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An important class of nonlinear weighted least-squares problems arises fromthe assimilation of observations in atmospheric and ocean models. Invariational data assimilation, inverse error covariance matrices define theweighting matrices of the least-squares problem. For observation errors, adiagonal matrix (i.e., uncorrelated errors) is often assumed for simplicityeven when observation errors are suspected to be correlated. While accountingfor observationerror correlations should improve the quality of the solution,it also affects the convergence rate of the minimization algorithms used toiterate to the solution. If the minimization process is stopped before reachingfull convergence, which is usually the case in operational applications, thesolution may be degraded even if the observation-error correlations arecorrectly accounted for. In this article, we explore the influence of theobservation-error correlation matrix (R) on the convergence rate of apreconditioned conjugate gradient (PCG) algorithm applied to a one-dimensionalvariational data assimilation (1D-Var) problem. We design the idealised 1D-Varsystem to include two key features used in more complex systems: we use thebackground error covariance matrix (B) as a preconditioner (B-PCG); and we usea diffusion operator to model spatial correlations in B and R. Analytical andnumerical results with the 1D-Var system show a strong sensitivity of theconvergence rate of B-PCG to the parameters of the diffusion-based correlationmodels. Depending on the parameter choices, correlated observation errors caneither speed up or slow down the convergence. In practice, a compromise may berequired in the parameter specifications of B and R between staying close tothe best available estimates on the one hand and ensuring an adequateconvergence rate of the minimization algorithm on the other.

Impact of correlated observation errors on the convergence of the conjugate gradient algorithm in variational data assimilation