Three data assimilation methods are compared for their ability to produce the best analysis: (i) 4DVar, four-dimensional variational data assimilation using linear and adjoint models with either a (perfect) 3D climatological background-error covariance or a 3D ensemble background-error covariance; (ii) EDA, an ensemble of 4DEnVars, which is a variational method using a 4D ensemble covariance; and (iii) the deterministic ensemble Kalman filter (DEnKF, also using a 4D ensemble covariance).
The accuracy of the deterministic analysis from each method was measured for both perfect and imperfect toy model experiments. With a perfect model, 4DVar with the climatological covariance is easily beaten by the ensemble methods, due to the importance of flow-dependent background-error covariances. When model error is present, 4DVar is more competitive and its relative performance is improved by increasing the observation density. This is related to the model error representation in the background-error covariance.
The dynamical time-consistency of the 4D ensemble background-error covariance is degraded by the localization, since the localization function and the nonlinear model do not commute. As a result, 4DVar with the ensemble covariance performs significantly better than the other ensemble methods when severe localization is required, i.e. for a small ensemble.