Dusanka Zupanski CIRA/Colorado State University Fort Collins, Colorado Ensemble Kalman Filter Guest Lecture at AT 753: Atmospheric Water Cycle 21 April 2006, CSU/ATS Dept., Fort Collins, CO Dusanka Zupanski, CIRA/CSU Acknowledgements: M. Zupanski, C. Kummerow, S. Denning, and M. Uliasz, CSU A. Hou and S. Zhang, NASA/GMAO

 Why Ensemble Data Assimilation?  Kalman filter and Ensemble Kalman filter  Maximum likelihood ensemble filter (MLEF)  Examples of MLEF applications  Future research directions Dusanka Zupanski, CIRA/CSU OUTLINE

Why Ensemble Data Assimilation? Dusanka Zupanski, CIRA/CSU Three main reasons :  Need for optimal estimate of the atmospheric state + verifiable uncertainty of this estimate;  Need for flow-dependent forecast error covariance matrix; and  The above requirements should be applicable to most complex atmospheric models (e.g., non-hydrostatic, cloud-resolving, LES).

Example 1: Fronts Example 2: Hurricanes (From Whitaker et al., THORPEX web-page) Benefits of Flow-Dependent Background Errors

Are there alternatives? Dusanka Zupanski, CIRA/CSU Two good candidates:  4d-var method: It employs flow-dependent forecast error covariance, but it does not propagate it in time.  Kalman Filter (KF): It does propagate flow- dependent forecast error covariance in time, but it is too expensive for applications to complex atmospheric models. EnKF is a practical alternative to KF, applicable to most complex atmospheric models.   A bonus benefit: EnKF does not use adjoint models!

Typical EnKF Dusanka Zupanski, CIRA/CSU Forecast error Covariance P f (ensemble subspace) DATA ASSIMILATION Observations First guess Optimal solution for model state x=(T,u,v,f, ) ENSEMBLE FORECASTING Analysis error Covariance P a (ensemble subspace) INFORMATION CONTENT ANALYSIS T b,u b,v b,f b, , ,  Hessian preconditioning Non-Gaussian PDFs Maximum Likelihood Ensemble Filter

Dusanka Zupanski, CIRA/CSU Data Assimilation Equations Equations in model space: Prior (forecast) error covariance of x (assumed known): - Dynamical model for model state evolution (e.g., NWP model) - Model state vector of dim Nstate ; - Model error vector of dim Nstate - Dynamical model for state dependent model error Model error covariance (assumed known): - Mathematical expectation ; GOAL: Combine Model and Data to obtain optimal estimate of dynamical state x - Time step index

- Observations vector of dim Nobs ; Observation error covariance, includes also representatives error (assumed known): - Observation operator Equations in data space: - Observation error Data Assimilation Equations Dusanka Zupanski, CIRA/CSU - Time step index (denoting observation times) Data assimilation should combine model and data in an optimal way. Optimal solution z can be defined in terms of optimal initial conditions x a (analysis), model error w, and empirical parameters , , .

Approach 1: Kalman filterExtended Kalman filterEnKF Approach 1: Optimal solution (e.g., analysis x a ) = Minimum variance estimate, or conditional mean of Bayesian posterior probability density function (PDF) (e.g., Kalman filter; Extended Kalman filter; EnKF) - PDF Dusanka Zupanski, CIRA/CSU How can we obtain optimal solution? Two approaches are used most often: Extended Kalman filterEnsemble Kalman filter For non-liner M or H the solution can be obtained employing Extended Kalman filter, or Ensemble Kalman filter. Kalman filter Assuming liner M and H and independent Gaussin PDFs  Kalman filter solution (e.g., Jazwinski 1970) x a is defined as mathematical expectation (i.e., mean) of the conditional posterior p ( x|y ), given observations y and prior p ( x ).

Dusanka Zupanski, CIRA/CSU Approach 2: variationalMLEF Approach 2: Optimal solution (e.g., analysis x a ) = Maximum likelihood estimate, or conditional mode of Bayesian posterior p ( x | y ) (e.g., variational methods; MLEF) For independent Gaussian PDFs, this is equivalent to minimizing cost function J: Solution can be obtained (with ideal preconditioning) in one iteration for liner H and M. Iterative solution for non-linear H and M : - Preconditioning matrix = inverse Hessian of J x a = Maximum of posterior p ( x | y ), given observations and prior p ( x ).

VARIATIONAL MLEF Milija Zupanski, CIRA/CSU Ideal Hessian Preconditioning

Dusanka Zupanski, CIRA/CSU x mode x mean x p(x)p(x) Non-Gaussian x mode = x mean x p(x)p(x) Gaussian MEAN vs. MODE For Gaussian PDFs and linear H and M results of all methods [KF, EnKF (with enough ensemble members), and variational] should be identical, assuming the same P f, R, and y are used in all methods. Minimum variance estimate= Maximum likelihood estimate!

KF, EnKF, 4d-var, all created equal? Does this really happen?!?

TEST RESULTS EMPLOYING A LINEAR MODEL AND GAUSSIAN PDFs (M.Uliasz) (D. Zupanski)

Dusanka Zupanski, CIRA/CSU - Optimal estimate of x (analysis) Kalman filter solution Analysis step: - Background (prior) estimate of x - Analysis (posterior) error covariance matrix ( Nstate x Nstate ) Forecast step: ; - Update of forecast error covariance - Kalman gain matrix ( Nstate x Nobs ) Often neglected

Ensemble Kalman Filter (EnKF) solution EnKF as first introduced by Evensen (1994) as a Monte Carlo filter. Analysis solution defined for each ensemble member i : Mean analysis solution: Analysis error covariance in ensemble subspace: Analysis step: Analysis ensemble perturbations: Sample analysis covariance Equations given here following Evensen (2003)

Ensemble Kalman Filter (EnKF) Forecast step: Forecast error covariance calculated using ensemble perturbations: Ensemble forecasts employing a non-linear model M ; Sample forecast covariance Non-linear forecast perturbations

There are many different versions of EnKF  Monte Carlo EnKF (Evensen 1994; 2003)  EnKF (Houtekamer et al. 1995; 2005; First operational version)  Hybrid EnKF (Hamill and Snyder 2000)  EAKF (Anderson 2001)  ETKF (Bishop et al. 2001)  EnSRF (Whitaker and Hamill 2002)  LEKF (Ott et al. 2004)  MLEF (Zupanski 2005; Zupanski and Zupanski 2006) Minimum variance solution Maximum likelihood solution Why maximum likelihood solution? It is more adequate for employing non- Gaussian PDFs (e.g., Fletcher and Zupanski 2006).

Current status of EnKF applications  EnKF is operational in Canada, since January 2005 (Houtekamer et al.). Results comparable to 4d-var.  EnKF is better than 3d-var (experiments with NCEP T62 GFS) - Whitaker et al., THORPEX presentation ).  Very encouraging results of EnKF in application to non- hydrostatic, cloud resolving models (Zhang et al., Xue et al.).  Very encouraging results of EnKF for ocean (Evensen et al.), climate (Anderson et al.), and soil hydrology models (Reichle et al.). Theoretical advantages of ensemble-based DA methods are getting confirmed in an increasing number of practical applications.

Examples of MLEF applications Dusanka Zupanski, CIRA/CSU

Dusanka Zupanski, CIRA/CSU - Dynamical model for standard model state x Maximum Likelihood Ensemble Filter - Dynamical model for model error (bias) b - Dynamical model for empirical parameters  Define augmented state vector z Find optimal solution (augmented analysis) z a by minimizing J (MLEF method): And augmented dynamical model F ,. (Zupanski 2005; Zupanski and Zupanski 2006)

Both the magnitude and the spatial patterns of the true bias are successfully captured by the MLEF. 40 Ens 100 Ens True  R Cycle 1 Cycle 3 Cycle 7 Bias estimation: Respiration bias  R, using LPDM carbon transport model (Nstate=1800, Nobs=1200, DA interv=10 days) Domain with larger bias (typically land) Domain with smaller bias (typically ocean)

Dusanka Zupanski, CIRA/CSU Information measures in ensemble subspace Shannon information content, or entropy reduction Degrees of freedom (DOF) for signal (Rodgers 2000): - information matrix in ensemble subspace of dim Nens x Nens - are columns of Z - control vector in ensemble space of dim Nens - model state vector of dim Nstate >>Nens Errors are assumed Gaussian in these measures. (Bishop et al. 2001; Wei et al. 2005; Zupanski et al. 2006, subm. to MWR) - eigenvalues of C for linear H and M

Dusanka Zupanski, CIRA/CSU GEOS-5 Single Column Model: DOF for signal (Nstate=80; Nobs=80, seventy 6-h DA cycles, assimilation of simulated T,q observations) DOF for signal varies from one analysis cycle to another due to changes in atmospheric conditions. 3d-var approach does not capture this variability. Small ensemble size (10 ens), even though not perfect, captures main data signals. T obs (K)q obs (g kg -1 ) Data assimilation cycles Vertical levels RMS Analysis errors for T, q: ens ~ 0.45K; 0.377g/kg 20ens ~ 0.28K; 0.265g/kg 40ens ~ 0.23K; 0.226g/kg 80ens ~ 0.21K; 0.204g/kg No_obs ~ 0.82K; 0.656g/kg

Non-Gaussian (lognormal) MLEF framework: CSU SWM (Randall et al.) Beneficial impact of correct PDF assumption – practical advantages Dusanka Zupanski, CIRA/CSU Cost function derived from posterior PDF ( x-Gaussian, y-lognormal): Lognormal additional nonlinear term Normal (Gaussian) Courtesy of M. Zupanski

Future Research Directions  Covariance inflation and localization need further investigations: Are these techniques necessary?  Model error and parameter estimation need further attention: Do we have sufficient information in the observations to estimate complex model errors?  Information content analysis might shed some light on DOF of model error and also on the necessary ensemble size.  Non-Gaussian PDFs have to be included into DA (especially for cloud variables).  Characterize error covariances for cloud variables.  Account for representativeness error. Dusanka Zupanski, CIRA/CSU

References for further reading Anderson, J. L., 2001: An ensemble adjustment filter for data assimilation. Mon. Wea. Rev., 129, 2884–2903. Fletcher, S.J., and M. Zupanski, 2006: A data assimilation method for lognormally distributed observational errors. Q. J. Roy. Meteor. Soc. (in press). Evensen, G., 1994: Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics. J. Geophys. Res., 99, (C5), Evensen, G., 2003: The ensemble Kalman filter: theoretical formulation and practical implementation. Ocean Dynamics. 53, Hamill, T. M., and C. Snyder, 2000: A hybrid ensemble Kalman filter/3D-variational analysis scheme. Mon. Wea. Rev., 128, 2905–2919. Houtekamer, Peter L., Herschel L. Mitchell, 1998: Data Assimilation Using an Ensemble Kalman Filter Technique. Mon. Wea. Rev., 126, Houtekamer, Peter L., Herschel L. Mitchell, Gerard Pellerin, Mark Buehner, Martin Charron, Lubos Spacek, and Bjarne Hansen, 2005: Atmospheric data assimilation with an ensemble Kalman filter: Results with real observations. Mon. Wea. Rev., 133, Ott, E., and Coauthors, 2004: A local ensemble Kalman filter for atmospheric data assimilation. Tellus., 56A, 415–428. Tippett, M. K., J. L. Anderson, C. H. Bishop, T. M. Hamill, and J. S. Whitaker, 2003: Ensemble square root filters. Mon. Wea. Rev., 131, 1485–1490. Whitaker, J. S., and T. M. Hamill, 2002: Ensemble data assimilation without perturbed observations. Mon. Wea. Rev., 130, 1913–1924. Zupanski D. and M. Zupanski, 2006: Model error estimation employing an ensemble data assimilation approach. Mon. Wea. Rev. 134, Zupanski, M., 2005: Maximum likelihood ensemble filter: Theoretical aspects. Mon. Wea. Rev., 133, 1710–1726 Dusanka Zupanski, CIRA/CSU

Dusanka Zupanski, CIRA/CSU Thank you.