## 1. Introduction

Data assimilation is a method for combining model forecast data with observational data in order to forecast more accurately the state of a system. One of the most popular data assimilation methods used in modern numerical weather prediction (NWP) is four-dimensional variational data assimilation (4DVar) (Sasaki, 1970; Talagrand, 1981; Lewis *et al.*, 2006), which seeks initial conditions such that the forecast best fits both the observations and the background state (which is usually obtained from the previous forecast) within an interval called the assimilation window. Currently, in most operational weather centres, systems and states of dimension or higher are considered, whereas there are considerably fewer observations, usually (for reviews on data assimilation methods see Daley, 1991; Nichols, 2010).

Linearized 4DVar can be shown to be equivalent to Tikhonov, or *L*_{2}-norm regularization, a well-known method for solving ill-posed problems (Johnson *et al.*, 2005). Such problems appear in a wide range of applications (Engl *et al.*, 1996) such as geosciences and image restoration: the process of estimating an original image from a given blurred image. From the latter work it is known that by replacing the *L*_{2}-norm penalty term with an *L*_{1}-norm penalty function, image restoration becomes edge-preserving as the process does not penalize the edges of the image. The *L*_{1}-norm penalty regularization then recovers sharp edges in the image more precisely than the *L*_{2}-norm penalty regularization (Hansen, 1998; Hansen *et* *al.*, 2006). Edges in images lead to outliers in the regularization term and hence *L*_{1}-norms for the regularization terms give a better result in image restoration. This is the motivation behind our approach for variational data assimilation.

The edge-preserving property of *L*_{1}-norm regularization can be used for models that develop shocks, which is the case for moving weather fronts. In NWP and ocean forecasting, it is recognized that the 4DVar assimilation method may not give a good analysis where there is a sharp gradient in the flow, such as a front (Bennett, 2002; Lorenc, 1981). If the front is displaced in the background estimate, then the assimilation algorithm may smear the front and also underestimate the true amplitude of the shock (Johnson, 2003). In these cases the error covariances propagated implicitly by 4DVar are not representative of the correct error structures near the front. If model error is present, then there are systematic errors between the incorrect model trajectories and the observed data and therefore the strong constraint 4DVar, which assumes a perfect model, is not able to represent these errors correctly. Here we apply an *L*_{1}-norm penalty approach to several numerical examples containing sharp fronts for cases with model error. We show that the *L*_{1}-norm penalty approach applied to the gradient of the analysis vector (we call this mixed total variation (TV) *L*_{1}–*L*_{2}-norm penalty regularization) performs better than the standard *L*_{2}-norm regularization in 4DVar. With the use of the gradient operator and the *L*_{1} norm, localization of the gradient is enforced, which is important in tracking fronts. As an example we use the linear advection equation where sharp fronts and shocks are present. We use a numerical scheme that introduces some form of *model error* into the systems and find that, using an *L*_{1}-norm regularization term, applied to the gradient of the solution, fronts are resolved more accurately than with the standard *L*_{2}-norm regularization of 4DVar. Further investigation remains to be done in order to evaluate the technique in an operational setting.

Section 2 gives an introduction to 4DVar and shows its relation to Tikhonov regularization. In section 3 we introduce the new algorithms and in section 4 we explain how we solve the *L*_{1}-norm regularization problem and the mixed TV *L*_{1}–*L*_{2}-norm regularization problem. Sections 5 and 6 describe experiments using a linear advection model where the new regularization approaches are compared with standard 4DVar for cases with model error. Under these conditions it is seen that mixed TV *L*_{1}–*L*_{2}-norm regularization outperforms 4DVar where sharp fronts are present. In the final section we present conclusions and discuss future work.