A dual weighted trust-region adaptive POD 4D-Var applied to a Finite-Volume shallow-water Equations Model on the sphere†

In this paper we study solutions of an inverse problem for a global shallow water model controlling its initial conditions speci ed from the 40-yr ECMWF Re-Analysis (ERA-40) datasets, in presence of full or incomplete observations being assimilated in a time interval(window of assimilation) with or without background error covariance terms. As an extension of the work in Chen et al., 2009, we attempt to obtain a reduced-order model of above inverse problem, based on proper orthogonal decomposition (POD), referred to as POD 4-D Var for a nite volume global shallow water equations model based on the Lin-Rood ux-form semi-Lagrangian semi-implicit time integration scheme. Di erent approaches of POD implementation of the reduced inverse problem are compared, including a dual-weighted method for snapshot selection coupled with a trust-region POD adaptivity approach. Numerical results with various observational densities and background error covariance operator are also presented. The POD 4-D Var model results combined with the trust region adaptivity exhibit similarity in terms of various error metrics to the full 4-D Var results, but are obtained using a signi cantly lesser number of minimization iterations and require lesser CPU time. Based on our previous and current work, we conclude that POD 4-D Var certainly warrants further studies,with promising potential of its extension to operational 3-D numerical weather prediction models. Copyright c © 2004 John Wiley & Sons, Ltd.