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. The training of the neural network is typically done offline, once a large enough dataset 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 neural network 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 centres. The simplified method is implemented in the ECMWF Object-Oriented Prediction System, with the help of a newly developed Fortran neural network 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.

Model error is one of the main obstacles to improved accuracy and reliability in state-of-the-art analysis and forecasting applications, both in Numerical Weather Prediction (NWP) and in Climate Prediction, conducted with comprehensive high resolution General Circulation Models. In a data assimilation framework, recent advances in the context of weak constraint 4D-Var have shown that it is possible to estimate and correct for a large fraction of systematic model error which develops in the stratosphere over short-range forecast ranges. The recent explosion of interest in Machine Learning/Deep Learning technologies has been driven by their remarkable success in disparate application areas. This raises the question of whether model error estimation and correction in operational NWP and Climate Prediction can also benefit from these techniques. In this work, we aim to start to give an answer to this question. Specifically, we show that Artificial Neural Networks (ANN) can reproduce the main results obtained with weak constraint 4D-Var in the operational configuration of the IFS model of ECMWF. We show that the use of ANN models inside the weak constraint 4D-Var framework has the potential to extend the applicability of the weak constraint methodology for model error correction to the whole atmospheric column. Finally, we discuss the potential and limitations of the Machine Learning/Deep Learning technologies in the core NWP tasks. In particular, we reconsider the fundamental constraints of a purely data driven approach to forecasting and provide a view on how to best integrate Machine Learning technologies within current data assimilation and forecasting methods.