## 1. Introduction

Oceanic and atmospheric processes are characterized by motions of different scales which often manifest themselves by distinct maxima in the spectral density of a background model state and in the spectra of background error covariance (BEC) matrices. Numerical modelling of the multi-scale BEC structures in variational data assimilation is a challenging task which has recently drawn a significant attention due continuous growth of computer power. In particular, it is desirable to formulate the scale-dependent BEC operators (Dance, 2004), which can account for smaller-scale components present in very high-resolution models. The term ‘multi-scale correlation function’ is also used in the theory of turbulence and reflects covariances of multifractal nature characterized by the power-law decay of correlations (e.g. Mandelbrot, 1997).

A straightforward way to construct multi-scale BEC operators is to use suitable superpositions of the single-scale correlation functions for modelling the BEC matrix elements (e.g. Hristopulos, 2003; Gaspari *et al.*, 2006). In this approach, the resulting spectrum is difficult to control directly by the free parameters of the correlation functions and care should be taken to maintain positive definiteness of the correlation matrix.

A promising approach is to introduce scale separation in the BEC models by splitting the covariance matrix into several additive single-scale components (e.g. Wu *et al.*, 2002; Purser *et al.*, 2003) and perform assimilation on a sequence of grids with increasingly fine resolution (Li *et al.*, pers. comm. 2012).

A multi-scale BEC operator can also be constructed using a polynomial of the discretized diffusion operator for representing the inverse covariance. This approach has been studied by many authors (e.g. Sasaki, 1970; Wahba and Wendelberger, 1980; Purser, 1986; McIntosh, 1990; Xu, 2005). Its attractive features are the flexibility in controlling the BEC spectrum and the low cost of computing the action of the inverse BEC matrix on a state vector. In practice, however, applications of this approach were limited to BEC operators with Gaussian-shaped correlation functions and their approximations (e.g. Weaver *et al.*, 2003; Di Lorenzo *et al.*, 2007). Among the reasons for that limited applicability is poor conditioning of the BEC operators generated by high-degree polynomials and the necessity to link polynomial coefficients with the shape of the BEC spectrum. In the recent studies of Hristopulos and Elogne (2007, 2009) and Yaremchuk and Smith (2011), correlation functions associated with an arbitrary quadratic polynomial of the homogeneous diffusion operator were obtained and relationships between the polynomial coefficients and the magnitude/length scale of the corresponding spectral peak have been provided.

In this note the result of Yaremchuk and Smith (2011) is extended for the case of an arbitrary polynomial, generating a multiple-peak BEC spectrum. Besides, it is shown that the action of the BEC operator can be reduced to a sequence of inversions of the quadratic functions of the diffusion operator, thereby relaxing the above-mentioned conditioning problem.