1. Prepared by Dusanka Zupanski and …… Maximum Likelihood Ensemble Filter: application to carbon problems

2. Maximum Likelihood Ensemble Filter (MLEF) (Zupanski 2005; Zupanski and Zupanski 2005) Developed using ideas from :  Variational data assimilation (3DVAR, 4DVAR)  Iterated Kalman Filters  Ensemble Transform Kalman Filter (ETKF, Bishop et al. 2001) Characteristics of the MLEF  Calculates optimal estimates of: - model state variables (e.g., carbon fluxes, sources, sinks) - empirical parameters (e.g., light response, allocation, drought stress) - model error (bias) - boundary conditions error (lateral, top, bottom boundaries)  Calculates uncertainty of all estimates  Fully non-linear approach. Adjoint models are not needed.  Provides more information about PDF (higher order moments could be calculated from ensemble perturbations)  Non-derivative minimization (first variation instead of first derivative is used). Dusanka Zupanski, CIRA/CSU Zupanski@CIRA.colostate.edu

3. MLEF APPROACH Change of variable (preconditioning) - control vector in ensemble space of dim Nens Minimize cost function J - model state vector of dim Nstate >>Nens Dusanka Zupanski, CIRA/CSU Zupanski@CIRA.colostate.edu - information matrix of dim Nens  Nens

4. MLEF APPROACH (continued) Analysis error covariance Dusanka Zupanski, CIRA/CSU Zupanski@CIRA.colostate.edu Forecast error covariance Forecast model M essential for propagating in time (updating) columns of P f.

5. Ideal Hessian Preconditioning VARIATIONAL MLEF Milija Zupanski, CIRA/CSU ZupanskiM@CIRA.colostate.edu

6. STATE AUGMENTATION APPROACH as a part of the MLEF Example: parameter estimation - augmented state variable - augmented forecast model Assumption: parameter remains constant, or changes slowly with time SAME FRAMEWORK IS USED FOR MODEL BIAS ESTIMATION (use bias instead of a parameter to augment state variable) Parameters are randomly perturbed only in the first cycle. In later cycles, the MLEF updates ensemble perturbations.

7.  TRANSCOM - …. -….(Ravi, perhaps you can include a couple of bullets for Transcom)  SiB Parameter estimation - Estimate control parameters on the fluxes - MLEF calculates uncertainties of all parameters (in terms of P a and P f )  LPDM - Estimate monthly mean carbon fluxes, empirical parameters - Estimate uncertainties of the mean fluxes and empirical parameters  SiB-CASA-RAMS - Use various observations of weather, eddy-covariance fluxes, CO2 - Estimate carbon fluxes, empirical parameters (e.g., light response, allocation, drought stress, phonological triggers) - Time evolution of state variables, provided by the coupled model, is critical for updating P f Applications of the MLEF to carbon studies Dusanka Zupanski, CIRA/CSU Zupanski@CIRA.colostate.edu

8. TRANSCOM Dusanka Zupanski, CIRA/CSU Zupanski@CIRA.colostate.edu Ravi, you might want to add more detail about TRANSCOM

9. Preliminary results using RAMS Dusanka Zupanski, CIRA/CSU Zupanski@CIRA.colostate.edu  Hurricane Lili case  35 1-h DA cycles: 13UTC 1 Oct 2002 – 00 UTC 3 Oct  30x20x21 grid points, 15 km grid distance (in the Gulf of Mexico)  Control variable: u,v,w,theta,Exner, r_total (dim=54000)  Model simulated observations with random noise (7200 obs per DA cycle)  Nens=50  Iterative minimization of J (1 iteration only) RMS errors of the analysis (control experiment without assimilation) Hurricane entered the model domain. Impact of assimilation more pronounced.

10. Example: Total humidity mixing ratio, level=200 m, cycle 31 Locations of min and max centers are much improved in the experiment with assimilation. TRUTHNO ASSIMILATION ASSIMILATION

11. SUMMARY Dusanka Zupanski, CIRA/CSU Zupanski@CIRA.colostate.edu  The MLEF is currently being evaluated in various atmospheric science applications, showing encouraging results.  The MLEF is suitable for assimilation of numerous new carbon observations, employing complex non-linear coupled models.  Work in carbon applications has just started. Results will be presented in the future.