ λ1t)) Threshold Exceedances: M0 : Y~GPD(λ, ξ) M1 : Y~GPD(λ0(1+ λ1t), ξ) Results: Trend on inter-arrivals P(M0|X)=0.11 P(M0|Y)=0.79 Floods frequency decreases Floods intensity is stationary P(M1|X)=0.89 P(M1|Y)=0.21 Extreme Value Analysis, August 15-19, 2005

  13. The Drome river at Luc-en-Diois 0.9-quantile estimate by model Averaging Extreme Value Analysis, August 15-19, 2005

  14. Let denotes the annual maxima at site i at time t Perspectives: regional trend detection Motivations Regional model can improve estimators accuracy Climate change impacts should be regionally consistent Models Extreme Value Analysis, August 15-19, 2005

  15. cumulated probability Gaussian Transformation Multivariate Gaussian model Perspectives: regional trend detection Likelihoods The multivariate distribution of annual maxima is needed… Independence hypothesis: Gaussian copula approximation: Extreme Value Analysis, August 15-19, 2005

  16. Perspectives: regional trend detection Example of preliminary results Data: 6 stations with 31 years of common data Independence hypothesis M0 model estimation(regional in red, at-site in black): Extreme Value Analysis, August 15-19, 2005

  17. Perspectives: regional trend detection M1 model estimation: Extreme Value Analysis, August 15-19, 2005

  18. Conclusion Advantages of Bayesian analysis • Prior knowledge integration • Model choice uncertainty is taken into account • No asymptotic assumption • Robustness of MCMC methods to deal with high dimensional problems But… • Part of subjectivity? • A better understanding of extreme’s dependence is still needed Extreme Value Analysis, August 15-19, 2005

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