p.1 The Ensemble Kalman Filter: Theoretical formulation and practical implementation Geir Evensen Norsk Hydro Research Centre, Bergen, Norway Based on Evensen, Ocean Dynamics, Vol 5, No
p. The Ensemble Kalman Filter (EnKF) Represents error statistics using an ensemble of model states Evolves error statistics by ensemble integrations Variance minimizing analysis scheme operating on the ensemble Monte Carlo, low rank, error subspace method Converges to the Kalman Filter with increasing ensemble size Fully nonlinear error evolution, contrary to EKF Assumption of Gaussian statistics in analysis scheme
p. The error covariance matrix Define ensemble covariances around the ensemble mean The ensemble mean is the best-guess. The ensemble spread defines the error variance. The covariance is determined by the smoothness of the ensemble members. A covariance matrix can be represented by an ensemble of model states (not unique).
p. Dynamical evolution of error statistics Each ensemble member evolve according to the model dynamics which is expressed by a stochastic differential equation d The probability density then evolve according to Kolmogorov s equation! This is the fundamental equation for evolution of error statistics and can be solved using Monte Carlo methods.
# $ & p.5 Analysis scheme (1) Given an ensemble of model forcasts Create an ensemble of observations. %$ # # with the first guess observations, a vector of observation noise. The measurement error covariance matrix is $ $ &('
* *, + *, + ' ' p.6 Analysis scheme () Update each ensemble member according to # ) ' where /. - ' * ) ' This is equivalent to updating the mean * # ) ' The posterior error covariance becomes 1
7 6 5 9 7 5 8 8 6 7 5 6 ; p.7 Analysis of the Analysis scheme (1) Define the ensemble matrix :; ) The ensemble mean is (defining 8 The ensemble perturbations becomes 1 becomes The ensemble covariance matrix =
# ;? # ; p.8 Analysis equation (), define 5> Given a vector of measurements %$ # # stored in 7 > 5 # # The ensemble perturbations are stored in 7 > 5 $ $ $ thus, the measurement error covariance matrix becomes &
. p.9 Analysis equation () The analysis equation can now be written /. & Defining the innovations previous definitions: and using i.e., analysis expressed entirely in terms of the ensemble
Practical computation of analysis (1) Use pseudo inverse A B A C. A B. A Computational cost is: D ; to form, E D for eigenvalue decomposition. Unafordable for large D! p.1
Practical computation of analysis () Note that 9 F, thus Compute SVD, HG I, giving HG I HI G HG H G Computational cost is E D ; for SVD. p.11
Practical computation of analysis () The analysis equation can now be written G B. G The computation goes as follows with J K ; B. G 5 7> D J 5 7 D ; J L G 5 > 7 D ; J M L 5 7 D ; ; M 5 6 7 N ; ; All D ; computations reduced to D ;. p.1
9 1 1 O M 8 M M L p.1 Practical computation of analysis () The final update can be written as O thus, the analysis is a weakly nonlinear combination of the forecast ensemble. Note also ;
O O p.1 Remarks on the analysis equation (1) Covariances only needed between observed variables at measurement locations. Covariances never computed but indirectly used to determine. Analysis may be interpreted as: combination of ensemble members, or, forecast pluss combination of covariance functions. Covariances only needed to compute Accuracy of analysis is determined by the accuracy of the properties of the ensemble error space.
p.15 Remarks on the analysis equation () For a linear model, any choice of analysis which is also a solution of the model. Filtering of covariance functions introduces nondynamical modes in the analysis. O will result in an
O p.16 Local analysis A local analysis is computed grid point by grid point using only nearby measurements. Introduces nondynamical modes in the analysis. Different for each grid point. Allows us to reach a larger class of solutions.
P # P 7 R > 5. p.17 Nonlinear measurements Measurement equation $ P Define ensemble of model prediction of the measurements Q The analysis then becomes Q Q Q Q Analysis based on covariances between P and.
O p.18 Ensemble Kalman Smoother (EnKS) Starts with EnKF solution. Computes updates backward in time; sequentially for each measurement time, using covariances in time, no backward integrations. The analysis becomes for. : KVU TS K U _^ ][ WZY \[ XWZY
p.19 Some recent applications of the EnKF Haugan and Evensen (), Ocean Dynamics. Mitchell et al. (), MWR. Brusdal et al (), JMS. Natvik and Evensen (a,b), JMS. Keppenne and Rienecker (), JMS. DIADEM project TOPAZ project (topaz.nersc.no) MERSEA project
h Time correlated model noise Scalar model. gfu a a `U. `U e`u bdc U. U p.
i Results ( ) 1-1 - - - 1 5 6 7 8 9 1 Time p.1
i Results ( j) 1-1 - - - 1 5 6 7 8 9 1 Time p.
Estimate of model noise, EnKF 1 1-1 - -1 - - - 1 5 6 7 8 9 1 Time p.
Estimate of model noise, EnKS 1-1 - 1 5 6 7 8 9 1 Time p.
k Model forced by estimated model error k e `U bdc U. U ml ml p.5
Š Š z Parameter and bias estimation in model n U Introduces poorly known parameter z ryx ˆ~ w z wqryx upu opo { upu z s ryx qr ƒ { s r { ~} { t ryx opo v upu qr s r t r opo p.6
p.7 Estimate and model error, EnKF 5 1 1-1 - -1 - - - 1 5 Time
p.8 Estimate and model error, EnKS 5 1-1 - 1 5 Time
p.9 Estimate, model error and bias, EnKF 5 1 1-1 - -1 - - - 1 5 Time
p. Estimate, model error and bias, EnKS 5 1-1 - 1 5 Time
p.1 Estimated bias and std dev 1-1 - 1 5 Time