The conditioning of variational data assimilation with correlated observation errors

Work with Jemima Tabeart, Sarah Dance, Nancy Nichols, Joanne Waller (University of Reading) and Stefano Migliorini, Fiona Smith (Met Office). 
In environmental prediction variational data assimilation (DA) is a method for using observational data to estimate the current state of the system. The DA problem is usually solved as a very large nonlinear least squares problem, in which the fit to the measurements is balanced against the fit to a previous model forecast. These two terms are weighted by matrices describing the correlations of the errors in the forecast and in the observations. Until recently most operational weather and ocean forecasting systems assumed that the errors in the observations are uncorrelated. However, as we move to higher resolution observations then it is becoming more important to specify observation error correlations. In this work we look at the effect this has on the conditioning of the optimization problem. In the context of a linear system we develop bounds on the condition number of the problem in the presence of correlated observation errors. We show that the condition number is very dependent on the minimum eigenvalue of the observation error correlation matrix. We then present results using the Met Office data assimilation system, in which different methods for reconditioning the correlation matrix are tested. We investigate the effect of these different methods on the conditioning and the final solution of the problem.