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

An important class of nonlinear weighted least-squares problems arises from the assimilation of observations in atmospheric and ocean models. In variational data assimilation, inverse error covariance matrices define the weighting matrices of the least-squares problem. For observation errors, a diagonal matrix (i.e., uncorrelated errors) is often assumed for simplicity even when observation errors are suspected to be correlated. While accounting for observationerror correlations should improve the quality of the solution, it also affects the convergence rate of the minimization algorithms used to iterate to the solution. If the minimization process is stopped before reaching full convergence, which is usually the case in operational applications, the solution may be degraded even if the observation-error correlations are correctly accounted for. In this article, we explore the influence of the observation-error correlation matrix (R) on the convergence rate of a preconditioned conjugate gradient (PCG) algorithm applied to a one-dimensional variational data assimilation (1D-Var) problem. We design the idealised 1D-Var system to include two key features used in more complex systems: we use the background error covariance matrix (B) as a preconditioner (B-PCG); and we use a diffusion operator to model spatial correlations in B and R. Analytical and numerical results with the 1D-Var system show a strong sensitivity of the convergence rate of B-PCG to the parameters of the diffusion-based correlation models. Depending on the parameter choices, correlated observation errors can either speed up or slow down the convergence. In practice, a compromise may be required in the parameter specifications of B and R between staying close to the best available estimates on the one hand and ensuring an adequate convergence rate of the minimization algorithm on the other.