An analysis is provided to show that Courtier's et al.(1994) method for estimating the Hessian preconditioning is not applicable to important categories of cases involving nonlinearity. An extension of the method to cases with higher nonlinearity is proposed in the present paper by designing an algorithm which reduces errors in Hessian estimation induced by lack of validity of the tangent linear approximation. The new preconditioning method was numerically tested in the framework of variational data assimilation experiments using both the NASA Semi-Lagrangian semi-implicit (SLSI) global shallow-water (S-W) equations model and the adiabatic version of NASA/DAO GEOS-1 GCM. Our results show that the new preconditioning method speeds-up convergence rate of minimization when applied to variational data assimilation cases characterized by strong nonlinearity. Finally we address issues related to computational cost of our new algorithm. These include the optimal determination of the number of random realizations p necessary for Hessian estimation methods. We tested a computationally efficient method which uses a coarser grid point model to estimate Hessian for application to a fine resolution mesh. The tests yielded encouraging results. ii
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