Four-dimensional variational assimilation in the unstable subspace and the optimal subspace dimension

Key a priori information used in 4D-Var is the knowledge of the system's evolution equations. In this article 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 for four-dimensional variational assimilation in the unstable subspace (4D-Var – AUS), which consists of 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 favour of the present approach are supported by numerical experiments in a simple perfect nonlinear 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 a minimum for N approximately equal to the dimension of the unstable and neutral manifold. For all assimilation windows, from 1 to 5 d, 4D-Var – AUS performs better than standard 4D-Var. In the presence of observational noise, the 4D-Var solution, while being closer to the observations, is farther away from the truth. The implementation of 4D-Var – AUS does not require the adjoint integration.