Extreme Value Analysis, August 15-19, 20051

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, 20052

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, 20053

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, 20054

Introduction

Probabilistic Model(s)

M1: X~p(θ)M2: X~p’(θ’)

…

ObservationsX=(x1, …, xn)

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’),…

Decisionp(M1|X), p(M2|X),…

Frequency analysisp(q(T))

Estimation = …θ̂

Iθ

Bayesian Analysis

Extreme Value Analysis, August 15-19, 20055

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, 20056

The Ardeche river at St Martin d’Ardeche

2240 km2

High slopes and granitic rocks on the top of the catchmentVery intense precipitations (September-December)

Extreme Value Analysis, August 15-19, 20057

The Ardeche river at St Martin d’Ardeche

Discharge data

Extreme Value Analysis, August 15-19, 20058

The Ardeche river at St Martin d’Ardeche

Model: Annual Maxima follow a GEV distribution

Likelihood:

1 11

1 ( ) ( )( ; , , ) 1 exp 1

x xf x

33 47

1

42 54 182

1 2 3

( | , , ) ( ; , , )

( ; , , ) ( ; , , ) ( ; , , )

ii

p f x

F S F S F S

X

Extreme Value Analysis, August 15-19, 20059

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, 200510

The Ardeche river at St Martin d’Ardeche

Results: uncertainties reduction

1

2

3

Extreme Value Analysis, August 15-19, 200511

The Drome river at Luc-en-Diois

Data: 93 flood events between 1907 and 2003

Extreme Value Analysis, August 15-19, 200512

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(M1|X)=0.89

P(M0|Y)=0.79

P(M1|Y)=0.21

Floods frequency decreasesFloods intensity is stationary

Extreme Value Analysis, August 15-19, 200513

The Drome river at Luc-en-Diois

0.9-quantile estimate by model Averaging

Extreme Value Analysis, August 15-19, 200514

Perspectives: regional trend detection

Motivations

Models

Regional model can improve estimators accuracyClimate change impacts should be regionally consistent

( )1 : ~ ( , (1 ), )i

t i iM X GEV t

( )0 : ~ ( , , )i

t i iM X GEV

Let denotes the annual maxima at site i at time t( )itX

Extreme Value Analysis, August 15-19, 200515

Perspectives: regional trend detection

Likelihoods

The multivariate distribution of annual maxima is needed…

Independence hypothesis:

(1) ( ) (1) ( )( ,..., ; ) ( ,..., ; )p pi i

i

p f x xX X θ θ

(1) ( ) (1) ( )1( ,..., ) ( ) ... ( )p p

pf x x f x f x

(1) ( ) (1) ( )1

1 (1) 1 ( ) 1 1 (1) 1 ( )1 11/ 2

( ,..., ) ( ) ... ( )

( ) ,..., ( ) ( ) ,..., ( )exp

2

p pp

Tp p

p p

f x x f x f x

F x F x I F x F x

Gaussian copula approximation:

cumulated probability

Multivariate Gaussian model

Gaussian Transformation

Extreme Value Analysis, August 15-19, 200516

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, 200517

Perspectives: regional trend detection

M1 model estimation:

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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