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Ensemble Kalman Filter Methods

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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 Outline slide 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 DAAD P00007 Computational support from NASA Halem and Columbia super-computers, CIRA and Atmospheric Science Dept. Linux clusters Dusanka Zupanski, CIRA/CSU Milija Zupanski CIRA/CSU

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 Outline slide Dusanka Zupanski, CIRA/CSU Milija Zupanski CIRA/CSU

3 Typical KF DATA ASSIMILATION LINEARISED FORECAST MODEL 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 Milija Zupanski CIRA/CSU

4 Typical EnKF DATA ASSIMILATION NON-LINEAR ENSEMBLE OF FORECAST MODELS
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 Milija Zupanski CIRA/CSU

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 Milija Zupanski CIRA/CSU

6 Comparisons of KF and 3d-var within the same algorithm.
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 Milija Zupanski CIRA/CSU

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 Outline slide 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 Milija Zupanski CIRA/CSU

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 Milija Zupanski CIRA/CSU

9 3d-var does not capture this variability (straight line).
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). Outline slide 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 Milija Zupanski CIRA/CSU

10 Inadequate Pf (ensemble members far from the truth):
Is this applicable to CSU/RAMS? (Nstate= ; 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, Analysis Outline slide T_brightness, Observations DOF=49.39, end ineffective use of the observations (the analysis is close to the background). Dusanka Zupanski, CIRA/CSU Milija Zupanski CIRA/CSU

11 Adequate Pf (ensemble members close to the truth):
Is this applicable to CSU/RAMS? (Nstate= ; 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 Outline slide T_brightness, Observations DOF=14.73, and effective use of the observations (the analysis is close to the truth). Dusanka Zupanski, CIRA/CSU Milija Zupanski CIRA/CSU

12 Conclusions and Future Research Directions
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. Outline slide Dusanka Zupanski, CIRA/CSU Milija Zupanski CIRA/CSU

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

14 Large Pf Inadequate Pf Large Pf
Is the increased amount of information a simple consequence of a large magnitude of Pf? Large Pf Outline slide Inadequate Pf Large Pf Dusanka Zupanski, CIRA/CSU Milija Zupanski CIRA/CSU

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) Outline slide Adequate Pf Reduced information content of data, but good analysis quality (effective use of observed information) Dusanka Zupanski, CIRA/CSU Milija Zupanski CIRA/CSU

16 Benefits of Flow-Dependent Background Errors
(From Whitaker et al., THORPEX web-page) Example 1: Fronts Here are a couple of contrived, but illuminating examples of how flow-dependant background error covariances can really help. #1 3dvar doesn’t know ob is in a front. #2 3dvar doesn’t know ob is in a hurricane. Example 2: Hurricanes Milija Zupanski CIRA/CSU

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