Robust method for EnKF in the presence of observation outliers/Multivariate localization methods for EnKF Mikyoung Jun Department of Statistics Texas A&M University May 19, 2015 M. Jun (TAMU) Multivariate Localization/Robust Methods for EnKF May 19 1 / 32
Setup ◮ y t ∈ R p : observations at time t ◮ x t ∈ R n : unobservable state at time t ◮ Nonlinear system eq. x t = M ( x t − 1 ) + e t ◮ Observation eq. y t = H t x t + ǫ t ◮ System error e t ∼ N n ( 0 , Q t ) ◮ Observation error ǫ t ∼ N p ( 0 , R t ) ◮ System and observation errors are uncorrelated. ◮ M , H t , Q t , R t assumed to be known. M. Jun (TAMU) Multivariate Localization/Robust Methods for EnKF May 19 2 / 32
Definitions ◮ M -member background ensemble x b = { x b ( k ) : k = 1 , . . . , M } ∈ R n × M x b = 1 � M ◮ background mean ¯ k = 1 x b ( k ) M ◮ ensemble-based estimate of the background error covariance P b = k = 1 X b ( k ) [ X b ( k ) ] T , where X b ( k ) = x b ( k ) − ¯ � M 1 x b M − 1 x a = ¯ x b + K ( y − H ¯ x b ) ◮ analysis mean ¯ ◮ analysis covariance P a = ( I − KH ) P b ◮ Kalman gain K = P b H T ( HP b H + R ) − 1 M. Jun (TAMU) Multivariate Localization/Robust Methods for EnKF May 19 3 / 32
Part 1: Multivariate Localization for EnKF Outline Part 1: Multivariate Localization for EnKF Part 2: Robust EnKF in the presence of observation outliers M. Jun (TAMU) Multivariate Localization/Robust Methods for EnKF May 19 4 / 32
Part 1: Multivariate Localization for EnKF Motivation Joint work with Soojin Roh , Istvan Szunyogh , and Marc Genton ◮ Localization: Schur (elementwise) product of P b and a localization matrix from a compactly supported correlation function ρ ( · ) ◮ In statistics, localization function is called “taper” or compactly supported covariance function; needs to be positive definite ◮ For multivariate state variables, current practice is to apply the same localization function to each “block” of P b ◮ Does it not matter? ( K = P b H T ( HP b H + R ) − 1 ) ◮ Kang et al. (2011, JGR) zeroes out covariances between physically unrelated variables (not about positive-definiteness) ◮ M. Jun (TAMU) Multivariate Localization/Robust Methods for EnKF May 19 5 / 32
Part 1: Multivariate Localization for EnKF Motivation ◮ Problem of rank deficiency: � L � L ◮ Localization matrix L L ◮ Problem is more serious when P b ij ’s are “significantly” non-zero ◮ We need ρ ( · ) = { ρ ij ( · ) } i , j = 1 ,..., N : matrix-valued correlation (positive definite) function, N : number of state variables – multivariate version of Gaspari-Cohn functions? ◮ In statistics literature, not many known such “valid” ρ (parametric) functions are available yet M. Jun (TAMU) Multivariate Localization/Robust Methods for EnKF May 19 6 / 32
Part 1: Multivariate Localization for EnKF One simple idea ◮ Use ρ ij ( · ) = β ij · ρ ( · ) with | β ij | < 1, | β ji | < 1, and β ii = β jj = 1. � 1 � β is positive-definite and of full rank for any β with | β | < 1. ◮ β 1 ◮ For ρ , use any localization functions in Gaspari and Cohn (1999). ◮ Example: 8 − 1 4 ( | d | / c ) 5 + 1 2 ( d / c ) 4 + 5 8 ( | d | / c ) 3 − 5 3 ( d / c ) 2 + 1 , 0 ≤| d |≤ c ; > < ρ ( d ; c ) = (1) 12 ( | d | / c ) 5 − 1 1 2 ( d / c ) 4 + 5 8 ( | d | / c ) 3 + 5 3 ( d / c ) 2 − 5 ( | d | / c )+ 4 − 2 3 c / | d | , c ≤| d |≤ 2 c ; > 0 , : 2 c ≤| d | and β 11 = β 22 = 1, 0 ≤ β ij ≤ 1. ◮ This multivariate localization function is separable in the sense that � 1 � β multivariate component (in the above example, ) β 1 and localization function (in the above example, ρ ) are factored M. Jun (TAMU) Multivariate Localization/Robust Methods for EnKF May 19 7 / 32
Part 1: Multivariate Localization for EnKF Another idea ◮ Use one of a few multivariate compactly supported correlation functions available in statistics literature. ◮ e.g. Bivariate Askey function (Porcu et al. 2012) � ν + µ ij � 1 − d ρ ij ( d ; ν, c ) = β ij , c + ◮ c > 0, µ 12 = µ 21 ≥ 1 2 ( µ 11 + µ 22 ) , ν ≥ [ 1 2 s ] + 2, and s is space dimension. � Γ( 1 + µ 12 ) Γ( 1 + ν + µ 11 )Γ( 1 + ν + µ 22 ) ◮ | β ij | ≤ , β ii = β jj = 1 Γ( 1 + ν + µ 12 ) Γ( 1 + µ 11 )Γ( 1 + µ 22 ) ◮ | β ij | ≤ 1 if µ 11 = µ 22 . M. Jun (TAMU) Multivariate Localization/Robust Methods for EnKF May 19 8 / 32
Part 1: Multivariate Localization for EnKF Gaspari−Cohn ● 0.8 Askey ( ν =1) covariance Askey ( ν =2) Askey ( ν =3) ● 0.4 0.0 0 10 20 30 40 50 distance M. Jun (TAMU) Multivariate Localization/Robust Methods for EnKF May 19 9 / 32
Part 1: Multivariate Localization for EnKF Experiment with bivariate Lorenz Model ◮ X k and Y j , k are equally spaced on a latitude circle ( j = 1 , . . . , J and k = 1 , . . . , K ). ◮ With boundary conditions X k ± K = X K , Y j , k ± K = Y j , k , Y j − J , k = Y j , k − 1 , and Y j + J , k = Y j , k + 1 , J dX k � = − X k − 1 ( X k − 2 − X k + 1 ) − X k − ( ha / b ) Y j , k + F , dt j = 1 dY j , k = − abY j + 1 , k ( Y j + 2 , k − Y j − 1 , k ) − aY j , k + ( ha / b ) X k dt M. Jun (TAMU) Multivariate Localization/Robust Methods for EnKF May 19 10 / 32
Part 1: Multivariate Localization for EnKF Experiment details ◮ True model states generated by a long time step integration of the model ◮ Initialize ensembles by adding Gaussian noise to the true state ◮ We discard first 3000 time steps ◮ Simulated observations are generated by adding mean zero noise (variance 0.02 for X and 0.005 for Y ) to the truth ◮ 20 ensemble members are used (we tested 40 ensembles as well) ◮ Covariance inflation of 1.015 ◮ RMSE calculated using last 1000 time steps and we repeat 50 times to produce boxplots M. Jun (TAMU) Multivariate Localization/Robust Methods for EnKF May 19 11 / 32
Part 1: Multivariate Localization for EnKF Bivariate Lorenz Model ◮ 36 variables of X , 360 variables of Y , a = 10, b = 10, h = 2 6 X Y 4 state 2 X Y 0 0 50 100 150 200 250 300 350 variable locations longitudinal profiles M. Jun (TAMU) Multivariate Localization/Robust Methods for EnKF May 19 12 / 32
Part 1: Multivariate Localization for EnKF Experiment set up ◮ Two scenarios for observation 1 Observe 20 % of X and 90 % of Y at locations where X is not observed. 2 Fully observe X and Y ◮ Four localization schemes S1 No localization S2 No localization and let P b 12 = P b 21 = 0. S3 localize P b 11 and P b 22 , but let P b 12 = P b 21 = 0. S4 localize P b 11 , P b 22 , P b 12 , P b 21 M. Jun (TAMU) Multivariate Localization/Robust Methods for EnKF May 19 13 / 32
Part 1: Multivariate Localization for EnKF Localization (S4) 1 Gaspari-Cohn function: ρ ij ( d ; c ) = β ij ρ ( d ; c ) , i , j = 1 , 2, where − 1 4 ( | d | / c ) 5 + 1 2 ( d / c ) 4 + 5 8 ( | d | / c ) 3 − 5 3 ( d / c ) 2 + 1 , 0 ≤| d |≤ c ; ρ ( d ; c ) = 3 c / | d | , 12 ( | d | / c ) 5 − 1 1 2 ( d / c ) 4 + 5 8 ( | d | / c ) 3 + 5 3 ( d / c ) 2 − 5 ( | d | / c )+ 4 − 2 c ≤| d |≤ 2 c ; 0 , 2 c ≤| d | and β 11 = β 22 = 1, 0 ≤ β ij ≤ 1. (support=2 c ) 2 Bivariate Askey function � ν + µ ij � 1 − | d | ρ ij ( d ; c ) = β ij , i , j = 1 , 2 c + with µ 11 = 0, µ 22 = 2, µ ij = 1, ν = 3, and β 11 = β 22 = 1, 0 ≤ β ij ≤ 0 . 7. (support= c ) M. Jun (TAMU) Multivariate Localization/Robust Methods for EnKF May 19 14 / 32
Part 1: Multivariate Localization for EnKF Results for X in scenario 1 support 50 support 70 4 4 Gaspari−Cohn Askey 3 3 RMSE RMSE ● 2 2 ● ● ● ● ● ● ● 1 1 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 ● 0 S1 S2 S3 S4 S4 S4 S4 S4 S4 S1 S2 S3 S4 S4 S4 S4 S4 S4 5e −3 1e −2 0.1 5e −3 1e −2 0.1 0.4 0.7 1 0.4 0.7 1 support 100 support 160 4 4 3 3 RMSE RMSE ● 2 ● ● 2 ● ● ● ● ● ● ● ● ● ● ● ● 1 ● 1 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 0 S1 S2 S3 S4 S4 S4 S4 S4 S4 S1 S2 S3 S4 S4 S4 S4 S4 S4 5e −3 1e −2 0.1 5e −3 1e −2 0.1 0.4 0.7 1 0.4 0.7 1 M. Jun (TAMU) Multivariate Localization/Robust Methods for EnKF May 19 15 / 32
Part 1: Multivariate Localization for EnKF Results for X in scenario 1 ◮ S1 vs S2: ignoring cross-covariance is better than not doing localization ◮ S3 is worse than S1 ◮ S4 with β = 0 . 01 performs the best regardless of localization radius ◮ Askey seems to be better than the Gaspari-Cohn function M. Jun (TAMU) Multivariate Localization/Robust Methods for EnKF May 19 16 / 32
Recommend
More recommend
