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This paper studies multiplicative inflation: the complementary scaling of the state covariance in the ensemble Kalman filter (EnKF). Firstly, error sources in the EnKF are catalogued and discussed in relation to inflation; nonlinearity is given particular attention as a source of sampling error. In response, the "finite‐size" refinement known as the EnKF‐N is re‐derived via a Gaussian scale mixture, again demonstrating how it yields adaptive inflation. Existing methods for adaptive inflation estimation are reviewed, and several insights are gained from a comparative analysis. One such adaptive inflation method is selected to complement the EnKF‐N to make a hybrid that is suitable for contexts where model error is present and imperfectly parametrized. Benchmarks are obtained from experiments with the two‐scale Lorenz model and its slow‐scale truncation. The proposed hybrid EnKF‐N method of adaptive inflation is found to yield systematic accuracy improvements in comparison with the existing methods, albeit to a moderate degree. This paper studies multiplicative inflation: the complementary scaling of the state covariance in the ensemble Kalman filter (EnKF). Firstly, error sources in the EnKF are catalogued and discussed in relation to inflation; nonlinearity is given particular attention as a source of sampling error. In response, the "finite‐size" refinement known as the EnKF‐N is re‐derived via a Gaussian scale mixture, again demonstrating how it yields adaptive inflation. An extension is proposed that hybridizes the EnKF‐N with an existing adaptive inflation scheme, making it suitable also in the presence of model error. [ABSTRACT FROM AUTHOR]

Recent studies have demonstrated that it is possible to combine machine learning with data assimilation to reconstruct the dynamics of a physical model partially and imperfectly observed. The surrogate model can be defined as an hybrid combination where a physical model based on prior knowledge is enhanced with a statistical model estimated by a neural network (NN). The training of the NN is typically done offline, once a large enough data set of model state estimates is available. By contrast, with online approaches the surrogate model is improved each time a new system state estimate is computed. Online approaches naturally fit the sequential framework encountered in geosciences where new observations become available with time. In a recent methodology paper, we have developed a new weak‐constraint 4D‐Var formulation which can be used to train a NN for online model error correction. In the present article, we develop a simplified version of that method, in the incremental 4D‐Var framework adopted by most operational weather centers. The simplified method is implemented in the European Center for Medium‐Range Weather Forecasts (ECMWF) Object‐Oriented Prediction System, with the help of a newly developed Fortran NN library, and tested with a two‐layer two‐dimensional quasi geostrophic model. The results confirm that online learning is effective and yields a more accurate model error correction than offline learning. Finally, the simplified method is compatible with future applications to state‐of‐the‐art models such as the ECMWF Integrated Forecasting System. Plain Language Summary: We have recently proposed a general framework for combining data assimilation (DA) and machine learning (ML) techniques to train a neural network for online model error correction. In the present article, we develop a simplified version of this online training method, compatible with future applications to more realistic models. Using numerical illustrations, we show that the new method is effective and yields a more accurate model error correction than the usual offline learning approach. The results show the potential of incorporating DA and ML tightly, and pave the way toward an application to the Integrated Forecasting System used for operational numerical weather prediction at the European Centre for Medium‐Range Weather Forecasts. Key Points: Variants of weak‐constraint 4D‐Var can be used to train neural networks for online model error correctionOnline learning yields a more accurate model error correction than offline learningThe new, simplified method, developed in the incremental 4D‐Var framework, can be easily applied in operational weather models [ABSTRACT FROM AUTHOR]

The iterative ensemble Kalman filter (IEnKF) in a deterministic framework was introduced in Sakov et al. [Mon. Wea. Rev. 140: 1988–2004 ( )] to extend the ensemble Kalman filter (EnKF) and improve its performance in mildly up to strongly nonlinear cases. However, the IEnKF assumes that the model is perfect. This assumption simplified the update of the system at a time different from the observation time, which made it natural to apply the IEnKF for smoothing. In this study, we generalize the IEnKF to the case of an imperfect model with additive model error. The new method called IEnKF‐Q conducts a Gauss–Newton minimization in ensemble space. It combines the propagated analysed ensemble anomalies from the previous cycle and model noise ensemble anomalies into a single ensemble of anomalies, and by doing so takes an algebraic form similar to that of the IEnKF. The performance of the IEnKF‐Q is tested in a number of experiments with the Lorenz'96 model, which show that the method consistently outperforms both the EnKF and the IEnKF naively modified to accommodate additive model noise. [ABSTRACT FROM AUTHOR]

The idea of using machine learning (ML) methods to reconstruct the dynamics of a system is the topic of recent studies in the geosciences, in which the key output is a surrogate model meant to emulate the dynamical model. In order to treat sparse and noisy observations in a rigorous way, ML can be combined with data assimilation (DA). This yields a class of iterative methods in which, at each iteration, a DA step assimilates the observations and alternates with a ML step to learn the underlying dynamics of the DA analysis. In this article, we propose to use this method to correct the error of an existing, knowledge‐based model. In practice, the resulting surrogate model is a hybrid model between the original (knowledge‐based) model and the ML model. We demonstrate the feasibility of the method numerically using a two‐layer, two‐dimensional, quasi‐geostrophic channel model. Model error is introduced by the means of perturbed parameters. The DA step is performed using the strong‐constraint 4D‐Var algorithm, while the ML step is performed using deep learning tools. The ML models are able to learn a substantial part of the model error and the resulting hybrid surrogate models produce better short‐ to mid‐range forecasts. Furthermore, using the hybrid surrogate models for DA yields a significantly better analysis than using the original model. [ABSTRACT FROM AUTHOR]

In recent years, there has been increased interest in applying data assimilation (DA) methods, originally designed for state estimation, to the model selection problem. In this setting, previous studies introduced the contextual formulation of model evidence, or contextual model evidence (CME), and showed that CME can be efficiently computed using a hierarchy of ensemble‐based DA procedures. Although these studies analysed the DA methods most commonly used for operational atmospheric and oceanic prediction worldwide, they did not study these methods in conjunction with localization to a specific domain. Yet, any application of ensemble DA methods to realistic, very high‐dimensional geophysical models requires the implementation of some form of localization. The present study extends CME estimation to ensemble DA methods with domain localization. Domain‐localized CME (DL‐CME) developed in this article is tested for model selection with two models: (a) the Lorenz 40‐variable midlatitude atmospheric dynamics model (Lorenz‐95); and (b) the simplified global atmospheric SPEEDY model. CME is compared to the root‐mean‐square error (RMSE) as a metric for model selection. The experiments show that CME systematically outperforms RMSE in model selection skill, and that this skill improvement is further enhanced by applying localization to the CME estimate using DL‐CME. The potential use and range of applications of CME and DL‐CME as a model selection metric are also discussed. [ABSTRACT FROM AUTHOR]