2004 SIAM Annual Meeting Minisymposium on Data Assimilation and Predictability for Atmospheric and Oceanographic Modeling July 15, 2004, Portland, Oregon ISSUES IN FURTHER DEVELOPMENT OF ENSEMBLE DATA ASSIMILATION Milija Zupanski Cooperative Institute for Research in the Atmosphere Colorado State University Fort Collins, CO

Outline  Probabilistic analysis-prediction - ensemble framework  Gaussian Probability Density Function (PDF) framework - non-Gaussian PDFs - nonlinearity  Maximum Likelihood Ensemble Filter (MLEF)  Model Errors

Why ensemble data assimilation ?  Analysis-prediction problem is probabilistic Inherent uncertainties in observed and predicted values: - Observation errors - Model errors - Turbulence, Convection Kolmogorov equation: - Transport of Probability Density Function (PDF) - General mathematical framework for analysis-prediction Chaotical Atmosphere/Ocean/Land processes: - Existence of low-dimensional attractor subspace suggests the need for ‘likelihood’, rather than deterministic knowledge of prediction  Highly nonlinear processes and interactions in real atmosphere/ocean - Ensemble DA methodologies are best equipped to handle nonlinearities  Practical aspects - Parallel computing, code development

Forward Kolmogorov Equation Prediction: Estimate of the forecast PDF Data Assimilation: Estimate of the initial PDF PredictionData Assimilation p – probability density function (PDF); f – dynamical model; g – stochastic forcing (model error)

Implications of Kolmogorov Equation Framework  THERE IS A SINGLE PROBABILISTIC ANALYSIS-PREDICTION SYSTEM Current systems: - only weak coupling between analysis and prediction - modeled forecast PDF information in data assimilation - practical DA algorithms estimate only a single PDF parameter (e.g., PDF mode) - analysis PDF estimate is commonly NOT produced New systems: - fully coupled: complete feedback between prediction and analysis - estimate of: (i) analysis PDF, and (ii) forecast PDF - possibility to estimate various PDF parameters: mode, mean, covariance,...

What do we want from PDF? Event A: Event B:  Likelihood of an event occurring - optimal PDF parameter estimate - uncertainty of the estimate  PDF parameters - conditional mean - conditional mode - covariance -...  Conditional probability using Bayes formula: Gaussian PDF Maxwell PDF

Practical limitations of PDF parameter estimation  Statistical PDF parameters estimation methods: Minimum variance: Ensemble mean - Monte Carlo (ensemble) Kalman Filter (EnKF) – stochastic filters - Ensemble Square-Root filters (EnSRF)– deterministic filters Maximum likelihood: Ensemble mode (deterministic control) - variational data assimilation - Maximum Likelihood Ensemble Filter (MLEF)  LARGE NUMBER OF DEGREES OF FREEDOM (DIMENSIONS) - computational burden: memory allocation, efficiency  REDUCING THE NUMBER OF DEGREES OF FREEDOM - statistical sampling of PDF - ensemble framework: span dynamically important (e.g., unstable) subspace

EKF/EnKF/EnSRF as a quadratic optimization process Consider a Gaussian conditional PDF Subject to P f - forecast error covariance R - observation error covariance H - nonlinear observation operator H - linearized observation operator (Jacobian) y - observation vector x - analysis vector x b - first-guess vector Form a quadratic cost function: J= - ln(Pr)  Search for x (e.g., analysis) which maximizes the conditional probability (e.g., minimizes the cost function)

Linear KF analysis solution (with Gaussian PDF assumption) (1) One-step solution of quadratic optimization problem: Linear solution framework: EKF, EnKF, EnSRF solution form obtained by assuming linear observation operators (2) Direct solution of EKF/EnKF/EnSRF: linear H=> step-length  =1 Maximum likelihood and minimum variance estimates identical for Gaussian PDF

Nonlinearity Issue 1: Observation and model operators are highly nonlinear  Options: (1) Use linear form of the solution, combined with nonlinear models in covariance calculation - current EnKF, EnSRF algorithms (2)Directly search for nonlinear solution by minimizing non-quadratic cost function - Maximum Likelihood Ensemble Filter (MLEF) - Nonlinear prediction model M used in P f - Nonlinear observation operator H used in P f H T and HP f H T Remaining question : - How restrictive is the linear form of the KF, EnKF solution ? - Should nonlinearity of H be included in a more consistent manner ?

Non-Gaussian PDF assumption  Fundamental problem: Inconsistent PDF assumption - Operators are nonlinear (observation, model), Gaussian assumption violated - Gaussian assumption known to be incorrect for some variables (e.g., precipitation, clouds, etc.) - Current mathematical framework used in realistic data assimilation relies heavily on Gaussian PDF assumption (e.g., cost function, PDF)  Need general mathematical framework: Non-Gaussian PDFs A solution: Within the Max Likelihood (MLEF) approach, optimize arbitrary non-Gaussian conditional PDF  Remaining problem: Multi-modal PDFs

MeanMode Statistical PDF parameters Dynamical state PDF Dynamical state PDF Uni-modal Bi-modal Mean Mode

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 Unconstrained minimization, well suited for larger residuals (C-G, LBFGS) Hessian preconditioning using the ETKF transformation Major assumption: Inverse Hessian = Analysis error covariance => satisfactory if solution is close to the minimum References 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/milija/papers/MLEF_model_err.pdf] 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]

Maximum Likelihood Ensemble Filter (MLEF) Experiment: Nonlinear advection, dispersion, diffusion Periodic boundary conditions Two solitary waves (solitons) Model domain: 101 grid-points Observation error: 0.05 units 10 observations (perfect model + perturbation) 3 minimization iterations in each MLEF analysis cycle Korteweg-de Vries-Burgers (KdVB) Equation - conditional PDF mode by minimization of cost function

MLEF data assimilation with KdVB model (quadratic obs operator, 10 ensembles, 10 obs) Analysis error covariance NO OBS MLEF Model dynamics helps in localization of analysis error covariance ! H(x)=x 2 NO MIN MLEF RMS error

Model errors in Ensemble Data Assimilation (EnsDA) More important than ever before ! - Forecast error covariance information relies on model forecasts: if incorrect, the forecast error covariance is incorrect ! Model bias, empirical parameters, physics, truncation errors, … Improve the spread of ensemble forecasts Optimal estimate of model error Optimal estimate of model error covariance Can be used to learn about the sources of model error

Model error estimation State augmentation approach: - adopted in MLEF (and NCEP’s Eta 4DVAR) x 0 – initial conditions ; b – model bias ;  – empirical parameters Augmented control variable: Augmented error covariance:

Model error estimation – cont. State augmentation approach: - initial conditions + model bias x 0 – initial conditions ; b – model bias Augmented control variable: Augmented error covariance:

From Zupanski and Zupanski 2004, MWR [Available at ftp://ftp/cira.colostate.edu/milija/MLEF_model_err.pdf] MLEF data assimilation with KdVB model Augmented analysis error covariance matrix Cross-covariance between model bias and initial conditions: P x0,b Auto-covariance for model bias: P b.b Auto-covariance for initial conditions: P x0,x0

Conclusions  Unified probabilistic analysis-prediction system is important in addressing the atmospheric and oceanographic issues: - sampling of analysis-prediction PDF (ensemble framework) - complete feed-back between ensemble data assimilation and ensemble forecasting  Treatment of nonlinearities of prediction model and observation operator can be improved with cost function minimization (MLEF)  Model errors (bias, empirical parameters) need to be included in realistic ensemble data assimilation applications  Need non-Gaussian PDF framework

Future development  Non-Gaussian PDF framework within MLEF approach - Control theory application - Direct optimization of non-Gaussian conditional PDFs - Nonlinear observation and model operators - Global shallow-water model on geodesic grid - Optimization algorithms, Hessian preconditioning  Applications with NCEP’s Global Forecasting System - Comparison between the conditional mean and conditional mode ensemble data assimilation - Real measurements, operational prediction model - Practical aspects: fine resolution control, coarse resolution ensembles