[1] We present a data assimilation system to estimate surface fluxes of CO 2 and other trace gases from observations of their atmospheric abundances. The system is based on ensemble data assimilation methods under development for Numerical Weather Prediction (NWP) and is the first of its kind to be used for CO 2 flux estimation. The system was developed to overcome computational limitations encountered when a large number of observations are used to estimate a large number of unknown surface fluxes. The ensemble data assimilation approach is attractive because it returns an approximation of the covariance, does not need an adjoint model or other linearization of the observation operator, and offers the possibility to optimize fluxes of chemically active trace gases (e.g., CH 4 , CO) in the same framework. We assess the performance of this new system in a pseudodata experiment that resembles the real problem we will apply this system to. The sensitivity of the method to the choice of several parameters such as the assimilation window size and the number of ensemble members is investigated. We conclude that the system is able to provide satisfactory flux estimates for the relatively large scales resolved by our current observing network and that the loss of information in the approximated covariances is an acceptable price to pay for the efficient computation of a large number of surface fluxes. The full potential of this data assimilation system will be used for near-real time operational estimates of North American CO 2 fluxes. This will take advantage of the large amounts of atmospheric data that will be collected by NOAA-CMDL in conjunction with the implementation of the North American Carbon Program (NACP).Citation: Peters, W., J. B. Miller, J. Whitaker, A. S. Denning, A. Hirsch, M. C. Krol, D. Zupanski, L. Bruhwiler, and P. P. , An ensemble data assimilation system to estimate CO 2 surface fluxes from atmospheric trace gas observations, J. Geophys.

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A methodology for model error estimation is proposed and examined in this study. It provides estimates of the dynamical model state, the bias, and the empirical parameters by combining three approaches: 1) ensemble data assimilation, 2) state augmentation, and 3) parameter and model bias estimation. Uncertainties of these estimates are also determined, in terms of the analysis and forecast error covariances, employing the same methodology.The model error estimation approach is evaluated in application to Korteweg-de Vries-Burgers (KdVB) numerical model within the framework of maximum likelihood ensemble filter (MLEF). Experimental results indicate improved filter performance due to model error estimation. The innovation statistics also indicate that the estimated uncertainties are reliable. On the other hand, neglecting model errors-either in the form of an incorrect model parameter, or a model bias-has detrimental effects on data assimilation, in some cases resulting in filter divergence.Although the method is examined in a simplified model framework, the results are encouraging. It remains to be seen how the methodology performs in applications to more complex models.

[1] We evaluate the capability of an ensemble based data assimilation approach, referred to as Maximum Likelihood Ensemble Filter (MLEF), to estimate biases in the CO 2 photosynthesis and respiration fluxes. We employ an off-line Lagrangian Particle Dispersion Model (LPDM), which is driven by the carbon fluxes, obtained from the Simple Biosphere -Regional Atmospheric Modeling System (SiB-RAMS). The SiB-RAMS carbon fluxes are assumed to have errors in the form of multiplicative biases. Our goal is to estimate and reduce these biases and also to assign reliable posterior uncertainties to the estimated biases. Experiments of this study are performed using simulated CO 2 observations, which resemble real CO 2 concentrations from the Ring of Towers in northern Wisconsin. We evaluate the MLEF results with respect to the ''truth'' and the Kalman Filter (KF) solution. The KF solution is considered theoretically optimal for the problem of this study, which is a linear data assimilation problem involving Gaussian errors. We also evaluate the impact of forecast error covariance localization based on a new ''distance'' defined in the space of information measures. Experimental results are encouraging, indicating that the MLEF can successfully estimate carbon flux biases and their uncertainties. As expected, the estimated biases are closer to the ''true'' biases in the experiments with more ensemble members and more observations. The data assimilation algorithm has a stable performance and converges smoothly to the KF solution when the ensemble size approaches the size of the model state vector (i.e., the control variable of the data assimilation problem).

ABSTRACT:The Maximum Likelihood Ensemble Filter (MLEF) equations are derived without the differentiability requirement for the prediction model and for the observation operators. The derivation reveals that a new non-differentiable minimization method can be defined as a generalization of the gradient-based unconstrained methods, such as the preconditioned conjugate-gradient and quasi-Newton methods. In the new minimization algorithm the vector of first-order increments of the cost function is defined as a generalized gradient, while the symmetric matrix of second-order increments of the cost function is defined as a generalized Hessian matrix. In the case of differentiable observation operators, the minimization algorithm reduces to the standard gradient-based form.The non-differentiable aspect of the MLEF algorithm is illustrated in an example with one-dimensional Burgers model and simulated observations. The MLEF algorithm has a robust performance, producing satisfactory results for tested non-differentiable observation operators.