Implementation of the weak constraint 4D-Var in NEMOVAR

4D-Var is designed to handle observations that are distributed in time over a given period and to compare them with the model state at the appropriate time. Usually 4D-Var seeks the initial condition of the assimilation period such that the model trajectory best fits the observations within this interval. In most of the current 4D-Var implementations, while errors in observations and background state are accounted for, the numerical model representing the evolution of the atmospheric flow is assumed perfect, or at least the model errors are assumed small enough to be neglected compared to other errors in the system. This assumption is often called strong constraint 4D-Var. When going toward high resolution, However, in weak-constraint 4D-Var a sequence of model states are estimated (rather than just the initial state), with the consequence that the tangent linear assumption is relied upon only for the shorter time segment between successive state estimates, and not for propagation of information throughout the assimilation window. The weak-constraint 4D-Var system is truly four-dimensional in the sense that the model state vector is determined at a succession of times within the assimilation window. The present report study the implementation of weak constaint 4D-Var in NEMOVAR