1. MODEL ERROR ESTIMATION EMPLOYING DATA ASSIMILATION METHODOLOGIES Dusanka Zupanski Cooperative Institute for Research in the Atmosphere Colorado State University Fort Collins, CO 80523-1375 ATS Colloquium series 29 January 2004 ftp://ftp.cira.colostate.edu/Zupanski/presentations ftp://ftp.cira.colostate.edu/Zupanski/manuscripts

2. OUTLINE:  Data assimilation methods  State augmentation approach  Model error estimation employing variational and EnsDA frameworks  Experimental results employing various models Q: What is data assimilation?  Conclusions and future work Dusanka Zupanski, CIRA/CSU Zupanski@CIRA.colostate.edu

3. DATA ASSIMILATION (ESTIMATION THEORY) Discrete stochastic-dynamic model Dusanka Zupanski, CIRA/CSU Zupanski@CIRA.colostate.edu Discrete stochastic observation model w k-1 – model error (stochastic forcing) M – non-linear dynamic (NWP) model G – model (matrix) reflecting the state dependence of model error  k – measurement + representativeness error M  D H– non-linear observation operator ( M  D )

4. VARIATIONAL APPROACH (1) State estimate (optimal solution): KALMAN FILTER APPROACH (2) Estimate of the uncertainty of the solution: ENSEMBLE KALMAN FILTER or EnsDA APPROACH In EnsDA solution is defined in ensemble subspace (reduced rank problem) ! DATA ASSIMILATION INCLUDES THE FOLLOWING: KALMAN FILTER APPROACH

5. Dusanka Zupanski, CIRA/CSU Zupanski@CIRA.colostate.edu State augmentation approach (a model bias example) Control variable for the analysis cycle k: Solve EnKF equations (or EnsDA) equations in terms of control variable z and forecast model F : Parameter estimation is a special case of state augmentation approach!

6. 4DVAR framework Forecast error covariance Data assimilation (Init. Cond. and Model Error adjust.) Observations First guess Init. Cond. and Model Error opt. estimates Forecast error covariance Data assimilation (Init. Cond. and Model Error adjust.) Observations First guess Init. Cond. and Model Error opt. estimates Ens. forecasting Analysis error Covariance (in ensemble subspace) EnsDA framework In EnsDA framework model error does not depend on assumptions regarding forecast error covariance; data assimilation problem is solved in ensemble subspace

7. Dusanka Zupanski, CIRA/CSU Zupanski@CIRA.colostate.edu ETA 4DVAR: Surface pressure model error time evolution (every 2-h over a 12-h data assimilation interval) From Zupanski et al. 2004 (submitted to MWR)

8. Dusanka Zupanski, CIRA/CSU Zupanski@CIRA.colostate.edu RAMS 4DVAR: Exner function model error time evolution (lev=5km), every 2-h From Zupanski et al. 2004 (submitted to MWR)

9. Dusanka Zupanski, CIRA/CSU Zupanski@CIRA.colostate.edu Eta: Horizontal wind model error time evolution (lev=250hPa), every 2-h

10. Dusanka Zupanski, CIRA/CSU Zupanski@CIRA.colostate.edu RAMS: Horizontal wind model error time evolution (lev=250hPa), every 2-h

11. Dusanka Zupanski, CIRA/CSU Zupanski@CIRA.colostate.edu RAMS: Horizontal wind model error, vertical cross-section, every 2-h

12. Dusanka Zupanski, CIRA/CSU Zupanski@CIRA.colostate.edu EnsDA experiments with Korteweg-de Vries-Burgers (KdVB) model - one-dimensional model - includes non-linear advection, diffusion and dispersion From Zupanski and Zupanski 2004 (submitted to MWR) PARAMETER estimation impact

13. Dusanka Zupanski, CIRA/CSU Zupanski@CIRA.colostate.edu EnsDA experiments with KdVB model

14. Kalman Filter statistical verification tools Innovation statistics   2 statistics – can be used to test the stability of an ensemble filter Innovation vector =  obs – first guess P f – produced by Ensemble Filter algorithm R – input to Ensemble Filter algorithm The conditional mean of  2 (normalized by obs) should be equal to one  Innovation histogram – Probability Density Function of normalized innovation vectors For Gaussian distribution, and with linear observation operator H, the innovation histogram should be equal to standard normal distribution N (0,1)

15. EnsDA experiments with KdVB model (PARAMETER estimation impact) 10 obs101 obs

16. EnsDA experiments with KdVB model (PARAMETER estimation impact) It would be BEST to have a perfect model, but since this is not the case, it is necessary to estimate model error and use it to correct the model!

17. EnsDA experiments with KdVB model A feasible solution to reduce the number of degrees of freedom is to define bias in terms of small number of parameters. BIAS estimation results BIAS estimation may require many observations and large ensemble size !

18. Dusanka Zupanski, CIRA/CSU Zupanski@CIRA.colostate.edu From Zupanski and Zupanski 2004 (submitted to MWR) EnsDA experiments with KdVB model Analysis error covariance matrix (UNCERTAINTY estimate)

19. We are employing more and more models! NASA’s GEOS column model Work in progress in collaboration with: -A. Hou and S. Zhang (NASA/GMAO) -C. Kummerow (CSU/Atmos. Sci.) Preliminary results including parameter estimation: R 1/2 =  R 1/2 = 2  Choice of observation errors directly impacts innovation statistics. Observation error covariance R is the only given input to the system!

20. Q: What is data assimilation? A:  Method of defining optimal initial conditions (classic definition)  Model error estimation method  Model development tool (estimate and correct model errors during the model development phase)  PDF estimation Dusanka Zupanski, CIRA/CSU Zupanski@CIRA.colostate.edu

21. CONCLUSIONS  To employ full data assimilation power, model error estimation should be included  EnsDA approaches are very promising since they can provide not only optimal estimate of the atmospheric state, but the uncertainty of the estimate as well FUTURE WORK  Estimate and correct model errors for various models (GEOS, RAMS, WRF, etc.) Dusanka Zupanski, CIRA/CSU Zupanski@CIRA.colostate.edu

22. Dusanka Zupanski, CIRA/CSU Zupanski@CIRA.colostate.edu

23. An example: Equivalence between variational and Kalman filter equations (for linear models and Gaussian statistics)

24. Dusanka Zupanski, CIRA/CSU Zupanski@CIRA.colostate.edu An example (continued): Important difference: variational methods DO NOT provide forecast error covariance update (update of P)! Using the matrix equality (e. g., Jaswinski 1970, Appendix 7b: We obtain the Kalman filter analysis equation: