Outline

1. Applications of optimal control and EnKF to Flow Simulation and ModelingFlorida State University,23-24 February, 2005, Tallahassee, FloridaThe Maximum Likelihood Ensemble Filter (MLEF): An ensemble analysis/prediction system based on Control TheoryMilija ZupanskiCooperative Institute for Research in the AtmosphereColorado State UniversityFort Collins, CO 80523-1375ZupanskiM@CIRA.colostate.eduIn 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(106-108) • - 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) Forecast error covariance Minimize cost function Analysis error covariance • If h is linear, perfect Hessian preconditioning • If h is nonlinear, need xa~ true xmin to havereliable estimate of Pa • No sample error covariances: ensemble perturbations used to define the analysis increment subspace, not as random samples

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

9. Model error in MLEF • State augmentation approach x0 – initial conditions ; b – model bias ; g – empirical parameters Augmented control variable: Augmented error covariance:

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

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) R1/2 = 1/2 e R1/2 = e 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 [xa-xb] Height RMS error [xa-xt] 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. Use Bayes formula for multiple evidence [Yi – Evidence (observation type); X – Hypothesis (analysis)] How to exploit information content from specified observation types using ensembles? Denote: Maximum Likelihood Approach with multiple evidence:

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

16. Hessian Preconditioning in MLEF

17. Hessian Preconditioning in MLEF

18. Hessian Preconditioning in MLEF