Addressing the nonlinear problem of low order clustering in deterministic filters by using mean-preserving non-symmetric solutions of the ETKF

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1 Addressing the nonlinear problem of low order clustering in deterministic filters by using mean-preserving non-symmetric solutions of the ETKF Javier Amezcua, Dr. Kayo Ide, Dr. Eugenia Kalnay 1

2 Outline Mean-Preserving Non-Symmetric Ensemble Transform Kalman Filter (MPNS-ETKF) Ensemble Square Root Filters (EnSRFs) Solutions of the ETKF The low order clustering problem Using the MPNS-ETKF to solve this problem Simple univariate nonlinear model Lorenz 1963 model Lorenz 1996 and the importance of symmetry in R-localization 2

3 Stochastic vs. deterministic filters Stochastic EnKF (Evensen, 1994; Burgers et al, 1996; Houtekamer and Mitchell, 1998) Perturbed observations, the covariance equation fulfilled only statistically. Deterministic (EnSRF: Tippett et al, 2003) Serial (Whitaker and Hamill, 2002) EAKF (Anderson, 2001) ETKF (Bishop et al, 2001), LETKF (Hunt et al, 2007) X a = X b W a, W a R M M 3

4 Ensemble Transform Kalman Filter The original formulation (Bishop et al, 2001) W a = C I + Γ 1 2 The columns of this matrix contains eigenvectors. Hence, it is an orthonormal matrix. CΓC T = YbT R 1 Y b M 1 Is a diagonal matrix with eigenvalues. The one-sided ETKF fulfills the covariance equation, but we also require the mean of the perturbations to be zero. X a 1 = 0 This solution doesn t fulfill this in general (Livings et al, 2007) and leads to bad performance (Sakov and Oke, 2008). 4

5 ETKFs that preserves the mean Spherical simplex (Wang et al, 2004) W a = C I + Γ 1 2C T Symmetric square root in the LETKF (Hunt et al, 2007) W a = I + CΓC T 1 2 Where CΓC T = Y bt R 1 Y b M 1. In this [symmetric] case W a is the closest to I; hence, X a is the closest to X b (Ott et al, 2004). 5

6 General solution to the ETKF A general ETKF is: W a = C I + Γ 1 2S T where S R M M must be orthonormal and meanpreserving. There are cheap ways to construct S (Bishop, pers. comm.). A particular form (which will be important in R-localization) is to rotate the symmetric solution: W a = C I + Γ 1 2C T Σ T where Σ T Σ = I and Σ T 1 = 1 Some choices: S = I gives the one sided S = C gives LETKF 6

7 Outline Mean-Preserving Non-Symmetric Ensemble Transform Kalman Filter (MPNS-ETKF) Ensemble Square Root Filters (EnSRFs) Solutions of the ETKF The low order clustering problem Using the MPNS-ETKF to solve this problem Simple univariate nonlinear model Lorenz 1963 model Lorenz 1996 and the importance of symmetry in R-localization 7

8 Kalman filtering The filtering problem linear x t = f x t 1 +w t y t o = h x t + v t Gaussian The conditions aren t usually perfectly fulfilled. How well they are approximated depends upon: The length of the assimilation window. The magnitude of the model error covariance and the observational error covariance. 8

9 The nonlinear effect of ensemble clustering Using the Ikeda model, Lawson and Hansen (2004) realized that the performance of the EnSRF (they used the serial) is more sensitive to nonlinearity. EnKF EnSRF M-1 members 1 member Background ensemble, Analysis ensemble Lawson and Hansen (2004) 9

10 The nonlinear effect of ensemble clustering The higher order moments are the most affected. EnKF linear EnSRF linear The analysis RMSE was not affected much. The spread is conserved. EnKF nonlinear EnSRF nonlinear The stochastic EnKF is more robust, but it introduces more sampling error (Whitaker and Hamill, 2002). 10 Lawson and Hansen (2004)

11 Outline Mean-Preserving Non-Symmetric Ensemble Transform Kalman Filter (MPNS-ETKF) Ensemble Square Root Filters (EnSRFs) Solutions of the ETKF The low order clustering problem Using the MPNS-ETKF to solve this problem Simple univariate nonlinear model Lorenz 1963 model Lorenz 1996 and the importance of symmetry in R-localization 11

12 A univariate quadratic model Consider the following nonlinear ODE (similar to Anderson, 2010): x = x + bx 2 An Euler Forward discretization leads to the following map: x t+1 = 1 + x t + b x 2 t Time step for the integration: Nonlinearity coefficient 0,0.5 Let s consider the unstable fixed point x = 0 to be the truth. 12

13 Progressive deformation of the ensemble due to quadratic evolution b=0 b=0.2 T I M E x b=0.5 x b=0.7 x x 13

14 Experimental settings x t+1 = 1 + x t + b x t 2 Several combinations of settings were chosen, the results shown here use: M = 20 ensemble members. σ 2 = 1, observational error. The initial ensemble was centered in x = 0 with p 0 = 1. The observation/assimilation frequency was varied. The degree on nonlinearity was varied. 14

15 Observation/assimilation window: 2 ETKF b = 0 x x ETKF b = 0.2 When using the symmetric ETKF, the clustering appears as a consequence of the strong nonlinearity time 15

16 Observation/assimilation window: 2 x MPNS-ETKF b = 0 MPNS-ETKF b = 0.2 With the nonsymmetric ETKF, the clustering doesn t appear even in the presence of strong nonlinearity. x WHY? time 16

17 ensemble values ensemble values What happens in the assimilation? Nonlinear case with obs/assim window: 5 ETKF MPNS-ETKF Background ensemble Analysis ensemble assimilation cycle There is a constant scrambling of the ensemble that prevents any deformation (due to nonlinearities) in the ensemble to remain. 17

18 Experiments with Lorenz 1963 The system: x 1 = σ x 2 x 1 x 2 = x 1 r x 3 x 2 x 3 = x 1 x 2 bx 3 Solved with RK4 t = 0.01 Identical twin experiment observing all variables 18

19 skewness Higher order moments: skewness The figure is very similar for background and analysis. The figure for kurtosis is omitted since it is redundant. R = 2I, 8 steps R = 2I, 24 steps 19 S NS S NS S NS S NS M = 3 M = 10 M = 25 M = 40 S NS S NS S NS S NS M = 3 M = 10 M = 25 M = 40

20 How important is symmetry in R- Localization? In the LETKF, the symmetric solution guarantees a smooth transition in the analysis values among neighboring gridpoints. Using a different S (rotation matrix) for each gridpoint introduces noise to the system. S 1 S 2 S 3 Will using a fixed S F be enough? Each local analysis would be oriented in a different direction. NOT AUTOMATICALLY! 20

21 Symmetry and R-localization How smooth is the transition for the weights among neighboring gridpoints? Let s see for gridpoint 15. LETKF: W a = C I + Γ 1 2C T One-sided ETKF W a = C I + Γ 1 2 MPNSETKF W a = C I + Γ 1 T 2S F MPNSETKF W a = C I + Γ 1 T 2S variable The symmetric solution automatically guarantees smoothness! 21

22 Symmetry and R-localization The same is true for all gridpoints. LETKF: W a = C I + Γ 1 2C T One-sided ETKF W a = C I + Γ 1 2 MPNSETKF W a = C I + Γ 1 2S F T MPNSETKF W a = C I + Γ 1 2S variable T 22

23 Nonsymmetry and R-Localization An R-localized analysis must be locally symmetric. It can be globally rotated afterwards. W a = C I + Γ 1 2 C T Σ T S F T X a global = X b globalσ T This rotation can still bring benefits in the higher order moments. Skewness reduction 23

24 Conclusions Non-symmetric, in particular randomly rotated EnSRFs are a good alternative to stochastic filters when nonlinearity causes ensemble clustering. The process can be considered a type of resampling. Special care must be paid when using R-Localization. The symmetry is needed to form the local analyses, but the global analysis can be rotated afterwards. 24