## 1. Introduction

Data-assimilation methods based on variational or Kalman-filter techniques are widely used to determine initial states for numerical weather prediction (NWP). This is done by combining direct and indirect observations of meteorological fields, as well as a background (*a priori*) state corresponding to a short-range forecast. Specifying realistic background-error covariances is particularly crucial for correctly weighting and propagating observed information (Bannister, 2008). In a typical NWP system, the state vector currently has a dimension of about 10^{8}, leading to a background-error covariance matrix (denoted by **B**) containing about 10^{16} elements. It is thus impossible to compute such a matrix explicitly, firstly because it cannot be stored in current computers and secondly because of a lack of available information (Dee, 1995). In variational systems, a solution is to model the **B** matrix in the form of sparse operators applied sequentially. In particular, a spectral block-diagonal approach is convenient to represent spatial covariances and scale dependences in an economical way (Rabier *et al.*, 1998; Derber and Bouttier, 1999). However, it relies on a horizontal homogeneity assumption, which prevents the representation of horizontal and temporal variations of covariances (Berre and Desroziers, 2010). This can be relaxed partly (Fisher, 2003) by representing local standard deviations in grid-point space and spatial correlations in wavelet space, and by using flow-dependent balance equations. Such a model is now commonly calibrated by using data provided by ensemble variational assimilation experiments (Fisher, 2003; Belo Pereira and Berre, 2006). The calibration is often performed off-line to obtain a climatological estimate of **B**, by using an ensemble experiment carried out over a period of a few weeks. However, real-time ensemble variational assimilation systems are also being developed, and they have been running operationally at Météo-France since 2008 (Berre *et al.*, 2007; Berre *et al.*, 2009) and also at the European Centre for Medium-Range Weather Forecasts (ECMWF) since 2010 (Bonavita *et al.*, 2011). Such real-time ensembles are currently used to calculate flow-dependent background-error standard deviations, and also to provide perturbed initial states for ensemble prediction systems. Researchers are also considering, in future, the use of such real-time ensembles to calculate flow-dependent correlations in wavelet space for instance. However, the first operational implementations such as at ECMWF are currently restricted to climatological correlation estimates in wavelet space. Moreover, these climatological wavelet-based correlations have not been much documented yet, not only in terms of diagnostics regarding associated geographical variations (in particular for vertical correlations) but also in terms of impact on the forecast quality. The goal of this article is thus to document the wavelet representation of geographical variations of three-dimensional (3D) correlations in a global NWP context, and also their impact on forecast quality.

The structure of the article is as follows. In section 2, the formulation of **B** as a sequence of sparse operators is presented, with either spectral- or wavelet-based correlations. In section 3, a diagnostic comparison between spectral and wavelet approaches is shown, with regards to 3D correlations and sensitivity to the choice of calibration period. Section 4 is an evaluation of the impact of the wavelet **B** matrix on the global model forecast quality for two different periods. Finally, conclusions are given in section 5.