Ensemble forecasting/data assimilation and model error estimation algorithm Prepared by

1. Ensemble forecasting/data assimilation and model error estimation algorithm Prepared by Dusanka Zupanski and Milija Zupanski CIRA/CSU References Zupanski, M., 2004: The Maximum Likelihood Ensemble Filter. Theoretical aspects. Submitted to Mon. Wea. Rev. [Available at ftp://ftp.cira.colostate.edu/milija/papers/MLEF_MWR.pdf] Zupanski, D., and M. Zupanski, 2004: Model error estimation employing ensemble data assimilation approach. Submitted to Mon. Wea. Rev. [Available at ftp://ftp.cira.colostate.edu/Zupanski/manuscripts/MLEF_model_err.revised2.pdf] Dusanka Zupanski, CIRA/CSU Zupanski@CIRA.colostate.edu

2. Maximum Likelihood Ensemble Filter (MLEF): • MLEF developed using ideas from: • Variational data assimilation(3DVAR, 4DVAR) • Iterated Kalman Filters • Ensemble Transform Kalman Filter (ETKF) • Algorithm specifics: • Nonlinear cost function minimization – as in 3DVAR, 4DVAR • - an iterative solution of a non-linear problem • Unconstrained minimization used, well suited for larger residuals (CG, LBFGS) • Hessian preconditioning using the ETKF transformation • State augmentation approach for model error: • - modelbias, boundary conditions, empirical parameters • Model error covariance also estimated • Analysis error covariance estimated from inverse Hessian (provides consistent Data Assimilation – Ensemble Prediction System) Milija Zupanski, CIRA/CSU ZupanskiM@CIRA.colostate.edu

3. What can be obtained from EnsDA? • Optimal estimatesof model state variables (e.g., carbon fluxes, sources, sinks) • Optimal estimates of model empirical parameters • Model error (bias) estimation and correction • Boundary conditions error estimation and correction • Uncertainty of the estimated model state variables (analysis and forecast uncertainty) • Uncertainty of the estimated empirical parameters, model errors, boundary conditions errors • Information content of the observations (observability in ensemble subspace) • Adjoint-type sensitivity, without using an adjoint model (total variation in ensemble subspace is used instead) • Targeted observations strategies, based on the forecast uncertainty Dusanka Zupanski, CIRA/CSU Zupanski@CIRA.colostate.edu

4. Ensemble Data Assimilation (EnsDA) Algorithm Options Models: KdVB, GEOS, SWM, RAMS, … Observations: Synthetic or include various real observations Obs. operators: Include various forward operators Solution Type: Mode (max likelihood-MLEF) or Mean (ensemble mean ETKF) Estimator: Filter or Smoother Control variable: Initial conditions, Model bias, Model parameters Covariances: Localized, or Non-localized forecast error covariance Minimization: Minimization algorithm (C-G, L-BFGS) MPI: ParallelMPI run or a Single processor run Verifications: Innovation statistics (chi-square test, K-S test), RMS-errors Dusanka Zupanski, CIRA/CSU Zupanski@CIRA.colostate.edu

5. Ensemble Data Assimilation (EnsDA) Algorithm prep_ensda.sh WARM start: Copy files from previously completed cycle COLD start: Run randomly-perturbed ensemble forecasts to initialize fcst err cov cycle_ensda.sh icycle < N_cycles_max fcsterr_cov.sh - Prepare first-guess (background) vector - Prepare forecast error covariance (from ensembles) prep_obs.sh Given ‘OBSTYPE’ and ‘delobs’, select and copy available obs files assimilation.sh Iterative minimization of cost function, save current cycle output

6. Ensemble Data Assimilation (EnsDA) Algorithm assimilation.sh script assimilation.sh iter < ioutmax forward.sh: Transformation from model space to observation space Hessian preconditioning (only for iter=1) Gradient calculation (ensembles) Cost function calculation (diagnostic) Minimization (ensemble subspace) Step-length (line-search) Control variable update (transformation from ensemble subspace to model (physical) space) • - Analysis error covariance calculation • Save current cycle output files • Post-processing (chi-square, RMS, etc.)

7. Ensemble Data Assimilation (EnsDA) Algorithm Control variables Optional control variable components: - initial conditions - model bias - empirical parameters ../include/Cntrl_vrbl_list.h !--------------------------------------- max_num_of_cntrl_vrbls = 5 if(.not.allocated (cvar_list)) then allocate (cvar_list(1:max_num_of_cntrl_vrbls)) end if !--------------------------------------- ncv= 1 cvar_list(ncv)%ndim = 3 cvar_list(ncv)%start_index(1) = 1 cvar_list(ncv)%start_index(2) = 1 cvar_list(ncv)%start_index(3) = 1 cvar_list(ncv)%start_index(4) = 1 cvar_list(ncv)%end_index(1) = NNXP(1) cvar_list(ncv)%end_index(2) = NNYP(1) cvar_list(ncv)%end_index(3) = NNZP(1) cvar_list(ncv)%end_index(4) = 1 cvar_list(ncv)%name = 'T ' cvar_list(ncv)%stddev = 1.5 cvar_list(ncv)%description = 'Temperature ‘ !--------------------------------------- ../namelists/Cntrl_vrbl_’MODEL’.h 6 p T F F 0 t T F T 1 q T F T 3 u T F F 0 v T F F 0 param1 F T F 0 param2 F T F 0 !-- first line number is the number of control variables defined --! !-- vrbl name, ic flag, param flag, bias flag, number of biases ----! !--- recall that vrbl name has 9 characters !!!!! !--- All inputs have to be separated by at least one blank space !!!

8. Tasks • Prepare models (LPDM, RAMS-Sib-CASA, NASA’s Global model?) • Install LAPACK on your computer, unless it already exists. Compile with 32 bit object code option (LAPACK can be found at http://www.cs.colorado.edu/~lapack) • Install EnsDA algorithm on your computer • Include MODEL into EnsDA • - develop an interface between MODEL and EnsDA • - prepare all input files for the model • Prepare observations - synthetic observations of atmospheric and carbon variables - define data assimilation interval • Perform initial data assimilation experiments (with synthetic observations) - perfect model assumption experiments - parameter estimation experiments - model error estimation experiments

9. Tasks (continued) • Data assimilation and ensemble forecasting experiments with real observations - atmospheric observations (u,v,T,p,q, etc.) - carbon observations - LPDM model - RAMS-SiB-CASA model