Variable-storage quasi-Newton operators as inverse forecast/analysis error covariance matrices in variational data assimilation

Two approximations of the Hessian matrix as limited-memory operators are built from the limited-memoryBFGS inverse Hessian approximationprovided by the minimization code, in view of the speci cation of the inverse analysis/forecast error covariance matrix in variational data assimilation. Some numerical experiments and theoretical considerations lead to reject the limited-memory DFP Hessian approximation and to retain the BFGS one for the applications foreseen. Conditioning issues are explored and a preconditioning strategy via a change of control variable is proposed, based on a suitable Cholesky factorization of the limited-memory inverse Hessian matrix. This factorization is implemented as the composition of linear operators. The memory requirements and the number of oatingpoint operations required by the method are given and con rmed by numerical experiments. The method is found to have a strong potential for variational data assimilation systems using high resolution ocean or atmosphere general circulation models. Key-words: limited-memory Hessian, variational data assimilation, analysis error covariances, forecast error covariances