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

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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

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

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

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

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

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

The Ardeche river at St Martin d’Ardeche

Discharge data

Extreme Value Analysis, August 15-19, 2005

The Ardeche river at St Martin d’Ardeche

Model: Annual Maxima follow a GEV distribution

Likelihood:

Extreme Value Analysis, August 15-19, 2005

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

The Ardeche river at St Martin d’Ardeche

Results: uncertainties reduction

1

3

2

Extreme Value Analysis, August 15-19, 2005

The Drome river at Luc-en-Diois

Data: 93 flood events between 1907 and 2003

Extreme Value Analysis, August 15-19, 2005

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

The Drome river at Luc-en-Diois

0.9-quantile estimate by model Averaging

Extreme Value Analysis, August 15-19, 2005

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

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

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

Perspectives: regional trend detection

M1 model estimation:

Extreme Value Analysis, August 15-19, 2005

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