Intercomparison of the primal and dual formulations of variational data assimilation

Two approaches can be used to solve the variational data assimilation problem. The primal form corresponds to the 3D/4D-Var used now in many operational NWP centres. An alternative approach, called dual or 3D/4D-PSAS, consists in solving the problem in the dual of observation space. Both forms use the same basic operators so that once one method is developed, it should be possible to obtain the other easily provided these operators have a modular form. It has been shown that, with proper conditioning of the minimization problem, the two algorithms should have similar convergence rates and computational performances. In the presence of nonlinearities, the incremental form of 3D/4D-Var extends the equivalence to the so-called 3D/4D-PSAS. The first objective of this paper is to present results obtained with the variational data assimilation of the Meteorological Service of Canada to show the equivalence between the 3D-Var and the PSAS systems. This exercise has forced us to have a close look at the modularity of the operational 3D/4D-Var which then makes it possible to obtain the 3D-PSAS scheme. This paper then focuses on these two quadratic problems that show similar convergence rates. However, the minimization of 3D-PSAS is examined more thoroughly as some parameters are shown to be determining elements in the minimization process. Lastly, preconditioning properties are studied and the Hessians of the two problems are shown to be directly related to one another through their singular vectors, which makes it possible to cycle the Hessian of the PSAS form. Copyright (C) 2008 Royal Meteorological Society.