Ensemble Kalman Filter Methods

1. Ensemble Kalman Filter Methods Dusanka Zupanski CIRA/Colorado State University Fort Collins, Colorado NOAA/NESDIS Cooperative Research Program (CoRP) Third Annual Science Symposium 15-16 August 2006, Hilton Fort Collins, CO Collaborators: M. Zupanski, L. Grasso, M. DeMaria, S. Denning, M. Uliasz, R. Lokupityia, C. Kummerow, G. Carrio, T. Vonder Haar, D. Randall, CSU A. Hou and S. Zhang, NASA/GMAO Grant support: NASA Grant NNG05GD15G, NASA NNG04GI25G,NOAA Grant NA17RJ1228, and DoD Grant DAAD19-02-2-0005 P00007 Computational support from NASA Halem and Columbia super-computers, CIRA and Atmospheric Science Dept. Linux clusters Dusanka Zupanski, CIRA/CSU Zupanski@CIRA.colostate.edu

2. OUTLINE • Kalman filter, ensemble Kalman filter and variational methods • Maximum Likelihood Ensemble Filter (MLEF) • KF vs. 3d-var, as special cases of the MLEF • Information content analysis of data (e.g., TRMM, GPM, GOES-R) • NASA/GEOS-5 single column model (complex, 1-d model) • CSU/RAMS non-hydrostatic model (complex, 3-d model) • Conclusions and future research directions Dusanka Zupanski, CIRA/CSU Zupanski@CIRA.colostate.edu

3. Typical KF Forecast error Covariance Pf (full-rank space) Observations First guess DATA ASSIMILATION Analysis error Covariance Pa (full-rank space) Optimal solution for model state x=(T,u,v,w, q, …) LINEARISED FORECAST MODEL Dusanka Zupanski, CIRA/CSU Zupanski@CIRA.colostate.edu

4. Typical EnKF Forecast error Covariance Pf (reduced-rank ensemble subspace) Observations First guess DATA ASSIMILATION Analysis error Covariance Pa (reduced-rank ensemble subspace) Optimal solution for model state x=(T,u,v,w, q, …) NON-LINEAR ENSEMBLE OF FORECAST MODELS Dusanka Zupanski, CIRA/CSU Zupanski@CIRA.colostate.edu

5. Typical variational method Prescribed Forecast error Covariance Pf (full-rank space) Observations First guess DATA ASSIMILATION Analysis error Covariance Pa (full-rank space) Optimal solution for model state x=(T,u,v,w, q, …) NON-LINEAR FORECAST MODEL Dusanka Zupanski, CIRA/CSU Zupanski@CIRA.colostate.edu

6. Maximum Likelihood Ensemble Filter (MLEF)(Zupanski 2005; Zupanski and Zupanski 2006) • Linear full-rank MLEF =KF (Full-rank means Nens=Nstate) ; for =1 MLEF= KF valid under Gaussian error assumption. For Non-Gaussian case, ask M. Zupanski, S. Fletcher and collaborators. • Non-linear full-rank MLEF, without updating of Pf=3d-var  Comparisons of KF and 3d-var within the same algorithm. Dusanka Zupanski, CIRA/CSU Zupanski@CIRA.colostate.edu

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

8. KF vs. 3d-var: GEOS-5 Single Column Model (Nstate=80; Nobs=40, Nens=80, seventy 6-h DA cycles, assimilation of simulated T,q observations) Dusanka Zupanski, CIRA/CSU Zupanski@CIRA.colostate.edu

9. GEOS-5 Single Column Model: DOF for signal(Nstate=80; Nobs=40, Nens=80 or Nens=10, seventy 6-h DA cycles, assimilation of simulated T,q observations) Inadequate Pf Large Pf DOF for signal varies from one analysis cycle to another due to changes in atmospheric conditions. 3d-var does not capture this variability (straight line). T true (K) q true (g kg-1) Small ensemble size (10 ens), even though not perfect, captures main data signals. Vertical levels Data assimilation cycles Dusanka Zupanski, CIRA/CSU Zupanski@CIRA.colostate.edu

10. Is this applicable to CSU/RAMS? (Nstate=2138400; Nobs=5940, Nens=50, assimilation of simulated GOES-R 10.35 brightness temperature observations, hurricane Lili case) Inadequate Pf (ensemble members far from the truth): T_brightness, Analysis T_brightness, Background T_brightness, Observations DOF=49.39, end ineffective use of the observations (the analysis is close to the background). Dusanka Zupanski, CIRA/CSU Zupanski@CIRA.colostate.edu

11. Is this applicable to CSU/RAMS? (Nstate=2138400; Nobs=5940, Nens=50, assimilation of simulated GOES-R 10.35 brightness temperature observations, hurricane Lili case) Adequate Pf (ensemble members close to the truth): T_brightness, Background T_brightness, Analysis T_brightness, Observations DOF=14.73, and effective use of the observations (the analysis is close to the truth). Dusanka Zupanski, CIRA/CSU Zupanski@CIRA.colostate.edu

12. Conclusions and Future Research Directions Conclusions • Flow-dependent forecast error covariance is of fundamental importance for both analysis and information measures. • Ensemble-based data assimilation methods employ flow-dependent forecast error covariance. • Information matrix defined in ensemble subspace is practical to calculate in many applications due to small ensemble size. Future work • Evaluate DOF in the presence of model error. • Apply the information content analysis to WRF model and real satellite observations. Dusanka Zupanski, CIRA/CSU Zupanski@CIRA.colostate.edu

13. Thank you. Dusanka Zupanski, CIRA/CSU Zupanski@CIRA.colostate.edu

14. Is the increased amount of information a simple consequence of a large magnitude of Pf? Large Pf Inadequate Pf Large Pf Dusanka Zupanski, CIRA/CSU Zupanski@CIRA.colostate.edu

15. The GEOS-5 results indicated the following impact of Pf Inadequate Pf  Increased information content of data, but poor analysis quality (ineffective use of observed information)  Adequate Pf Reduced information content of data, but good analysis quality (effective use of observed information) Dusanka Zupanski, CIRA/CSU Zupanski@CIRA.colostate.edu

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