Extreme Value Analysis, August 15-19, 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, 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, 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, Introduction Probabilistic Model(s) M 1 : X~p(θ ) M 2 : X ~p’(θ’ ) … Observations X=(x 1, …, x n ) Posterior distribution(s) p(θ|X), p’(θ’|X) Prior distribution(s) π ( θ), π’ ( θ’), … Bayes Theorem Likelihood(s) p (X| θ), p’(X| θ’),… Decision p(M 1 |X), p(M 2 |X),… Frequency analysis p(q(T)) Estimation = … Bayesian Analysis

Extreme Value Analysis, August 15-19, 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, The Ardeche river at St Martin d’Ardeche 2240 km 2 High slopes and granitic rocks on the top of the catchment Very intense precipitations (September-December)

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

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

Extreme Value Analysis, August 15-19, 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 (q 10 ) Gradex method: use extreme rainfall distribution and expert’s judgment about response time of the watershed (q 200 -q 10 ) Record floods analysis: use discharges data on an extended geographical scale (q 1000 ) The prior distribution on quantiles is then transformed in a prior distribution on parameters

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

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

Extreme Value Analysis, August 15-19, The Drome river at Luc-en-Diois Models: Inter-arrivals duration: M 0 : X~Exp(λ) M 1 : X~Exp(λ 0 (1+ λ 1 t)) Threshold Exceedances: M 0 : Y~GPD(λ, ξ) M 1 : Y~GPD(λ 0 (1+ λ 1 t), ξ) Results: Trend on inter-arrivals P(M 0 |X)=0.11 P(M 1 |X)=0.89 P(M 0 |Y)=0.79 P(M 1 |Y)=0.21 Floods frequency decreases Floods intensity is stationary

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

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

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

Extreme Value Analysis, August 15-19, Perspectives: regional trend detection Example of preliminary results Data: 6 stations with 31 years of common data Independence hypothesis M 0 model estimation (regional in red, at-site in black):

Extreme Value Analysis, August 15-19, Perspectives: regional trend detection M 1 model estimation:

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