The current evolution of computer architectures towards increasing parallelism requires a corresponding evolution towards more parallel data assimilation algorithms. In this article, we consider parallelization of weak-constraint four-dimensional variational data assimilation (4D-Var) in the time dimension. We categorize algorithms according to whether or not they admit such parallelization and introduce a new, highly parallel weakconstraint 4D-Var algorithm based on a saddle-point representation of the underlying optimization problem. The potential benefits of the new saddle-point formulation are illustrated with a simple two-level quasi-geostrophic model.

Variational data assimilation problems arising in meteorology and oceanography require the solution of a regularized nonlinear least-squares problem. Practical solution algorithms are based on the incremental (Truncated Gauss-Newton) approach, which involves the iterative solution of a sequence of linear least-squares (quadratic minimization) sub-problems. Each sub-problem can be solved using a primal approach, where the minimization is performed in a space spanned by vectors of the size of the model control vector, or a dual approach, where the minimization is performed in a space spanned by vectors of the size of the observation vector. The dual formulation can be advantageous for two reasons. First, the dimension of the minimization problem with the dual formulation does not increase when additional control variables, such as those accounting for model error in a weak-constraint formulation, are considered. Second, whenever the dimension of observation space is significantly smaller than that of the model control space, the dual formulation can reduce both memory usage and computational cost.In this paper, a new dual-based algorithm called Restricted B-preconditioned Lanczos (RBLanczos) is introduced, where B denotes the background-error covariance matrix. RBLanczos is the Lanczos formulation of the Restricted B-preconditioned Conjugate Gradient (RBCG) method. RBLanczos generates mathematically equivalent iterates to those of RBCG and the corresponding B-preconditioned Conjugate Gradient and Lanczos algorithms used in the primal approach. All these algorithms can be implemented without the need for a square-root factorization of B. RBCG and RBLanczos, as well as the corresponding primal algorithms, are implemented in two operational ocean data assimilation systems and numerical results are presented. Practical diagnostic formulae for monitoring the convergence properties of the minimization are also presented.

This paper discusses convergence issues for the saddle variational formulation of the weakly constrained 4D‐Var method in data assimilation, a method whose main interests are its parallelizable nature and its limited use of the inverse of the correlation matrices. It is shown that the method, in its original form, may produce erratic results or diverge because of the inherent lack of monotonicity of the produced objective function values. Convergent, variationally coherent variants of the algorithm are then proposed which largely retain the desirable features of the original proposal, and the circumstances in which these variants may be preferable to other approaches is briefly discussed.

The standard formulation of 4D-Var assumes random zero-mean errors for all sources of information used in the analysis. This assumption is usually not well verified in real-world applications. The performance of a weak-constraint 4D-Var formulation ("forcing" formulation) is studied in this paper in a simplified experimental setting using additive model errors of different length-scales and observing systems of different coverage and accuracy. A set of twin experiments is carried out and results show that weak-constraint 4D-Var can accurately estimate the actual model errors and the initial state only when background and model errors have different spatial scales and when the observations are unbiased and spatially homogeneous. We also present preliminary results from a different weak-constraint 4D-Var formulation ("state" formulation) which could in principle overcome some of these limitations, but at the cost of a substantial increase of computational and memory requirements. These findings help identify the potential but also the intrinsic limitations of the weak-constraint 4D-Var approach. They also help to clarify the experimental results seen in the operational ECMWF analysis system where the analysis and first-guess temperature bias is reduced by up to 50% in the stratosphere.

The numerical solution of saddle point systems has received a lot of attention over the past few years in a wide variety of applications such as constrained optimization, computational fluid dynamics and optimal control, to name a few. In this paper, we focus on the saddle point formulation of a large-scale variational data assimilation problem, where the computations involving the constraint blocks are supposed to be much more expensive than those related to the (1, 1) block of the saddle point matrix. New low-rank limited memory preconditioners exploiting the particular structure of the problem are proposed and analysed theoretically. Numerical experiments performed within the Object-Oriented Prediction System are presented to highlight the relevance of the proposed preconditioners.