## 1. Introduction

The background error covariance matrix (BECM) is a key component of any assimilation system (Kalnay, 2003). It represents the statistical error of the background state with respect to the true state. The background state usually comes from a model forecast and its statistics can therefore be evaluated, but the true state is by definition unknown. The aim of the assimilation process is to approximate it using a numerical model, a background state and some observations. BECM accounts for the background error variance (or standard deviation) and for the background error correlation. These two components can be assessed separately. The background error variance has to be compared to the observation error variance in order to evaluate their relative weight in the analysis produced by the assimilation process. The background error correlation combines spatial correlation and correlation between the different model variables. The spatial correlation spreads the information brought in by observations at their locations to the neighbouring grid points. It is usually expressed through the introduction of length-scales. Correlation between the variables transfers the information to other model variables, if these are physically related to the variable directly linked to the measurements and if the physical relations between the variables are represented in the model.

Evaluation of the BECM is an essential challenge as it conditions the assimilation analysis. Nevertheless, the BECM comprises statistical information that is not directly obtainable since the true state is unknown. To get around this problem, in atmospheric chemistry quite often only the variance part of the BECM is considered. When the correlation part of the BECM is considered, its model is generally basic. As an example, in the ASSET intercomparison of ozone analyses (Geer *et* *al.*, 2006) only three of five systems based on chemical transport models (CTMs) accounted for background error horizontal correlations: the Royal Netherlands Meteorological Institute (KNMI) system (which operates the TM5 CTM) and the Mocage-Palm (later referred to as Valentina) system, both with a constant length-scale; and MIMOSA for which the correlations were flow dependent and specified in terms of distances and of the potential vorticity field. Moreover, the Kalman filter approach used in the MIMOSA and KNMI–TM5 systems allowed advection of background error variances. Since then, except for the recent work of Schwinger and Elbern (2010), evaluation of the BECM for global atmospheric chemistry has not been the object of thorough investigations. Previous work concerning the BECM is mainly focused on its formulation for atmospheric applications (Bannister 2008a, 2008b) or atmospheric chemistry (Pannekoucke and Massart, 2008; Elbern *et al.*, 2010).

Meanwhile, in meteorology background (and analysis) errors have been successfully assessed using ensemble data assimilation systems. Exploitation in a cycled assimilation system could produce an estimate of a flow-dependent BECM. The ensemble approach is not restrained to Kalman filter methodologies. Lorenc (2003) and Buehner (2005) illustrated how an ensemble-estimate BECM can be estimated within a variational assimilation system. Useful information could statistically be extracted from an adequately perturbed ensemble data assimilation system (Belo Pereira and Berre, 2006). The use of ensembles in combination with a variational assimilation scheme is relatively unexplored in atmospheric chemistry data assimilation. The main purpose of this study is to investigate the potential of an ensemble of atmospheric ozone analyses to provide useful flow-dependent estimates of the background error variance and correlations, in a four-dimensional variational assimilation suite. Moreover, we assess the sensitivity of the analyses to the choice of fixed or estimated variance and correlations. Our results are based on stratospheric and upper tropospheric ozone analyses realized during a one-year period starting from January 2008.