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The effect of preconditioning linear weighted least‐squares using an approximation of the model matrix is analyzed. The aim is to investigate from a theoretical point of view the inefficiencies of this approach as observed in the application of the weakly constrained 4D‐Var algorithm in geosciences. Bounds on the eigenvalues of the preconditioned system matrix are provided. It highlights the interplay of the eigenstructures of both the model and weighting matrices: maintaining a low bound on the eigenvalues of the preconditioned system matrix requires an approximation error of the model matrix which compensates for the condition number of the weighting matrix. A low‐dimension analytical example is given illustrating the resulting potential inefficiency of such preconditioners. The consequences of these results in the context of the state formulation of the weakly constrained 4D‐Var data assimilation problem are discussed. It is shown that the common approximations of the tangent linear model which maintain parallelization‐in‐time properties (identity or null matrix) can result in large bounds on the eigenvalues of the preconditioned matrix system.

A note on preconditioning weighted linear least‐squares, with consequences for weakly constrained variational data assimilation