Generalization of the dual variational data assimilation algorithm to a nonlinear layered quasi-geostrophic ocean model

In this paper, we present a generalization to nonlinear models of the four dimensional variational dual method, the 4D-PSAS algorithm. The idea of 4DPSAS (Physical Space Analysis System) is to perform the minimization in the space of the observations, rather than in the model space as in the primal 4D-VAR scheme. Despite the formal equivalence between 4D-VAR and 4D-PSAS in a linear situation (both for model equations and observation operators), the dual method has several important advantages: in oceanographic cases, the observation space is smaller than the model space, which should improve the minimization process; for no additional cost, it provides an estimation of the model error; and finally, it does not have any singularities when the covariance error matrices tends to zero. The idea of this paper is to extend this algorithm to a fully nonlinear situation, as it has been done in the previous years with other classical data assimilation schemes: the 4D-VAR and the Kalman filter. For this purpose, we consider a nonlinear multi-layer quasi-geostrophic ocean model, which mimics quite well the mid-latitude circulation. We recall the standard primal 4D-VAR scheme applied to this model, and then introduce an extended 4D-PSAS algorithm in the particular case of this nonlinear QG model. We report then the results of extensive numerical experiments that have been carried out to compare this extended algorithm to the classical variational formulation, and to study its sensitivity to many parameters such as the nonlinearities, the number of available observations, the presence of an unknown term in the assimilation model, and to study the detection of the model error. As a matter of fact, it is found that this extended algorithm has kept the same advantages as in the linear case (model error detection, smaller sensitivity to various perturbations, more efficient minimization process). All these experiments suggest that it is an efficient assimilation scheme for oceanographic problems.