A weak-constraint 4DEnsembleVar. Part I: formulation and simple model experiments

4DEnsembleVar is a hybrid data assimilation method which purpose is not only to use ensemble flow-dependent covariance information in a variational setting, but to altogether avoid the computation of tangent linear and adjoint models. This formulation has been explored in the context of perfect models. In this setting, all information from observations has to be brought back to the start of the assimilation window using the space-time covariances of the ensemble. In large models, localisation of these covariances is essential, but the standard time-independent localisation leads to serious problems when advection is strong. This is because observation information is advected out of the localisation area, having no influence on the update. This is part I of a two-part paper in which we develop a weak-constraint formulation in which updates are allowed at observational times. This partially alleviates the time-localisation problem. Furthermore, we provide-for the first time-a detailed description of strong-and weak-constraint 4DEnVar, including implementation details for the incremental form. The merits of our new weak-constraint formulation are illustrated using the Korteweg-de-Vries equation (propagation of a soliton). The second part of this paper deals with experiments in larger and more complicated models, namely the Lorenz (1996) model and a shallow water equations model with simulated convection.