p.1 The Ensemble Kalman Filter: Theoretical formulation and practical implementation Geir Evensen Norsk Hydro Research Centre, Bergen, Norway Based on Evensen 23, Ocean Dynamics, Vol 53, No 4
p.2 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.3 The error covariance matrix Define ensemble covariances around the ensemble mean The ensemble mean is the bestguess. 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.4 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, forecast error covariance, defining Create an ensemble of observations &% $# " " with, the real observations,, a vector of observation noise,. ' '
p.6 Analysis scheme (2) Update each ensemble member according to % " where + *) ' ( ( Thus the update of the mean becomes " The posterior error covariance becomes,
p.7 Example: Lorenz model Application with the chaotic Lorenz model Illustrates properties with higly nonlinear dynamical models. From Evensen (1997), MWR.
p.8 EnKF solution 2 EnKF: Estimate 15 1 5 5 1 15 2 5 1 15 2 25 3 35 4
p.9 EnKF error variance 3 EnKF: Error variance 25 2 15 1 5 5 1 15 2 25 3 35 4
p.1 EnKS solution 2 EnKS: Estimate 15 1 5 5 1 15 2 5 1 15 2 25 3 35 4
p.11 EnKS error variance 3 EnKS: Error variance 25 2 15 1 5 5 1 15 2 25 3 35 4
p.12 Summary: Lorenz model The EnKF and EnKS works well with highly nonlinear dynamical models. There is no linearization in the evolution of error statistics. Methods using tangent linear or adjoint operators have problems with the Lorenz equations: limited by the predictability time, limited by the validity time of tangent linear operator. Can we expect the same to be true for high resolution ocean and atmosphere models?
3 2 1 / % % 5 / / 3 1 / 4 8 4 / 2 3 1 2 7 p.13 Analysis equation (1) Define the ensemble matrix / +.% 67 ) The ensemble mean is (defining 4 / The ensemble perturbations become, becomes The ensemble covariance matrix 8 8
" 7 ; % % / " % < # % % 7 p.14 Analysis equation (2), define 19 Given a vector of measurements % :% $# " " stored in / 3 91 " % " +:% The ensemble perturbations are stored in / % 3 91 / # + % # thus, the measurement error covariance matrix becomes < < '
8 ) < < + p.15 Analysis equation (3) The analysis equation can now be written *) ' + Defining the innovations previous definitions: and using 8 8 8 8 8 i.e., analysis expressed entirely in terms of the ensemble
Analysis equation (4) Define = 8 and > = = < <. Use 8, 4 /. Use 4 / = 5?. 8 = = = < < ) + 8, 4 / = >) + 8,, 4 / = >) + 8, = >) + 8 (1) p.16
= = p.17 Remarks on the analysis equation (1) Covariances only needed between observed variables at measurement locations ( ). 8 = never computed but indirectly used to determine. Analysis may be interpreted as: combination of forecast ensemble members, or, forecast pluss combination of covariance functions. Accuracy of analysis is determined by: the accuracy of, the properties of the ensemble error space.
p.18 Remarks on the analysis equation (2) For a linear model, any choice of will result in an analysis which is also a solution of the model. Filtering of covariance functions introduces nondynamical modes in the analysis.
p.19 Examples of ensemble statistics Taken from Haugen and Evensen (22), Ocean Dynamics. OGCM (MICOM) for the Indian Ocean. Assimilation of SST and SLA data.
Spatial correlations Latitude 25 2 15 1 5 5 SSTDP(1) 1.9.8.7.6.5.4.3.2.1.1.2.3.4.5.6.7.8.9 1 Latitude 25 2 15 1 5 5 SSHDP(1) 1.9.8.7.6.5.4.3.2.1.1.2.3.4.5.6.7.8.9 1 1 1 15 15 2 4 6 8 1 12 Longitude 2 4 6 8 1 12 Longitude Latitude 25 2 15 1 5 5 SSTSSH 1.9.8.7.6.5.4.3.2.1.1.2.3.4.5.6.7.8.9 1 Latitude 25 2 15 1 5 5 SSHUVEL 1.9.8.7.6.5.4.3.2.1.1.2.3.4.5.6.7.8.9 1 1 1 15 15 2 4 6 8 1 12 Longitude 2 4 6 8 1 12 Longitude p.2
Correlation functions SSH SSH SSH SST SSH DP SST SSH SST SST SST DP p.21
Correlation functions SSH SSH SSH SST SSH DP SST SSH SST SST SST DP p.22
Time Depth: Temperature 1 2 Depth 3 4 OBSERVED AND SIMULATED TEMPERATURE 32 31.5 31 3.5 3 29.5 29 28 27 26 25 24 23 22 21 2 19 18 17 16 15 14 13 12 5 1 12 14 16 18 2 22 24 26 Days p.23
p.24 Ensemble Kalman Smoother (EnKS) Derived in Evensen and van Leeuwen (2), MWR. 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 + ) : @DC 8BA @ C ML 8 KI E G 8 JI FEHG
+ O V O S V O + p.25 Time correlated model noise Most schemes assume white model noise in time. Augment model state to include time correlations UTC ) + NC ) PRQ SNC C ) NC C White noise when and.
W Results ( ) 4 3 2 1 1 2 3 4 1 2 3 4 5 6 7 8 9 1 Time p.26
W Results ( X) 4 3 2 1 1 2 3 4 1 2 3 4 5 6 7 8 9 1 Time p.27
r r d r ^ r a p.28 Parameter and bias estimation in model Y C Introduces poorly known parameter d ]&c p:q b k gon d b\]&c `[`s Z[Z e `[` d ^ ]&c m\] l k ij e hij ^ ] e fg e _ ]&c Z[Z a `[` \] ^ ] _ ] Z[Z Two cases. ^ut. ^wv a poorly known parameter with 1. 2.
p.29 Estimate and model error, EnKF 45 3 4 2 3 1 2 1 1 2 1 3 4 2 1 2 3 4 5 Time
p.3 Estimate and model error, EnKS 5 4 3 2 1 1 2 1 2 3 4 5 Time
p.31 Estimate, model error and, EnKF 45 3 4 2 3 1 2 1 1 2 1 3 4 2 1 2 3 4 5 Time
p.32 Estimate, model error and, EnKS 5 4 3 2 1 1 2 1 2 3 4 5 Time
p.33 Estimated and std dev 2 1 1 2 1 2 3 4 5 Time
p.34 Summary The EnKF and EnKS can handle time correlated model errors. The EnKF and EnKS can be used for parameter estimation by augmenting the model state with the unknown parameters.
p.35 Oil reservoar application The EnKF has been used to estimate permeability in a reservoar simulation model using production well data. The greatest uncertainty is the reservoar permeability and porosity. The well data consists of pressures and oil, gas and water production rates.
Estimated bias and std dev p.36
x y 7 p.37 Local analysis May be useful when is large or Analysis is computed grid point by grid point using only nearby measurements. Inverts many small matrices instead of one large. Different for each grid point, thus, allows us to reach a larger class of solutions. Suboptimal solution which introduces nondynamical modes..
z 8 + " 8 z 3 91 + ) < < 8 8 p.38 Nonlinear measurements Measurement equation # z Define ensemble of model prediction of the measurements / { % / % % The analysis then becomes { % { { { Analysis based on covariances between z and.
p.39 TOPAZ Operational ocean prediction system for the Atlantic and Arctic oceans. topaz.nersc.no Based on HYCOM State space is 26 5 unknowns. Ensemble size is 1. Assimilates SSH, SST, Ice concentration, and ARGO T&S profiles. Total of more than 1 measurements in each assimilation step. Uses local analysis as well as nonlinear measurements.
TOPAZ p.4