The effect of numerical model error on data assimilation

The effects of numerical model error on 4D-Variational data assimilation (4D-Var) are investigated. 4D-Var is a method used to create an initialisation for a numerical model, that best replicates subsequent observations of the system it models. The numerical error introduced by this model is considered in the form of numerical dissipation and dispersion. We find that a solely numerically dispersive model results in destructive interference and the loss of some wavenumber components in the initialisation. We also determine upper bounds for the error in the initialisation due to numerical model error with and without observation errors. The bounds are found to depend on the regularity of the true initial condition. There is a critical number of discretisation points where both errors are minimised. Numerical results are presented to demonstrate the effectiveness of the upper bounds. Even if observation errors are additive, uncorrelated white noise, the numerical scheme has the potential to introduce correlated noise structures into the initialisation and its subsequent forecast. However, this effect is reduced through the use of a numerically non-dissipative model.