A new ensemble-based data assimilation method, named the maximum likelihood ensemble filter (MLEF), is presented. The analysis solution maximizes the likelihood of the posterior probability distribution, obtained by minimization of a cost function that depends on a general nonlinear observation operator. The MLEF belongs to the class of deterministic ensemble filters, since no perturbed observations are employed. As in variational and ensemble data assimilation methods, the cost function is derived using a Gaussian probability density function framework. Like other ensemble data assimilation algorithms, the MLEF produces an estimate of the analysis uncertainty (e.g., analysis error covariance). In addition to the common use of ensembles in calculation of the forecast error covariance, the ensembles in MLEF are exploited to efficiently calculate the Hessian preconditioning and the gradient of the cost function. A sufficient number of iterative minimization steps is 2–3, because of superior Hessian preconditioning. The MLEF method is well suited for use with highly nonlinear observation operators, for a small additional computational cost of minimization. The consistent treatment of nonlinear observation operators through optimization is an advantage of the MLEF over other ensemble data assimilation algorithms. The cost of MLEF is comparable to the cost of existing ensemble Kalman filter algorithms. The method is directly applicable to most complex forecast models and observation operators. In this paper, the MLEF method is applied to data assimilation with the one-dimensional Korteweg–de Vries–Burgers equation. The tested observation operator is quadratic, in order to make the assimilation problem more challenging. The results illustrate the stability of the MLEF performance, as well as the benefit of the cost function minimization. The improvement is noted in terms of the rms error, as well as the analysis error covariance. The statistics of innovation vectors (observation minus forecast) also indicate a stable performance of the MLEF algorithm. Additional experiments suggest the amplified benefit of targeted observations in ensemble data assimilation.

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.

[1] We use an ensemble-based data assimilation method, known as the maximum likelihood ensemble filter (MLEF), which has been coupled with a global atmospheric transport model to estimate slowly varying biases of carbon surface fluxes. Carbon fluxes for this test consist of hourly gross primary production and ecosystem, respiration over land, and air-sea gas exchange. Persistent multiplicative biases intended to represent incorrectly simulated biogeochemical or land-management processes such as stand age, soil fertility, or coarse woody debris were estimated for 1 year at 10°longitude by 6°latitude spatial resolution and with an 8-week time window. We tested the model using a pseudodata experiment with an existing observation network that includes flasks, aircraft profiles, and continuous measurements. Because of the underconstrained nature of the problem, strong covariance smoothing was applied in the first data assimilation cycle, and localization schemes have been introduced. Error covariance was propagated in subsequent cycles. The coupled model satisfactorily recovered the land biases in densely observed areas. Ocean biases, however, were poorly constrained by the atmospheric observations. Unlike in batch mode inversions, the MLEF has a capability of assimilating large observation vectors and hence is suitable for assimilating hourly continuous observations and satellite observations in the future. Uncertainty was reduced further in our pseudodata experiment than by previous batch methods because of the ability to assimilate a large observation vector. Propagation of spatial covariance and dynamic localization avoid the need for prescribed spatial patterns of error covariance centered at observation sites as in previous grid-scale methods.