Intercomparison of the primal and dual formulations of variational data assimilation

Abstract

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.

Convergence properties of the primal and dual forms of variational data assimilation

“…As shown by Courtier (1997) and discussed in El Akkraoui et al (2008), the variational data assimilation can be expressed either in its primal or dual form, where the latter solves the variational problem in the dual space instead of the model space in which the primal formulation is cast. The dual objective function to be minimized is:…”

“…Since the dual objective function has no immediate physical interpretation, the a posteriori probability distribution, as measured by the primal functional J(v), can be estimated by mapping each dual iterate u k to physical space (v k = L T u k ) and then evaluating J(v k ). The result of this process is shown in Figure 1, from El Akkraoui et al (2008), where the iterates associated with the minimization of F(u) were found to lead to less probable states than the background state at the beginning of the minimization. This is not desirable if the minimization has to stop after a finite number of iterations due to computational time constraints.…”

“…The PSAS is in fact the dual form of the primal variational problem. In El Akkraoui et al (2008), this equivalence was examined and a relationship between the Hessians of the two problems was introduced. When the Hessian is represented by a finite number of singular vectors, those can be mapped to the singular vectors of the Hessian of the other problem.…”

“…However, the dual form is commensurate with the number of observations and the dimension of the problem does not experience a significant increase. The issues raised by El Akkraoui et al (2008) regarding the convergence of the dual problem are then a source of concern for the development of a weak constraint 4D-Var in its dual form.…”

“…In El Akkraoui et al (2008), the equivalence of the primal and dual forms was established within the framework of an 'operational' 3D-Var. Although the two approaches lead to identical results at convergence, their results indicated that the dual form may exhibit unphysical states at the beginning of the minimization, which can add a significant number of iterations to reach convergence.…”