Application of Ensemble Kalman Filter in numerical models. UTM, CSIC, Barcelona, Spain 2. SMOS BEC, Barcelona, Spain 3. NURC, La Spezia, Italy 4

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1 UM Application of Ensemle Kalman Filter in numerical models Joaquim Ballarera,2, Baptiste Mourre 3, Sofia Kalaroni 2,4, Nina Hoareau 2,4, Marta Umert 4 UM, CSIC, Barcelona, Spain 2 SMOS BEC, Barcelona, Spain 3 NURC, La Spezia, Italy 4 ICM, CSIC, Barcelona, Spain

2 2 2 o o J ( x) y Hx R y Hx x x B x x he minimization of the cost function can e estimated iteratively (variational approach) or analytically (Kalman filter). a x x B H R H H R y Hx If the time evolution of the system is estimated y a numerical model, symolized y M, and if the model is lineal: x M x a k k, k k B M A M Q k k, k k k, k he time evolution of the error covariance of the state of the system depends uniquely of the initial error covariance.

3 a x x B H R H H R y Hx A B I H B H R H H B Kalman (960) x M x a k k, k k B M A M Q k k, k k k, k he Kalman Filter provides an uniased estimation of the state of the system y performing the weighted average of the prior information and the oservations. It is optimal if the system is linear and if we know the error of the model and the oservational error. Kalman receives the National Medal of Science (2008)

4 a x x B H R H H R y Hx A B I H B H R H H B x M x a k k, k k B M A M Q k k, k k k, k Practical prolems:. he size of the covariance matrices. Let n e the size of the state vector. If n=0 6, the covariance matrix takes 8 in doule precision. 2. he calculation of the time evolution of the error covariance if the model is nonlinear. Cognitive prolems: 3. he initial error covariance, A o. 4. he model error Q. x M x 0, 0, o 0, a o B M A M Q

5 Square Filters, an approach to ensure the symmetry of the error covariance, provide an introduction to the Ensemle approach: If we factorize, By defining, A E E k k k B M E E M Q k k, k k k k, k Ek Mk, k Ek then, B E E Q k k k With this definition, we have the advantage that the nonlinear model can e applied to every column of the E k matrix. he numer of model simulations required to advance the system is equal to the numer of columns of the matrix E k.

6 An alternative approach, the eigenvalue factorization of the error covariance matrix: B E Λ E B Projecting the KF equations onto the eigenvectors of the covariance matrix: A B Λ Λ HE R HE x k k E M (+ (0,)) E k A K E Λ HE R x x K y Hx A a o E Λ E A M x a k B E Λ E A k k k k k

7 In some applications, the ensemle is kept constant. It is given as the EOFs of the system (eigenvectors of the covariance matrix). Some directions may e sustituted adaptively. he cost is strongly reduced. (47 memers). Λ Λ y, H, E, x x k B B o k k k o k A B k k o k o Λ Λ HE R HE A o k o k K E Λ HE R x x K y Hx a o M x a k Ballarera-Poy et al. (200)

8 he ensemle can e allowed to evolve in time. he time evolution of the ensemle memers provide new directions of correction depending on the dynamics of the system. (30 memers) Hackert et al. (2007)

9 Even if assimilation is not performed, the ensemle can e used to measure the relative sensitivity to different parameters, forcing fields or oundary conditions. he error covariance matrices are now interpreted as sensitivity covariances. he larger the ensemle, the more roust the sensitivity estimation. x ( t) M( x, p, t) o x ( t) M( x, p, t) o x ( t) M( x, p, t) 2 o 2 x ( t) M( x, p, t) 2 o 2 Smaller sensitivity of the solution to the parameter p Larger sensitivity of the solution to the parameter p

10 Model SSS response Introduction Aplications Limits of EnKF Future Lines Wind stress Precipitation Open Boundary Data he impact of the inaccuracies in various forcing on the errors of salinity has een estimated using an ensemle method. Forcing and parameter changes Mourre et al. (2008)

11 Similarly, the ensemle covariance may e used as alternative approach to estimate correlation scales, that can e used in Ojective Analysis. Mourre et al. (2009)

12 prey parameters predator parameters An additional application of Ensemle Kalman filters is to estimate the parameters of the model. his approach is ased on the state augmentation x =(x,p), with dp/dt = days Figure 9. Prey parameter identification using 500 ensemles, predator parameter identification using 00 ensemles. ime is in days. days Umert and Ballarera-Poy (in preparation)

13 Some physical features as, for example atmosphere convection, may e virtually decorrelated from other variales of the system. In this case, the covariance etween them is zero. However, ensemle methods use a finite-size sample covariance, which merely approaches the true covariance. Most of the time, the ensemle covariance will have an infinite noise to signal ratio. It has een found that this situation degrades the data assimilation (Ballarera-Poy et al., 2009). Ballarera-Poy et al. (2009)

14 Constrained ensemle Oservations Unconstrained ensemle Ballarera-Poy et al. (2009)

15 With evolving and fixed ensemles (and with and without) nudging, we are assimilating Argo, SS and SSH. Role of SSS. Salinity profile at 35W-23N, 22 nd June 2005 EOF emperature LON=25W 5N 45N Kalaroni et al. (in preparation) Hoareau et al. (in preparation)