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.