Abstract
An analysis is provided to show that Courtier's et al. 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 that 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 National Aeronautics and Space Administration (NASA) semi-Lagrangian semi-implicit global shallow-water equations model and the adiabatic version of the NASA/Data Assimilation Office (DAO) Goddard Earth Observing System Version 1 (GEOS-1) general circulation model. The authors' 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, the authors address issues related to computational cost of the new algorithm presented in this paper. These include the optimal determination of the number of random realizations p necessary for Hessian estimation methods. The authors tested a computationally efficient method that uses a coarser gridpoint model to estimate the Hessian for application to a fine-resolution mesh. The tests yielded encouraging results.
