  A key a priori information used in 4DVar is the knowledge of the system's
evolution equations. In this paper we propose a method for taking full
advantage of the knowledge of the system's dynamical instabilities in order to
improve the quality of the analysis. We present an algorithm, four-dimensional
variational assimilation in the unstable subspace (4DVar-AUS), that consists in
confining in this subspace the increment of the control variable. The existence
of an optimal subspace dimension for this confinement is hypothesized.
Theoretical arguments in favor of the present approach are supported by
numerical experiments in a simple perfect non-linear model scenario. It is
found that the RMS analysis error is a function of the dimension N of the
subspace where the analysis is confined and is minimum for N approximately
equal to the dimension of the unstable and neutral manifold. For all
assimilation windows, from 1 to 5 days, 4DVar-AUS performs better than standard
4DVar. In the presence of observational noise, the 4DVar solution, while being
closer to the observations, if farther away from the truth. The implementation
of 4DVar-AUS does not require the adjoint integration.
