Dusanka Zupanski And Scott Denning Colorado State University Fort Collins, CO CMDL Workshop on Modeling and Data Analysis of Atmospheric CO 2 Observations in North America September 2004 ftp://ftp.cira.colostate.edu/Zupanski/presentations ftp://ftp.cira.colostate.edu/Zupanski/manuscripts Critical issues of ensemble data assimilation in application to carbon cycle studies

 Introduction: EnsDA approaches  Non-linear processes  Model error and parameter estimation  Uncertainty estimates  Correlated observations  Non-Gaussian PDFs  Conclusions and future work Dusanka Zupanski, CIRA/CSU OUTLINE:

Probabilistic approach to data assimilation and forecasting or Ensemble Data Assimilation (EnsDA) Dusanka Zupanski, CIRA/CSU Provides the following: (1) Optimal solution or state estimate (e. g., optimal CO 2 analysis) (2) Optimal estimates of model error and empirical parameters (3) Uncertainty of the analysis (a component of the analysis error covariance P a ) (4) Uncertainty of the estimated model error and parameters (components of the analysis error covariance P a ) (5) Estimate of forecast uncertainty (the forecast error covariance P f )

DATA ASSIMILATION (ESTIMATION THEORY) Discrete stochastic-dynamic model Dusanka Zupanski, CIRA/CSU Discrete stochastic observation model w k-1 – model error (stochastic forcing) M – non-linear dynamic (NWP) model G – model (matrix) reflecting the state dependence of model error  k – measurement + representativeness error M  D H– non-linear observation operator ( M  D )

(1) State estimate (optimal solution): (2) Estimate of the uncertainty of the solution: ENSEMBLE KALMAN FILTER or EnsDA APPROACH In EnsDA solution is defined in ensemble subspace (reduced rank problem) ! KALMAN FILTER APPROACH MAXIMUM LIKELIHOOD ESTIMATE (VARIATIONAL APPROACH ): MINIMUM VARIANCE ESTIMATE (KALMAN FILTER APPROACH ): DATA ASSIMILATION EQUATIONS:

Ensemble Data Assimilation (EnsDA) Dusanka Zupanski, CIRA/CSU (1) Maximum likelihood approach (involves an iterative minimization of a functional)  x mode (MLEF, Zupanski 2004) (2) Minimum variance approach (calculates ensemble mean)  x mean x mode x mean x PDF(x) x mode = x mean x PDF(x) Non-GaussianGaussian

Critical issues: Non-linear processes Dusanka Zupanski, CIRA/CSU - Use only non-linear models (tangent-linear, adjoint models are not needed) - Iterative minimization is beneficial for non-linear processes Example: KdVB model (M. Zupanski, 2004)

Critical issues: Model error and parameter estimation Dusanka Zupanski, CIRA/CSU - Estimate and correct all major sources of uncertainty: initial conditions, model error, boundary conditions, empirical parameters - Unified algorithm: EnsDA+state augmentation approach (Zupanski and Zupanski, 2004) Example: KdVB model

10 obs101 obs EnsDA experiments with KdVB model (PARAMETER estimation impact)

EnsDA experiments with KdVB model (BIAS estimation impact) Dusanka Zupanski, CIRA/CSU

Critical issues: Uncertainty estimates Dusanka Zupanski, CIRA/CSU - Analysis error covariance P a (analysis uncertainty) - Forecast error covariance P f (forecast uncertainty) - Both defined in ensemble sub-space KdVB model example:

Critical issues: Correlated observations Dusanka Zupanski, CIRA/CSU Problem: Numerous observations (~ ) are being projected onto a small ensemble sub-space (~ ) ! Loss of observed information! Remedies:  Process observations one by one (Anderson 2001, Bishop et al. 2001; Hamill et al. 2001). Or  Process observations successively over relatively small local areas (LEKF, Ott et al. 2004). Assumption in both approaches: Observations being processed separately are uncorrelated (independent)! This may not be justified for dense satellite observations.

Critical issues: Correlated observations Dusanka Zupanski, CIRA/CSU How does the observed information impact the uncertainty estimate of the optimal solution (analysis error covariance P a ) ? - square root of analysis error covariance (N state x N ens ) - square root of forecast error covariance (N state x N ens ) - impact of observations on the optimal solution (N ens x N ens ) The eigenvalue spectrum of (I+A) -1/2 may help understand the impact of observations, and perhaps find a better solution for correlated observations.

RAMS model example Dusanka Zupanski, CIRA/CSU A safe approach to prevent loss of observed information, assuming independent observations: N obs  N ens. If eigenvalues of (I+A) -1/2 spread over the entire interval [0,1], ensemble size (N ens ) is appropriate for a given observation number (N obs ).

CSU shallow-water model on geodesic grid When system can learn from its past, less information from observations is needed ! Smooth start (in cycle 1) can improve the performance of EnsDA Milija Zupanski, CIRA/CSU Analysis error smaller than obs error (Results from M. Zupanski et al.)

Non-Gaussian PDFs Non-linear Atmospheric- Hydrology- Carbon state variables and observations are likely to have non-Gaussian PDFs.  MLEF, as a maximum likelihood estimate, is a suitable tool for examining the impact of different PDFs.  Develop a non-Gaussian PDF framework (M. Zupanski) - allow for non-Gaussian observation errors - apply the Bayes theorem for multiple events Milija Zupanski, CIRA/CSU

CSU EnsDA algorithm is currently being examined in application to NASA’s GEOS column model in collaboration with: -A. Hou and S. Zhang (NASA/GMAO) -C. Kummerow (CSU/Atmos. Sci.) NASA’s GEOS column model example R 1/2 =  R 1/2 = 2  Prescribed observation errors directly impact innovation statistics. Since the observation error covariance R is the only input required by the system, it could be tuned! Dusanka Zupanski, CIRA/CSU In case we solved all critical issues, one problem remains: How to define observation error covariance matrix R, if it is not known?

 EnsDA approaches are very promising since they can provide not only optimal estimate of the state, but also the uncertainty of the optimal estimate.  The experience gained so far indicates that the EnsDA approach is suitable for addressing critical issues of data assimilation in Carbon cycle studies.  Model error and parameter estimation are necessary ingredients of a data assimilation algorithm.  Problems involved in Carbon data assimilation require a state-of- the art approach. We anticipate findings from different scientific disciplines (e. g., atmospheric science, ecology, hydrology) to be of mutual benefits.  It is especially important to gain experience with complex coupled models (e. g., RAMS-SiB-CASA), correlated (satellite) observations, and non-Gaussian PDFs in the future. Dusanka Zupanski, CIRA/CSU

Dusanka Zupanski, CIRA/CSU