1. Maximum Liklihood Ensemble Filter (MLEF) Dusanka Zupanski, Kevin Robert Gurney, Scott Denning, Milia Zupanski, Ravi Lokupitiya June, 2005 TransCom Meeting, Paris

2. CSU context Avoid adjoint formulation (4Dvar out) Amenable to parallel environment – cluster computing Avail of “in-house” expertise Flexible to linear, non-linear, coupled model frameworks Dusanka & Milia Zupanski (formerly at NCEP) – long history and developmental expertise in ensemble methods. Currently developing applications of MLEF.

3. Error covariances Forecast error covariance - If we had a “dynamical model” (we don’t, right now): variables ensembles But, we don’t have a dynamical model (just an observational operator), so the current timestep forecast error can be taken from the previous timestep’s analysis error. Analysis error covariance = (Hessian) -1

4. Hessian preconditioning Cost function The inverse Hessian: This very likely has a high condition number…..can we transform the Hessian into a space with a low(er) condition number? An ideal “pre-conditioner” looks like: This is ideal! CN=1

5. Transformation Now we have a tranformation operator we can use to go from out physical space to a low condition number space (the ensemble subspace): Control variable in ensemble space the i th column of C:

6. The C matrix Interesting! The transformation scales the forecast error covariance by a matrix that embodies the result of the perturbed fluxes in concentration space. An S x S matrix, invert using usual techniques cleaning up:

7. Impact

8. The last bit Insert the definition for (x-x b ) into cost function, determine the gradient, employ numerical scheme (conjugate gradient) and locate the minimum. The analysis error covariance is the inverse Hessian at the minimum: Go back and do it again……..

9. Practical sequence P f (random 1 st time then run a cold start) Compute C Compute gradient of cost function in ensemble space Conjugate gradient method to find X opt Compute P a from Hessian at X opt

10. Plans at CSU  TRANSCOM - Estimate CO 2 fluxes - Estimate the uncertainties of fluxes  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

11. Summary  MLEF is an iterative/numerical ensemble approach that avails of the efficiency gained through preconditioning.  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.