Linear ensemble transform filters: A unified perspective on ensemble Kalman and particle filters Yuan Cheng & Sebastian Reich University of Potsdam and University of Reading EnKF workshop 2014 Bergen, 23 June 2014 Yuan Cheng & Sebastian Reich (UP and UoR) Data assimilation 23 June 2014 1 / 29
Introduction Stochastic processes (here discrete time) Z 0 : N = ( Z 0 , Z 1 , . . . , Z N ) May depend on parameters, i.e. Z 0 : N | λ . Subject them to partial observations Y 1 : K = ( Y 1 , Y 2 , . . . , Y K } in order to assess and calibrate models. K < N (prediction), N = K (filtering), K > N (smoothing). Conditional PDFs π Z 0 : N ( z 0 : N | y 1 : K , λ ) or π Λ ( λ | y 1 : K ) through Bayesian inference and Monte Carlo methods. Yuan Cheng & Sebastian Reich (UP and UoR) Data assimilation 23 June 2014 2 / 29
Introduction Stochastic processes (here discrete time) Z 0 : N = ( Z 0 , Z 1 , . . . , Z N ) May depend on parameters, i.e. Z 0 : N | λ . Subject them to partial observations Y 1 : K = ( Y 1 , Y 2 , . . . , Y K } in order to assess and calibrate models. K < N (prediction), N = K (filtering), K > N (smoothing). Conditional PDFs π Z 0 : N ( z 0 : N | y 1 : K , λ ) or π Λ ( λ | y 1 : K ) through Bayesian inference and Monte Carlo methods. Yuan Cheng & Sebastian Reich (UP and UoR) Data assimilation 23 June 2014 2 / 29
Introduction Stochastic processes (here discrete time) Z 0 : N = ( Z 0 , Z 1 , . . . , Z N ) May depend on parameters, i.e. Z 0 : N | λ . Subject them to partial observations Y 1 : K = ( Y 1 , Y 2 , . . . , Y K } in order to assess and calibrate models. K < N (prediction), N = K (filtering), K > N (smoothing). Conditional PDFs π Z 0 : N ( z 0 : N | y 1 : K , λ ) or π Λ ( λ | y 1 : K ) through Bayesian inference and Monte Carlo methods. Yuan Cheng & Sebastian Reich (UP and UoR) Data assimilation 23 June 2014 2 / 29
Introduction McKean data analysis cycle A typical scenario Shadow or track an unknown reference solution z n + 1 = Ψ( z n ref ) , ref accessible through partial and noisy observations y n obs = h ( z n ref ) + ξ n , n ≥ 1 . We only know that z 0 ref is drawn from a random variable Z 0 . Yuan Cheng & Sebastian Reich (UP and UoR) Data assimilation 23 June 2014 3 / 29
Introduction McKean data analysis cycle Ensemble prediction relies on M independent realizations i = Z 0 ( ω i ) (MC or quasi-MC) from the initial Z 0 and associated z 0 trajectories z n + 1 = Ψ( z n i ; λ ) , n ≥ 0 , i = 1 , . . . , M . i i = z n + 1 Analysis step transforms the forecast ensemble { z f } i into an analysis ensemble { z a i } using Bayes theorem : π Z a ( z | y obs ) = π Y ( y obs | z ) π Z f ( z ) . π Y ( y obs ) Continue ensemble prediction with { z n + 1 = z a i } . i Yuan Cheng & Sebastian Reich (UP and UoR) Data assimilation 23 June 2014 4 / 29
Introduction McKean data analysis cycle Ensemble prediction relies on M independent realizations i = Z 0 ( ω i ) (MC or quasi-MC) from the initial Z 0 and associated z 0 trajectories z n + 1 = Ψ( z n i ; λ ) , n ≥ 0 , i = 1 , . . . , M . i i = z n + 1 Analysis step transforms the forecast ensemble { z f } i into an analysis ensemble { z a i } using Bayes theorem : π Z a ( z | y obs ) = π Y ( y obs | z ) π Z f ( z ) . π Y ( y obs ) Continue ensemble prediction with { z n + 1 = z a i } . i Yuan Cheng & Sebastian Reich (UP and UoR) Data assimilation 23 June 2014 4 / 29
Introduction McKean data analysis cycle Summary of the McKean approach to the analysis step : PDFs RVs MC Ref.: Del Moral (2004), CJC & SR (2013), YC & SR (2014). Yuan Cheng & Sebastian Reich (UP and UoR) Data assimilation 23 June 2014 5 / 29
Introduction McKean data analysis cycle Parametric statistics : The Gaussian choice (A) Fit a Gaussian N (¯ z f , P f ) to the forecast ensemble { z f i } and assume that h is linear. Then the analysis is also Gaussian N (¯ z a , P a ) with z a = ¯ z f − K ( H ¯ z f − y obs ) , P a = P f − KHP f . ¯ Here K denotes the Kalman gain matrix . Yuan Cheng & Sebastian Reich (UP and UoR) Data assimilation 23 June 2014 6 / 29
Introduction McKean data analysis cycle Non-parametric statistics : Empirical measures (B) Use the empirical measure M π f ( z ) = 1 � δ ( z − z f i ) M i = 1 to define the analysis measure M � π a ( z ) = w i δ ( z − z f i ) i = 1 with importance weights � − 1 2 ( h ( z f i ) − y obs ) T R − 1 ( h ( z f � exp i ) − y obs ) w i = � � � M − 1 2 ( h ( z f j ) − y obs ) T R − 1 ( h ( z f j = 1 exp j ) − y obs ) Yuan Cheng & Sebastian Reich (UP and UoR) Data assimilation 23 June 2014 7 / 29
Introduction McKean data analysis cycle Implementation of the McKean approach then either requires coupling two Gaussians (approach A) or two empirical measures (approach B). Approach A: ensemble Kalman filters (Evensen, 2006) Approach B: particle filters (Doucet et al, 2001). Optimal couplings in the sense of minimizing some cost function are known in both cases (CJC & SR, 2013). We next provide a unifying mathematical framework in form of linear ensemble transform filters (LETFs) (YC & SR, 2014). Yuan Cheng & Sebastian Reich (UP and UoR) Data assimilation 23 June 2014 8 / 29
Introduction McKean data analysis cycle Implementation of the McKean approach then either requires coupling two Gaussians (approach A) or two empirical measures (approach B). Approach A: ensemble Kalman filters (Evensen, 2006) Approach B: particle filters (Doucet et al, 2001). Optimal couplings in the sense of minimizing some cost function are known in both cases (CJC & SR, 2013). We next provide a unifying mathematical framework in form of linear ensemble transform filters (LETFs) (YC & SR, 2014). Yuan Cheng & Sebastian Reich (UP and UoR) Data assimilation 23 June 2014 8 / 29
Introduction McKean data analysis cycle Implementation of the McKean approach then either requires coupling two Gaussians (approach A) or two empirical measures (approach B). Approach A: ensemble Kalman filters (Evensen, 2006) Approach B: particle filters (Doucet et al, 2001). Optimal couplings in the sense of minimizing some cost function are known in both cases (CJC & SR, 2013). We next provide a unifying mathematical framework in form of linear ensemble transform filters (LETFs) (YC & SR, 2014). Yuan Cheng & Sebastian Reich (UP and UoR) Data assimilation 23 June 2014 8 / 29
Linear ensemble transform filters The analysis steps of an ensemble Kalman filter (EnKF) as well as the resampling step of a particle filter are of the form M � z a z f j = i s ij , i = 1 where { z f i } M i = 1 is the forecast ensemble and { z a i } M i = 1 is the analysis ensemble . (i) The matrix S = { s ij } ∈ R M × M depends on y obs and the forecast ensemble. (ii) S can be the realization of a matrix-valued RV S : Ω → R M × M , i.e. S = S ( ω ) . Yuan Cheng & Sebastian Reich (UP and UoR) Data assimilation 23 June 2014 9 / 29
Linear ensemble transform filters The analysis steps of an ensemble Kalman filter (EnKF) as well as the resampling step of a particle filter are of the form M � z a z f j = i s ij , i = 1 where { z f i } M i = 1 is the forecast ensemble and { z a i } M i = 1 is the analysis ensemble . (i) The matrix S = { s ij } ∈ R M × M depends on y obs and the forecast ensemble. (ii) S can be the realization of a matrix-valued RV S : Ω → R M × M , i.e. S = S ( ω ) . Yuan Cheng & Sebastian Reich (UP and UoR) Data assimilation 23 June 2014 9 / 29
Linear ensemble transform filters Optimal transportation The ensemble transform particle filter (ETPF) (SR, 2013) is determined by a coupling T ∈ R M × M between the discrete random variables Z f : Ω → { z f 1 , . . . , z f P [ z f M } with i ] = 1 / M and Z a : Ω → { z f 1 , . . . , z f P [ z f M } with i ] = w i , respectively. A coupling T has to satisfy t ij ≥ 0, M M � � t ij = 1 / M , t ij = w i . i = 1 j = 1 Yuan Cheng & Sebastian Reich (UP and UoR) Data assimilation 23 June 2014 10 / 29
Linear ensemble transform filters Optimal transportation The ensemble transform particle filter (ETPF) (SR, 2013) is determined by a coupling T ∈ R M × M between the discrete random variables Z f : Ω → { z f 1 , . . . , z f P [ z f M } with i ] = 1 / M and Z a : Ω → { z f 1 , . . . , z f P [ z f M } with i ] = w i , respectively. A coupling T has to satisfy t ij ≥ 0, M M � � t ij = 1 / M , t ij = w i . i = 1 j = 1 Yuan Cheng & Sebastian Reich (UP and UoR) Data assimilation 23 June 2014 10 / 29
Linear ensemble transform filters Optimal transportation Chosing a coupling that maximizes the correlation between forecast and analysis leads to an optimal transport problem with cost � � z f i − z f j � 2 t ij . J ( { t ij } = i , j Leads to the celebrated Monge-Kantorovitch problem : � z f − z a � 2 � Z f Z a ( z f , z a ) = arg π ∗ � inf π Zf Za ( z f , z a ) ∈ Π( π Zf ,π Za ) E Z f Z a as M → ∞ (McCann, 1996, SR, 2013). Let us denote the minimize by T ∗ , then the ETPF is given by M � z a z f i t ∗ j = M ij . i = 1 Yuan Cheng & Sebastian Reich (UP and UoR) Data assimilation 23 June 2014 11 / 29
Recommend
More recommend
