1. Applications of optimal control and EnKF to Flow Simulation and Modeling Florida State University, 23-24 February, 2005, Tallahassee, Florida The Maximum Likelihood Ensemble Filter (MLEF): An ensemble analysis/prediction system based on Control Theory Milija Zupanski Cooperative Institute for Research in the Atmosphere Colorado State University Fort Collins, CO 80523-1375 ZupanskiM@CIRA.colostate.edu In collaboration with: D. Zupanski, S. Fletcher, I.M. Navon, B. Uzunoglu, D. Randall, R. Heikes, D. Daescu, A. Hou, S. Zhang

2. Outline  General problem  Maximum Likelihood Ensemble Filter  What’s next?

3. General Problem Theoretical issues Single assimilation/prediction system - complete feed-back between the assimilation and prediction Universal mathematical form - one algorithm applicable to any model, minimum or no dependency upon the modeled physical phenomenon Nonlinearity - real-life problems are nonlinear, need to know how to solve nonlinear problems Non-differentiability - methodology that works without differentiability requirement Imperfect models - need to know how to account for errors of the prediction model and observation operators – nothing is perfect!

4. General Problem Practical issues High-dimensional systems ~O(10 6 -10 8 ) - real-life applications are highly-dimensional - ultimate goal of probabilistic analysis/prediction system is to solve practical problems - re-evaluate the feasibility of theoretical ideas Algorithm development and maintenance (upgrade) - models, data are constantly being developed and upgraded - need a simple and effective system, capable of quick adjustment to the user needs Numerical stability and robustness - system has to work, even if limited information is available! Computational issues Disk storage, I/O, matrix-matrix and matrix-vector operations Parallel computing – exploit development in computer science and technology

5. General problem: Solution options Major concern when looking for the solution: - nonlinearity: prediction model, observation operator (Option 1) Introduce nonlinearities to the closed-form (linear) KF solution - Ensemble Kalman Filters (EnKF) (Option 2) Directly solve the nonlinear problem using numerical solution methods (i.e. iterative minimization) - variational data assimilation, maximum likelihood ensemble filter Issues -Both approaches have well developed theory and practice for weakly nonlinear and differentiable problems -(1) may lead to oversimplification of the general problem, higher-order moments -(2) needs good Hessian preconditioning, robust minimization

6. Maximum Likelihood Ensemble Filter (MLEF) A control theory application to ensemble ensemble data assimilation -Estimate the conditional mode of the posterior Probability Density Function (PDF) -Use minimization algorithms (C-G, LBFGS) to minimize the cost-function -Augmented control variable: initial conditions, model error and bias, empirical parameters, boundary conditions -Ensembles used to estimate the uncertainty of the conditional mode -Posterior error covariance calculated from minimization algorithm -Under linear and Gaussian assumptions, identical to EnKF square-root filters (e.g., Ensemble Transform Kalman Filter – ETKF, Bishop et al. 2001)

7. Maximum Likelihood Ensemble Filter (MLEF) Minimize cost function Analysis error covariance Forecast error covariance If h is linear, perfect Hessian preconditioning If h is nonlinear, need x a ~ true x min to have reliable estimate of P a No sample error covariances: ensemble perturbations used to define the analysis increment subspace, not as random samples

8. Analysis Error Covariance Initial error covariance noisy, but quickly becomes spatially localized No need to force error covariance localization Cycle No. 4Cycle No. 7Cycle No. 10Cycle No. 1 MLEF with Korteweg-de Vries-Burgers model j i

9. Model error in MLEF State augmentation approach x 0 – initial conditions ; b – model bias ;  – empirical parameters Augmented control variable: Augmented error covariance:

10. MLEF with KdVB model Parameter estimation (diffusion coefficient) Error covariance block matrices Significant cross-correlation between initial conditions and model error IC-IC ME-MEIC-ME

11. MLEF with NASA’s GEOS column model Assimilation of PSAS analyses Work in progress under NASA’s TRMM project - D. Zupanski (CSU/CIRA) with A. Hou and Sara Zhang (NASA/GMAO) R 1/2 = 1/2  R 1/2 =  Choice of observation errors directly impacts innovation statistics. Observation error covariance R is the only given input to the system!

12. MLEF with CSU global shallow-water model Height analysis increment [x a -x b ] Height RMS error [x a -x t ] Impact of ensemble initialization: correlated initial ensemble perturbations can significantly improve algorithm performance Impact of error covariance localization: Dynamics has a positive impact on the smoothness and spatial localization of error covariance

13. Information content of observations

14. How to exploit information content from specified observation types using ensembles? Maximum Likelihood Approach with multiple evidence: Use Bayes formula for multiple evidence [Yi – Evidence (observation type); X – Hypothesis (analysis)] Denote:

15. Initial cycles carry more information Model still has a capability to learn from observations in later cycles MLEF application to calculate information content RAMS model example – GOES-R project Observation categories within the same cycle Multiple cycles Use Bayes formula for conditional probabilities with multiple observations Separate observations in sub-groups Calculate information content from each group

16. Hessian Preconditioning in MLEF