Ensemble Kalman Filter Methods Dusanka Zupanski CIRAColorado State

Ensemble Kalman Filter Methods Dusanka Zupanski CIRA/Colorado State University Fort Collins, Colorado NOAA/NESDIS Cooperative Research Program (Co. RP) Third Annual Science Symposium 15 -16 August 2006, Hilton Fort Collins, CO Collaborators: M. Zupanski, L. Grasso, M. De. Maria, 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 NNG 05 GD 15 G, NASA NNG 04 GI 25 G, NOAA Grant NA 17 RJ 1228, and Do. D Grant DAAD 19 -02 -2 -0005 P 00007 Computational support from NASA Halem and Columbia super-computers, CIRA and Atmospheric Science Dept. Linux clusters Dusanka Zupanski, CIRA/CSU Zupanski@CIRA. colostate. edu

OUTLINE Ø Kalman filter, ensemble Kalman filter and variational methods Ø Maximum Likelihood Ensemble Filter (MLEF) Ø KF vs. 3 d-var, as special cases of the MLEF Ø 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) Ø Conclusions and future research directions Dusanka Zupanski, CIRA/CSU Zupanski@CIRA. colostate. edu

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

Typical En. KF 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

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

Maximum Likelihood Ensemble Filter (MLEF) (Zupanski 2005; Zupanski and Zupanski 2006) Ø 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. Ø Non-linear full-rank MLEF, without updating of Pf = 3 d-var Comparisons of KF and 3 d-var within the same algorithm. Dusanka Zupanski, CIRA/CSU Zupanski@CIRA. colostate. edu

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

KF vs. 3 d-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

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. 3 d-var does not capture this variability (straight line). q true (g kg-1) Small ensemble size (10 ens), even though not perfect, captures main data signals. Vertical levels T true (K) Data assimilation cycles Dusanka Zupanski, CIRA/CSU Zupanski@CIRA. colostate. edu

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, Background T_brightness, Observations T_brightness, Analysis DOF=49. 39, end ineffective use of the observations (the analysis is close to the background). Dusanka Zupanski, CIRA/CSU Zupanski@CIRA. colostate. edu

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, Observations T_brightness, Analysis DOF=14. 73, and effective use of the observations (the analysis is close to the truth). Dusanka Zupanski, CIRA/CSU Zupanski@CIRA. colostate. edu

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

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

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

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

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