A spatio-temporal model for extreme precipitation simulated by a climate model

Transcription

1 A spatio-temporal model for extreme precipitation simulated by a climate model Jonathan Jalbert Postdoctoral fellow at McGill University, Montréal Anne-Catherine Favre, Claude Bélisle and Jean-François Angers Workshop: Uncertainty and Causality Assessment in Modeling Extreme and Rare Events April 27 th, 2016 NCAR, Boulder, CO.

2 Introduction The 2011 Richelieu River flood. Damages estimated at USD 100 million. c Ryan L. C. Quan - com/photos/ryan_quan/ c Bernard Brault - The Canadian Press The 2013 Alberta Floods. Damages estimated at USD 5 billion. Jonathan Jalbert A spatio-temporal model for extreme precipitation simulated by a climate model 2016/04/27 2 / 22

3 Introduction Questions naturally come up with these and other major floods: Are climate changes responsible for these particular events? Should flooding defences be modified? With climate, in situ experimentations are impossible. Climate models are therefore the only tools for providing partial answers to the precedent questions. The goal of the talk is to present a spatio-temporal statistical model especially suited for extreme precipitation simulated by a climate model: non-stationarity in transient time series; large spatial simulation domain; spatial dependence among grid points. Jonathan Jalbert A spatio-temporal model for extreme precipitation simulated by a climate model 2016/04/27 3 / 22

4 Introduction Questions naturally come up with these and other major floods: Are climate changes responsible for these particular events? Should flooding defences be modified? With climate, in situ experimentations are impossible. Climate models are therefore the only tools for providing partial answers to the precedent questions. The goal of the talk is to present a spatio-temporal statistical model especially suited for extreme precipitation simulated by a climate model: non-stationarity in transient time series; large spatial simulation domain; spatial dependence among grid points. Jonathan Jalbert A spatio-temporal model for extreme precipitation simulated by a climate model 2016/04/27 3 / 22

5 Data The dataset consists in the daily precipitation outputs from a run of the Canadian Regional Climate Model (CRCM). 12,570 land grid points; Daily precipitation series for the period [1961, 2100] at every grid point. Let Y ikl be the precipitation depth (mm) of day l of year k at grid point i, where 1 i 12, 570; 1 k 140; 1 l 365. Simulation domain Jonathan Jalbert A spatio-temporal model for extreme precipitation simulated by a climate model 2016/04/27 4 / 22

6 Data Let M ik be the annual maximum of year k at grid point i: M ik = max 1 l 365 Y ikl. For 66% of the grid points, the series of maxima (M ik : 1 k 140) exhibits temporal non-stationarity (the grid points in red in the following figure). Pointwise Mann-Kendall stationarity test (α = 5%) Jonathan Jalbert A spatio-temporal model for extreme precipitation simulated by a climate model 2016/04/27 5 / 22

7 Non-Stationarity The standard approach consists in modeling the maxima series as follows: M ik L GEV (µik, σ ik, ξ i ). (1) The non-stationarity is estimated only with the maxima series. Suppose that there exist sequences of constants ν ik and τ ik such that ( M ik = M ) ik ν ik : 1 k 140 ; (2) τ ik can be assumed identically distributed along index k for each grid point i. Therefore, M ik L GEV (µ i, σ i, ξ i ). (3) Jonathan Jalbert A spatio-temporal model for extreme precipitation simulated by a climate model 2016/04/27 6 / 22

8 Non-Stationarity The standard approach consists in modeling the maxima series as follows: M ik L GEV (µik, σ ik, ξ i ). (1) The non-stationarity is estimated only with the maxima series. Suppose that there exist sequences of constants ν ik and τ ik such that ( M ik = M ) ik ν ik : 1 k 140 ; (2) τ ik can be assumed identically distributed along index k for each grid point i. Therefore, M ik L GEV (µ i, σ i, ξ i ). (3) Jonathan Jalbert A spatio-temporal model for extreme precipitation simulated by a climate model 2016/04/27 6 / 22

9 Non-Stationarity Let Z ik be the vector of precipitation exceedences over the threshold u ik of year k at grid point i: and let Z ik = (Y ikl : Y ikl > u ik, 1 l 140) ; (4) ν ik = E(Z ik ) and τ 2 ik = Var(Z ik ). (5) The threshold has to be chosen in order that the transformation: M ik = M ik ν ik τ ik ; (6) removes the trend in the maxima series M i. The threshold is needed to isolate the trend in the tail from the one in the bulk of the distribution. Its definition does not rely on asymptotic convergence requirements as in the Peaks-Over-Threshold model. Jonathan Jalbert A spatio-temporal model for extreme precipitation simulated by a climate model 2016/04/27 7 / 22

10 Non-Stationarity Let Z ik be the vector of precipitation exceedences over the threshold u ik of year k at grid point i: and let Z ik = (Y ikl : Y ikl > u ik, 1 l 140) ; (4) ν ik = E(Z ik ) and τ 2 ik = Var(Z ik ). (5) The threshold has to be chosen in order that the transformation: M ik = M ik ν ik τ ik ; (6) removes the trend in the maxima series M i. The threshold is needed to isolate the trend in the tail from the one in the bulk of the distribution. Its definition does not rely on asymptotic convergence requirements as in the Peaks-Over-Threshold model. Jonathan Jalbert A spatio-temporal model for extreme precipitation simulated by a climate model 2016/04/27 7 / 22

11 Non-Stationarity We chose the 80th annual empirical quantiles at each grid point as the thresholds. Then, the non-stationarity hypothesis of the preprocessed maxima series was rejected for only 1.5% of the grid points (the grid points in red in the following figure). Pointwise Mann-Kendall stationarity test (α = 5%) for the preprocessed maxima series Jonathan Jalbert A spatio-temporal model for extreme precipitation simulated by a climate model 2016/04/27 8 / 22

12 Non-Stationarity Benefits of the proposed preprocessing approach are: if there exist constants a ik > 0 and b ik such that M ik b ik a ik L GEV (0, 1, ξi ); then there exists constants a ik > 0 and b ik such that M ik b ik a ik L GEV (0, 1, ξ i ). The model for the untransformed maxima is tractable: M ik L GEV (τik µ i + ν ik, τ ik σ i, ξ i ). (7) Jonathan Jalbert A spatio-temporal model for extreme precipitation simulated by a climate model 2016/04/27 9 / 22

13 Spatial dependence structure Following the idea of Cooley and Sain (2010), the spatial dependence is taken into account by modeling spatial variation in the GEV parameters mainly for two reasons: such a latent variable approach is very flexible; the local properties of extremal distributions (such as return levels) are well reproduced. However such an approach can neither model nor predict an event occurring simultaneously at several grid points. Jonathan Jalbert A spatio-temporal model for extreme precipitation simulated by a climate model 2016/04/27 10 / 22

14 Spatial latent model Local parameter estimations could definitely benefit from neighboring site values. Local estimates of the GEV location parameter. Jonathan Jalbert A spatio-temporal model for extreme precipitation simulated by a climate model 2016/04/27 11 / 22

15 Spatial latent model Local parameter estimations could definitely benefit from neighboring site values. Local estimates of the GEV scale parameter. Jonathan Jalbert A spatio-temporal model for extreme precipitation simulated by a climate model 2016/04/27 11 / 22

16 Spatial latent model Local parameter estimations could definitely benefit from neighboring site values. Local estimates of the GEV shape parameter. Jonathan Jalbert A spatio-temporal model for extreme precipitation simulated by a climate model 2016/04/27 11 / 22

17 Spatial latent model The most widely used spatial latent model is Gaussian field. For example, µ N n (η, Σ) ; (8) where µ = (µ 1,..., µ n ), η is a mean vector and Σ is a covariance matrix. Since the random variables µ lie on a regular lattice, a special case of Gaussian field is more appropriate: the Gaussian Markov random fields. Such a field inherits the Markov property: f [µi µ i =µ i ](µ i ) = f [µi µ δi =µ δi ](µ i ); (9) where δ i is the set of neighbors of grid point i. Jonathan Jalbert A spatio-temporal model for extreme precipitation simulated by a climate model 2016/04/27 12 / 22

18 Gaussian Markov random fields In Gaussian Markov random fields, the parametrization uses the precision matrix Q instead of the covariance matrix Σ because of the important simplification that comes from the conditional independence assumption: µ i µ j µ i, j q ij = 0; (10) where q ij is the element (i, j) of the precision matrix Q. ( ) The set of conditional distributions f [µi µ δi =µ δi ] : 1 i n must respect some constraints in order to yield a valid joint distribution: f [µi µ δi =µ δi ](µ i ) = N µ i η i 1 q ij (µ j η j ), 1 q ii q ii. (11) j δ i Jonathan Jalbert A spatio-temporal model for extreme precipitation simulated by a climate model 2016/04/27 13 / 22

19 Gaussian Markov random fields For a stationary field in space, the set of conditional distributions can be rewritten as follows: f [µi µ δi =µ δi ](µ i ) = N µ i η i + ρ (µ j η j ), ζ 2 ; (12) j δ i where 0 ρ 1 and ζ 2 > 0. Cor(µ i, µ j µ i, j = µ i, j ) = ρ, for j δ i. (13) It can be shown that marginal bivariate correlation coefficients between neighbors are necessarily less than : Cor(µ i, µ j ) 0.8, for j δ i. (14) The spatial correlation that can be modeled is therefore limited. 1 Besag and Kooperberg (1995) Jonathan Jalbert A spatio-temporal model for extreme precipitation simulated by a climate model 2016/04/27 14 / 22

20 Intrinsic Gaussian Markov random fields µ i Following Besag and Kooperberg (1995) and Rue and Held (2005), let n i = Card(δ i ), be the number of neighbors of grid point i. f [µi µ δi =µ δi ](µ i ) = N µ i 1 n i j δ i µ j, 1 ; κ µ n i (15) where κ µ > 0 is a precision parameter. The set of conditional distribution gives an improper joint distribution: ( f µ (µ) κ µ (n 1)/2 exp κ ) µ 2 µ W µ ; (16) where n i if j = i; w ij = 1 if j δ i ; (17) 0 otherwise. Jonathan Jalbert A spatio-temporal model for extreme precipitation simulated by a climate model 2016/04/27 15 / 22

21 Intrinsic Gaussian Markov random fields µ i Following Besag and Kooperberg (1995) and Rue and Held (2005), let n i = Card(δ i ), be the number of neighbors of grid point i. f [µi µ δi =µ δi ](µ i ) = N µ i 1 n i j δ i µ j, 1 ; κ µ n i (15) where κ µ > 0 is a precision parameter. The set of conditional distribution gives an improper joint distribution: ( f µ (µ) κ µ (n 1)/2 exp κ ) µ 2 µ W µ ; (16) where n i if j = i; w ij = 1 if j δ i ; (17) 0 otherwise. Jonathan Jalbert A spatio-temporal model for extreme precipitation simulated by a climate model 2016/04/27 15 / 22

22 Intrinsic Gaussian Markov random fields Under the Bayesian paradigm, the intrinsic Gaussian Markov random field used as a prior leads to a proper posterior if f κµ (κ µ ) is proper. E(µ i ) is undefined; Var(µ i ) =. (18) ( ) f [µi µ j ](δ) = N δ 0, 1 ; for j δ i. (19) κ µ The model is therefore: f [M ik (µi,σi,ξi )] (m ik ) L GEV (m ik µ i, σ i, ξ i ); f (µ,σ,ξ) (µ, σ, ξ) { (κ µ κ σ κ ξ ) n 1 2 exp κ µ 2 µ W µ κ σ 2 σ W σ κ } ξ 2 ξ W ξ ; Jonathan Jalbert A spatio-temporal model for extreme precipitation simulated by a climate model 2016/04/27 16 / 22

23 Intrinsic Gaussian Markov random fields Under the Bayesian paradigm, the intrinsic Gaussian Markov random field used as a prior leads to a proper posterior if f κµ (κ µ ) is proper. E(µ i ) is undefined; Var(µ i ) =. (18) ( ) f [µi µ j ](δ) = N δ 0, 1 ; for j δ i. (19) κ µ The model is therefore: f [M ik (µi,σi,ξi )] (m ik ) L GEV (m ik µ i, σ i, ξ i ); f (µ,σ,ξ) (µ, σ, ξ) { (κ µ κ σ κ ξ ) n 1 2 exp κ µ 2 µ W µ κ σ 2 σ W σ κ } ξ 2 ξ W ξ ; Jonathan Jalbert A spatio-temporal model for extreme precipitation simulated by a climate model 2016/04/27 16 / 22

24 Model fit κ ξ κ σ κ µ Iteration Jonathan Jalbert A spatio-temporal model for extreme precipitation simulated by a climate model 2016/04/27 17 / 22

25 Model fit A chain of length 6000 was generated where the first 1000 iterations were discarded as the burn-in period. It took less than 40 minutes of computation time on a 2.53 GHz processor. algorithms for sparse matrix; parallel MCMC. Jonathan Jalbert A spatio-temporal model for extreme precipitation simulated by a climate model 2016/04/27 18 / 22

26 Model fit According to the Deviance Information Criterion, the spatial modeling of ξ is relevant. Regional estimates of the GEV shape parameter. Jonathan Jalbert A spatio-temporal model for extreme precipitation simulated by a climate model 2016/04/27 19 / 22

27 Model fit 12 The grid point i containing the Richelieu River. 10 Estimated Quantiles Empirical Quantiles Jonathan Jalbert A spatio-temporal model for extreme precipitation simulated by a climate model 2016/04/27 20 / 22

28 Effective return levels Return level (mm) The grid point i containing the Richelieu River. 20-year 100-year Year Jonathan Jalbert A spatio-temporal model for extreme precipitation simulated by a climate model 2016/04/27 21 / 22

29 Conclusion The statistical model developed was well suited for climate model outputs, specifically for: transient time series; data that lie on a regular grid. The model s simplicity and intuitive interpretation along with its fast adjustment make it more appealing than Regional Frequency Analysis where uncertainty description is difficult. Nevertheless, the model could be enhanced by considering space-varying textures for the latent fields; by integrating several climate simulations for a better description of future climate uncertainty. Jonathan Jalbert A spatio-temporal model for extreme precipitation simulated by a climate model 2016/04/27 22 / 22

30 Conclusion The statistical model developed was well suited for climate model outputs, specifically for: transient time series; data that lie on a regular grid. The model s simplicity and intuitive interpretation along with its fast adjustment make it more appealing than Regional Frequency Analysis where uncertainty description is difficult. Nevertheless, the model could be enhanced by considering space-varying textures for the latent fields; by integrating several climate simulations for a better description of future climate uncertainty. Jonathan Jalbert A spatio-temporal model for extreme precipitation simulated by a climate model 2016/04/27 22 / 22