Online Model Error Correction With Neural Networks in the Incremental 4D-Var Framework

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.