  This paper proposes two practical implementations of Four-Dimensional
Variational (4D-Var) Ensemble Kalman Filter (4D-EnKF) methods for non-linear
data assimilation. Our formulations' main idea is to avoid the intrinsic need
for adjoint models in the context of 4D-Var optimization and, even more, to
handle non-linear observation operators during the assimilation of
observations. The proposed methods work as follows: snapshots of an ensemble of
model realizations are taken at observation times, these snapshots are employed
to build control spaces onto which analysis increments can be estimated. Via
the linearization of observation operators at observation times, a line-search
based optimization method is proposed to estimate optimal analysis increments.
The convergence of this method is theoretically proven as long as the dimension
of control-spaces equals model one. In the first formulation, control spaces
are given by full-rank square root approximations of background error
covariance matrices via the Bickel and Levina precision matrix estimator. In
this context, we propose an iterative Woodbury matrix formula to perform the
optimization steps efficiently. The last formulation can be considered as an
extension of the Maximum Likelihood Ensemble Filter to the 4D-Var context. This
employs pseudo-square root approximations of prior error covariance matrices to
build control spaces. Experimental tests are performed by using the Lorenz 96
model. The results reveal that, in terms of Root-Mean-Square-Error values, both
methods can obtain reasonable estimates of posterior error modes in the 4D-Var
optimization problem. Moreover, the accuracies of the proposed filter
implementations can be improved as the ensemble sizes are increased.
