A demonstration of 4D‐Var using a time‐distributed background term

The standard four‐dimensional variational data assimilation (4D‐Var) method used at several major operational centres minimises a cost function which consists of a background term evaluated at the start of the assimilation window and a sum of observation terms evaluated throughout the assimilation window. This method can be interpreted as a smoother in which the background term is evolved in time using the forecast model, and the observations assimilated sequentially. However, in operational practice, the initial estimate of the background error is climatological and highly simplified, so that it is not clear whether this interpretation is useful. We thus interpret the background term as a regularisation term, rather than as an estimate of the true background error. We demonstrate this approach using a toy model whose solutions contain two time‐scales: a slow scale which can be accurately predicted and a fast scale where the model errors are too large for deterministic prediction to be possible. This represents a common situation in operational predictions. We show that, if the regularisation term is only evaluated at the beginning of the assimilation window, as in standard 4D‐Var, it is less effective at controlling rapidly growing modes. This can be resolved by applying the regularisation with equal weight throughout the window. The effect is that, in poorly observed situations, better forecasts of the slow modes can be obtained by directing information from the observations onto the slow modes rather than the unpredictable fast modes. © Crown Copyright 2010. Reproduced with the permission of HMSO. Published by John Wiley & Sons, Ltd.