12 th International EnKF workshop Estimating model evidence using ensemble-based data assimilation with localization The model selection problem Sammy Metref , Juan Ruiz, Alexis Hannart, Alberto Carrassi, Marc Bocquet and Michael Ghil Project DADA June 12 th , 2017
A comparison of geopotential heights at 500hPa for 4 short range models
Outline Model evidence and data assimilation Contextual Model Evidence CME formulation The Domain Localized CME Localization in DA Localization and CME Numerical experiments Low-order atmospheric model Primitive Equations atmospheric model Conclusions
Model evidence and data assimilation The Domain Localized CME Numerical experiments Conclusions Model evidence For a model M simulating an unknown process such that: x k = M ( x k − 1 ) , (1) where M : R M → R M . And for an ideal infinite set of observations of the same process, y K : = { y K , y K − 1 , ..., y 1 , y 0 , ..., y −∞ } , such that: y k = H k ( x k ) + ǫ k , (2) where H k : R M → R d and ǫ k represents observation error. Model evidence (marginal likelihood of the observations) � p ( y K : |M ) = d x p ( y K : | x ) p ( x ) . (3)
Model evidence and data assimilation The Domain Localized CME Numerical experiments Conclusions Model evidence For a model M simulating an unknown process such that: x k = M ( x k − 1 ) , (1) where M : R M → R M . And for an ideal infinite set of observations of the same process, y K : = { y K , y K − 1 , ..., y 1 , y 0 , ..., y −∞ } , such that: y k = H k ( x k ) + ǫ k , (2) where H k : R M → R d and ǫ k represents observation error. Model evidence (marginal likelihood of the observations) � p ( y K : |M ) = d x p ( y K : | x ) p ( x ) . (3) Defined as a “climatological” model evidence
Model evidence and data assimilation The Domain Localized CME Numerical experiments Conclusions Model evidence using data assimilation We rather define a contextual model evidence i.e. conditioned on the present • p ( y K : |M ) → p ( y K : 1 | y 0 : ) [ M is dropped for clarity] In the context of present time, we marginalize over x 0 and not over x The Contextual Model Evidence (CME) � p ( y K : 1 | y 0 : ) = d x 0 p ( y K : 1 | x 0 ) p ( x 0 | y 0 : ) (4)
Model evidence and data assimilation The Domain Localized CME Numerical experiments Conclusions Model evidence using data assimilation We rather define a contextual model evidence i.e. conditioned on the present • p ( y K : |M ) → p ( y K : 1 | y 0 : ) [ M is dropped for clarity] In the context of present time, we marginalize over x 0 and not over x The Contextual Model Evidence (CME) � p ( y K : 1 | y 0 : ) = d x 0 p ( y K : 1 | x 0 ) p ( x 0 | y 0 : ) (4) with • the likelihood of the observations • the posterior density (state estimation DA product)
Model evidence and data assimilation The Domain Localized CME Numerical experiments Conclusions Estimating the CME using DA methods • ensemble Kalman filter • 4D ensemble methods (En-4D-Var/IEnKS) Carrassi et al. (2017)
Model evidence and data assimilation The Domain Localized CME Numerical experiments Conclusions Estimating the CME using DA methods • ensemble Kalman filter • 4D ensemble methods (En-4D-Var/IEnKS) Carrassi et al. (2017) Conclusions • Accurate estimation of the CME using DA • Accuracy related to DA method’s sophistication • Yet, not proportional ⇒ We use the EnKF formulation
Model evidence and data assimilation The Domain Localized CME Numerical experiments Conclusions CME formulation The CME’s EnKF formulation [ y 0 : is dropped for clarity] � � K � − 1 ( 2 π ) − d 2 | Σ k | − 1 2 [ y k − H k ( x f k )] T Σ − 1 k [ y k − H k ( x f 2 exp p ( y K : 1 ) ≈ k )] (5) k = 1 Hannart et al. (2016) ; Carrassi et al. (2017) with Σ k = H k P f k H T k + R k where H k : observation operator at time k , P f k : prior error covariance matrix at time k , H k : its linearization. R k : observation error covariance matrix,
Model evidence and data assimilation The Domain Localized CME Numerical experiments Conclusions CME formulation The CME’s EnKF formulation [ y 0 : is dropped for clarity] � � K � − 1 ( 2 π ) − d 2 | Σ k | − 1 2 [ y k − H k ( x f k )] T Σ − 1 k [ y k − H k ( x f 2 exp p ( y K : 1 ) ≈ k )] (5) k = 1 Hannart et al. (2016) ; Carrassi et al. (2017) with Σ k = H k P f k H T k + R k where H k : observation operator at time k , P f k : prior error covariance matrix at time k , H k : its linearization. R k : observation error covariance matrix, The objective of this study Problem in high dimension: Ensemble DA methods suffer from sampling errors in high dimension and are usually used with localization ⇒ Crucial to consider how to deal with localization in the CME formulation
Model evidence and data assimilation The Domain Localized CME Numerical experiments Conclusions Domain localization • Seperate analysis: DA performed for each model gridpoint s ∈ Γ • Box car: Only the neighboring obs. are used in the analysis i.e. with y | s , H | s , R | s restricted to a disk around s of radius ρ loc • Tapering: a (diagonal) localization matrix L applied such that − 1 � | s = L ◦ R − 1 | s = ( R − 1 | s ) i , j · ( L ) i , j R (6) ( L ) i , i is equal to 1 if i = s and decreases to 0 outside of the disk ρ loc S
Model evidence and data assimilation The Domain Localized CME Numerical experiments Conclusions Domain localization • Seperate analysis: DA performed for each model gridpoint s ∈ Γ • Box car: Only the neighboring obs. are used in the analysis i.e. with y | s , H | s , R | s restricted to a disk around s of radius ρ loc • Tapering: a (diagonal) localization matrix L applied such that − 1 � | s = L ◦ R − 1 | s = ( R − 1 | s ) i , j · ( L ) i , j R (6) ( L ) i , i is equal to 1 if i = s and decreases to 0 outside of the disk ρ loc S − 1 ⇒ Derive the CME for each gridpoint using y | s , H | s , � R | s
Model evidence and data assimilation The Domain Localized CME Numerical experiments Conclusions DL-CME At each gridpoint s ∈ Γ , it is possible to derive � � K � p ( y K : 1 | s ) ≈ d x k p ( y k | s | x k − 1 ) p ( x k − 1 | y k − 1 : | s ) d x 0 p ( y 1 | s | x 0 ) p ( x 0 | y 0 : ) k = 2 Local CME � � � K ( 2 π ) − � − 1 k ) T � d Σ k | − 1 2 | � Σ − 1 2 exp 2 ( y k | s − H k | s x f k ( y k | s − H k | s x f p ( y K : 1 | s ) ≈ k ) (7) k = 1 with � k | s + � R k | s and � Σ k = H k | s P f k H T d the size of y k | s .
Model evidence and data assimilation The Domain Localized CME Numerical experiments Conclusions DL-CME At each gridpoint s ∈ Γ , it is possible to derive � � K � p ( y K : 1 | s ) ≈ d x k p ( y k | s | x k − 1 ) p ( x k − 1 | y k − 1 : | s ) d x 0 p ( y 1 | s | x 0 ) p ( x 0 | y 0 : ) k = 2 Local CME � � � K ( 2 π ) − � − 1 k ) T � d Σ k | − 1 2 | � Σ − 1 2 exp 2 ( y k | s − H k | s x f k ( y k | s − H k | s x f p ( y K : 1 | s ) ≈ k ) (7) k = 1 with � k | s + � R k | s and � Σ k = H k | s P f k H T d the size of y k | s . Euristic global estimator Domain localized CME (DL-CME) �� � � p ( y K : 1 ) = exp w ( s ) ln { p ( y | s ) } (8) , s ∈ Γ with w ( s ) , scalar weights inversely proportional to the localization radius.
Model evidence and data assimilation The Domain Localized CME Numerical experiments Conclusions CME for model selection Two models: M 0 and M 1 and their respective model evidences: p 0 ( y ) = p ( y K : 1 | y 0 : , M 0 ) and p 1 ( y ) = p ( y K : 1 | y 0 : , M 1 ) Model selection indicator with global and domain localized CME: • G-CME: ∆ p ( M 0 , M 1 ) = ln { p 1 ( y ) } − ln { p 0 ( y ) } > 0, if M 1 correct • DL-CME: ∆ � p ( M 0 , M 1 ) = ln { � p 1 ( y ) } − ln { � p 0 ( y ) } > 0, if M 1 correct
Model evidence and data assimilation The Domain Localized CME Numerical experiments Conclusions CME for model selection Two models: M 0 and M 1 and their respective model evidences: p 0 ( y ) = p ( y K : 1 | y 0 : , M 0 ) and p 1 ( y ) = p ( y K : 1 | y 0 : , M 1 ) Model selection indicator with global and domain localized CME: • G-CME: ∆ p ( M 0 , M 1 ) = ln { p 1 ( y ) } − ln { p 0 ( y ) } > 0, if M 1 correct • DL-CME: ∆ � p ( M 0 , M 1 ) = ln { � p 1 ( y ) } − ln { � p 0 ( y ) } > 0, if M 1 correct The scope of the following experiments is to compare the G-CME’s and the DL-CME’s model selection abilities
Model evidence and data assimilation The Domain Localized CME Numerical experiments Conclusions L95 - Model selection problem Lorenz-95 model d x i d t = ( x i + 1 − x i − 2 ) x i − 1 − x i + F , (9) for i = 1 , ..., M = 40 and F represents the external forcing. The models The observations • M 1 : F ≡ F 1 = 8 M 1 traj. perturbed: ǫ ∈ N ( 0 , 1 ) • M 0 : F ≡ F 0 varying Obs. error cov. matrix: R = I 40 for T = 10 5 DA cycles Obs. grid: ∆ t = 0 . 05 and H k = I 40 DA setup LETKF - 10 members Localization radius: ρ loc = 5 (tuned for M 0 ) Inflation: tuned for each model
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