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Data assimilation algorithms combine prior and observational information,weighted by their respective uncertainties, to obtain the most likely posteriorof a dynamical system. In variational data assimilation the posterior iscomputed by solving a nonlinear least squares problem. Many numerical weatherprediction (NWP) centres use full observation error covariance (OEC) weightingmatrices, which can slow convergence of the data assimilation procedure.Previous work revealed the importance of the minimum eigenvalue of the OECmatrix for conditioning and convergence of the unpreconditioned dataassimilation problem. In this paper we examine the use of correlated OECmatrices in the preconditioned data assimilation problem for the first time. Weconsider the case where there are more state variables than observations, whichis typical for applications with sparse measurements e.g. NWP and remotesensing. We find that similarly to the unpreconditioned problem, the minimumeigenvalue of the OEC matrix appears in new bounds on the condition number ofthe Hessian of the preconditioned objective function. Numerical experimentsreveal that the condition number of the Hessian is minimised when thebackground and observation lengthscales are equal. This contrasts with theunpreconditioned case, where decreasing the observation error lengthscalealways improves conditioning. Conjugate gradient experiments show that in thisframework the condition number of the Hessian is a good proxy for convergence.Eigenvalue clustering explains cases where convergence is faster than expected.

New bounds on the condition number of the Hessian of the preconditioned variational data assimilation problem