Guaranteeing the convergence of the saddle formulation for weakly constrained 4D-Var data assimilation

This paper discusses convergence issues for the saddle variational formulation of the weakly constrained 4D-Var method in data assimilation, a method whose main interests are its parallelizable nature and its limited use of the inverse of the correlation matrices. It is shown that the method, in its original form, may produce erratic results or diverge because of the inherent lack of monotonicity of the produced objective function values. Convergent, variationally coherent variants of the algorithm are then proposed which largely retain the desirable features of the original proposal, and the circumstances in which these variants may be preferable to other approaches is briefly discussed.