Abstract
The justification for the standard four‐dimensional variational data assimilation (4D‐Var) method used at several major operational centres assumes a perfect forecast model, which is clearly unrealistic. However, the method has been very successful in practice. We investigate the reasons for this using a toy model with fast and slow time‐scales and with non‐random model error. The model error is chosen so that the solution remains predictable on both time‐scales. The fast modes are much less well observed than the slow modes. We show that poorly observed modes can be best forecast by using a regularization matrix in place of the background‐error covariance matrix, and using it to give a much stronger constraint than that implied by the true background error for these modes. The effect is that use can be made of observations over a longer time period. This allows the resulting forecast‐error growth to be reduced to much less than that of random perturbations generated using the analysis‐error covariance matrix and even less than the model error growth given sufficiently accurate observations. © Crown Copyright 2010. Published by John Wiley & Sons, Ltd.