  In this work, we aim at studying ensemble based optimal control strategies
for data assimilation. Such formulation nicely combines the ingredients of
ensemble Kalman filters and variational data assimilation (4DVar). In the same
way as variational assimilation schemes, it is formulated as the minimization
of an objective function, but similarly to ensemble filter, it introduces in
its objective function an empirical ensemble-based background-error covariance
and works in an off-line smoothing mode rather than sequentially like
sequential filters. These techniques have the great advantage to avoid the
introduction of tangent linear and adjoint models, which are necessary for
standard incremental variational techniques. They also allow handling a time
varying background covariance matrix representing the error evolution between
the estimated solution and a background solution. As this background error
covariance matrix -- of reduced rank in practice -- plays a key role in the
variational process, our study particularly focuses on the generation of the
analysis ensemble state with localization techniques. Besides, to clarify well
the differences between the different methods and to highlight the potential
pitfall and advantages of the different methods, we present key theoretical
properties associated to different choices involved in their setup. We compared
experimentally the performances of several variations of an ensemble technique
of interest with an incremental 4DVar method. The comparisons have been leaded
on the basis of a Shallow Water model and have been carried out both with
synthetic data and through a close experimental setup. The cases where the
system's components are either fully observed or only partially have been in
particular addressed.
