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