Before revision:  Finally, we show that the tendency correction opens the possibility to make online model error correction,i.e. improving the model progressively as new observations become available. The resulting algorithm can be seen as a new formulation of weak-constraint 4D-Var. We compare online and offline learning using the same framework with the two-scale Lorenz system, and show that with online learning, it is possible to extract all the information from sparse and noisy observations .
After revision:  Finally, we show that the tendency correction opens the possibility to make online model error correction,i.e. local forcings, and the couplings between layers. This paper more generally addresses the key question of online estimation of both global and local model parameters .