  This study develops a hybrid ensemble-variational approach for solving data
assimilation problems. The method, called TR-4D-EnKF, is based on a trust
region framework and consists of three computational steps. First an ensemble
of model runs is propagated forward in time and snapshots of the state are
stored. Next, a sequence of basis vectors is built and a low-dimensional
representation of the data assimilation system is obtained by projecting the
model state onto the space spanned by the ensemble deviations from the mean.
Finally, the low-dimensional optimization problem is solved in the
reduced-space using a trust region approach; the size of the trust region is
updated according to the relative decrease of the reduced order surrogate cost
function. The analysis state is projected back onto the full space, and the
process is repeated with the current analysis serving as a new background. A
heuristic approach based on the trust region size is proposed in order to
adjust the background error statistics from one iteration to the next.
Experimental simulations are carried out using the Lorenz and the
quasi-geostrophic models. The results show that TR-4D-EnKF is an efficient
computational approach, and is more accurate than the current state of the art
4D-EnKF implementations such as the POD-4D-EnKF and the Iterative Subspace
Minimization methods.
