## 1. Introduction

Four-dimensional variational data assimilation (4D-Var) has been used successfully at many major operational centres for several years. Examples are the Met Office (Rawlins *et al.*, 2007) and the European Centre for Medium-Range Weather Forecasts (ECMWF: Klinker *et al.*, 2000). In each case the operational introduction of 4D-Var has resulted in significant improvements in performance. However, the theoretical justification of 4D-Var, for instance given by Lorenc (1986), assumes a linear and perfect forecast model and Gaussian errors with zero mean in the background forecast and observations. Under these assumptions, 4D-Var gives a statistically optimal estimate of the state of the atmosphere. As a result, much research has been carried out since the operational introduction of 4D-Var to improve the formulation so that these assumptions can be relaxed. Examples are the nonlinear transform used to assimilate humidity into the ECMWF model by Hólm (2007) and the use of weak-constraint 4D-Var (Trémolet, 2006) to allow for model error.

It is difficult, however, to reconcile the theoretical limitations of 4D-Var with its practical success in situations far from those for which it is valid. This has motivated studies of which aspect of 4D-Var contributes most to its performance, for instance those by Lorenc and Rawlins (2005) and Laroche *et al.* (2007), which demonstrate that much of the improvement comes from incorporating a linearized version of the forecast model within the system. Diagnostic studies, for instance that by Cardinali *et al.* (2004), show that most of the information in an analysis comes from earlier observations via the background state. In the ECMWF 12 hour cycled system, Cardinali *et al.* showed that 85% of the information typically came from the background. The effect is that the analysis error is only slightly smaller than the background error.

Satisfactory performance of a cycled system requires that the growth of the analysis error during the assimilation window is on average compensated for by the reduction in error due to the observations. This is achieved in operational systems. Since the next background error is the analysis error evolved through the assimilation window, this can only be reconciled with Cardinali *et al.*'s results if the analysis error does not project strongly on to rapidly growing modes. The theory of 4D-Var shows that the analysis preferentially uses rapidly growing modes to fit the data; thus the analysis error in these modes is small. This is confirmed by toy-model experiments, e.g. Trevisan *at al.* (2010). The evidence from operational performance (Piccolo, 2010) is that this must be done efficiently, so that the subsequent error growth during the assimilation window is small. This is despite the fact that the background-error covariance matrix used in all current operational systems is essentially climatological, and thus contains considerable averaging. It is thus likely that this matrix underestimates the true errors in rapidly growing modes. This appears inconsistent with the observed efficiency of the analysis in correcting them.

In this article we illustrate how optimum forecast performance is obtained by forcing the analysis to use only slowly growing modes to fit the observations. This results in greater weight being given to observations from earlier assimilation cycles. Given sufficiently accurate observations, we show that the subsequent error growth in the forecast can also be reduced. We demonstrate this using the three-body model used for studies of 4D-Var by Watkinson (2006). This model supports rapidly growing perturbations, so is suitable for investigating the issue raised in the previous paragraph. The three bodies are referred to as sun, planet and moon. In this model there are two time-scales: a slow time-scale associated with the motion of the planet round the sun and a fast time-scale associated with the motion of the moon round the planet. We aim to make useful predictions of both modes. The case in which the fast time-scale is not accurately predicted by the model and has to be treated as ‘noise’ is discussed in a companion article (Cullen, 2010). We use two different versions of this model as the ‘truth’ model, which generates the trajectory from which the observations are drawn, and the ‘forecast’ model. This ensures that the forecast diverges from the truth unless the observations are successfully assimilated, as is the case in the real atmospheric system.

We can justify the use of a smaller background-error covariance matrix by thinking of 4D-Var as a method of regularizing the otherwise ill-posed problem of fitting a model state to the observations. This is described in Johnson *et al.* (2005b), where it is shown that 4D-Var corresponds to a Tikhonov regularization using the forecast background. We show that the 4D-Var algorithm can be re-interpreted as a regularization using a complete model trajectory, under the assumption that the model trajectory is accurately represented by the evolution of the Jacobian of the model starting from a given initial state. The studies of Lorenc and Rawlins and of Laroche *et al.* cited above suggest that the use of the trajectory is an essential part of the success of 4D-Var. The ‘optimal’ regularization would be the choice of background-error covariance matrix that minimized the short-range forecast error, which will not necessarily be the ‘true’ background-error covariance matrix.

There is a close link between this procedure and the use of a model-state control variable to represent model error in weak-constraint 4D-Var (Trémolet, 2006). If we consider an arbitrarily long window, so that the background becomes irrelevant, the weak-constraint method fits a time sequence of observations with a model trajectory to which small corrections are applied periodically. The regularization approach would seek the smallest corrections that would have to be made to a model trajectory to enable it to fit the observations to within the observational error over a long time period. The optimal-state estimation approach would make the corrections depend on the model error, so that large corrections could be made to modes where the model is inaccurate. We show that in situations where the model error growth is slower than the growth of perturbations under the action of the model, the regularization approach makes corrections to the trajectory that are smaller than the model error and is successful in improving the forecasts. However, unlike the long-window approach, the trajectory is computed sequentially rather than by solving a simultaneous minimization problem. It would be of interest to see whether further benefit could be obtained by applying the same regularization matrix within a simultaneous minimization problem.

All calculations have been carried out using Mathematica® 6.0 (Wolfram, 2007).