The effect of numerical model error on data assimilation

Strong constraint 4D-Variational data assimilation (4D-Var) is a method used to create an initialisation for a numerical model, that best replicates subsequent observations of the system it aims to recreate. The method does not take into account the presence of errors in the model, using the model equations as a strong constraint. This paper gives a rigorous and quantitative analysis of the errors introduced into the initialisation through the use of finite difference schemes to numerically solve the model equations. The 1D linear advection equation together with circulant boundary conditions, are chosen as the model equations of interest as they are representative of the advective processes relevant to numerical weather prediction, where 4D-Var is widely used. We consider the deterministic error introduced by finite difference approximations in the form of numerical dissipation and numerical dispersion and identify the relationship between these properties and the error in the 4D-Var initialisation. In particular, we find that a solely numerically dispersive scheme has the potential to introduce destructive interference resulting in the loss of some wavenumber components in the initialisation. Bounds for the error in the initialisation due to finite difference approximations are determined with and without observation errors. The bounds are found to depend on the smoothness of the true initial condition we wish to recover and the numerically dissipative and dispersive properties of the scheme. Numerical results are presented to demonstrate the effectiveness of the bounds. These lead to the conclusion that there exists a critical number of discretisation points when considering full sets of observations, where the effects of both the considered numerical model error and observational errors on the initialisation are minimised. The numerically dissipative and dispersive properties of the finite difference schemes also have the potential to alter the properties of the noise found in observations. Correlated noise structures may be introduced into the 4D-Var initialisation as a result. We determine when this occurs for observational errors in the form of additive white noise and find that the effect is reduced through the use of numerically non-dissipative finite difference schemes.