Abstract/Summary

Data assimilation combines information from observations of a dynamical system with a previous forecast, with each term weighted by its respective uncertainty. An important recent area of research has been the introduction of correlated observation error covariance (OEC) matrices in numerical weather prediction systems. The benefits of correlated OEC matrices are multiple: they permit the use of high density observation networks, allow the capture of small scale processes and help make best use of available data. However, their use is often associated with convergence problems for iterative methods. In this thesis we study the theoretical impact of introducing correlated OEC matrices on the conditioning of variational data assimilation problems. We develop new bounds on the condition number of the Hessian for two data assimilation formulations and illustrate our findings with numerical examples in an idealised framework. The minimum eigenvalue of the OEC matrix is a key term for both problems, which motivates the use of reconditioning methods to reduce the condition number of correlation matrices. We develop theory for two reconditioning methods: ridge regression and the minimum eigenvalue method. We show for the first time that standard deviations are increased by both methods. Ridge regression reduces absolute correlations, whereas the minimum eigenvalue method makes smaller changes to correlations and variances. We then present the first in-depth study of the ridge regression method for an operational data assimilation system, using the Met Office 1D-Var system. Reconditioning improves convergence, but alters the quality control procedure, which is used to select appropriate observations for further assimilation. The results in this thesis provide guidance on how to include correlation information in general variational data assimilation problems while ensuring computational efficiency.