  This paper discusses the practical use of the saddle variational formulation
for the weakly-constrained 4D-VAR method in data assimilation. 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, variationaly coherent variants of the algorithm are then
proposed whose practical performance is compared to that of other formulations.
This comparison is conducted on two data assimilation instances (Burgers
equation and the Quasi-Geostrophic model), using two different assumptions on
parallel computing environment. Because these variants essentially retain the
parallelization advantages of the original proposal, they often --- but not
always --- perform best, even for moderate numbers of computing processes.
