## 1. Introduction

Various methods have been proposed for modelling background-error correlations in geophysical applications of variational data assimilation (VDA) (see Bannister, 2008, for example, for a thorough review of methods used in atmospheric VDA). In ocean VDA, background-error correlation models based on the diffusion equation are popular. The method has its origins in the work of Derber and Rosati (1989), who proposed the use of an iterative Laplacian grid-point filter in order to approximate a Gaussian correlation operator. Egbert *et al.* (1994) described a close variant of the algorithm in which the Laplacian grid-point filter could be interpreted as a pseudo-time-step integration of a diffusion equation with an explicit scheme. Weaver and Courtier (2001) (hereafter WC01) described the algorithm in more detail and proposed various extensions to account for more general correlation functions than the quasi-Gaussian of the original Derber and Rosati (1989) algorithm. Correlation models based on explicit diffusion methods have been used in various VDA systems in oceanography (Weaver *et al.*, 2003; Di Lorenzo *et al.*, 2007; Muccino *et al.*, 2008; Daget *et al.*, 2009; Kurapov *et al.*, 2009; Moore *et al.*, 2011), meteorology (Bennett *et al.* 1996), and atmospheric chemistry (Geer *et al.*, 2006; Elbern *et al.*, 2010).

An explicit diffusion scheme is appealing because of its simplicity, but can be expensive if many iterations are required to keep the scheme numerically stable. This can occur when the local diffusion scale is ‘large’ relative to the local grid size. To keep the explicit scheme affordable, the correlation length-scales must be bounded even if statistics or physical considerations suggest that larger values would be more appropriate. This limitation can be overcome by reformulating the diffusion model using an implicit scheme which has the advantage of being unconditionally stable.

One-dimensional (1D) implicit diffusion operators have been used for representing temporal and vertical correlation functions (Bennett *et al.*, 1997; Chua and Bennett, 2001; Ngodock, 2005) and products of 1D implicit diffusion operators have been used for constructing two-dimensional (2D) and three-dimensional (3D) correlation models (Chua and Bennett, 2001; Zaron *et al.*, 2009). The correlation kernels associated with the 1D implicit diffusion operator belong to the family of *M*th-order autoregressive (AR) functions where *M* is the number of implicit iterations (Mirouze and Weaver, 2010; hereafter MW10). As discussed by MW10, the 1D implicit diffusion operator is closely linked to the recursive filter (Lorenc, 1992; Hayden and Purser, 1995), which has been developed extensively in meteorology for constructing correlation models in multiple dimensions (Wu *et al.*, 2002; Purser *et al.*, 2003a, 2003b; Liu *et al.*, 2007). The recursive filter has also been employed in ocean data assimilation systems (Martin *et al.*, 2007; Dobricic and Pinardi, 2008; Liu *et al.*, 2009).

The 1D implicit diffusion approach for constructing 2D and 3D correlation models can be convenient for computational reasons, but has limitations. For example, with few iterations, the product of 1D implicit diffusion operators produces a well-known spurious anisotropic response (Purser *et al.*, 2003a). Unphysical features can also appear near complex boundaries, such as coastlines or islands in an ocean model, where correlation functions cannot always be reasonably represented by a product of separable functions of the model's coordinates. Correlation models based on 2D or 3D implicit diffusion operators can overcome these limitations but are more difficult to implement since they involve the solution of a large linear system (matrices of dimension or larger in VDA). Some progress in the development of this approach has been made by Weaver and Ricci (2004) and Massart *et al.*(2012), who used sparse matrix methods to solve a 2D implicit diffusion problem directly, and by Carrier and Ngodock (2010) and S. Gratton (2011, personal communication), who used iterative methods based on conjugate gradient or multi-grid to approximate the solution of a 2D or 3D implicit diffusion problem.

Multidimensional implicit diffusion correlation operators can be interpreted in terms of smoothing norm splines, which were introduced to atmospheric data assimilation by Wahba and Wendelberger (1982) and Wahba (1982), and discussed within an oceanographic context by McIntosh (1990). In the norm spline approach, the background term of the cost function in VDA is formulated in terms of a linear combination of weighted derivative operators that penalize explicitly the amplitude and curvature of the solution. When the weighting coefficients are given by binomial coefficients, the inverse of the background-error correlation operator implied by the norm spline can be expressed as the inverse of an implicit diffusion operator. The direct penalty approach was popular in some of the early studies of four-dimensional VDA (Thacker, 1988; Sheinbaum and Anderson, 1990) but generally leads to a poorly conditioned minimization problem (Lorenc *et al.*, 2000). Effective preconditioning techniques for VDA require access to the background-error covariance operator itself. An interesting exception is the recent study of Yaremchuk *et al.* (2011), who propose a variational formulation in which the inverse of the background-error covariance is modelled directly using the inverse of a low-order (two-iteration) 3D implicit diffusion operator. No apparent conditioning problems were reported in their examples from an ocean VDA system.

The present paper has a dual purpose: first, to provide a review of the diffusion equation as a basis for constructing anisotropic and inhomogeneous correlation models for data assimilation; and second, to illustrate how fundamental parameters that control spatial smoothness properties of these models can be estimated using ensemble methods. Section 2 brings together key results from data assimilation and geostatistics on the isotropic diffusion problem. Diffusion is considered both on the sphere and in the *d*-dimensional Euclidean space. Analytical expressions for the isotropic correlation functions implied by appropriately normalized explicit and implicit diffusion in these spaces are presented and compared. The Daley length-scale is used as a standard parameter for comparing the different functions, and expressions relating it to the parameters of the diffusion-model are established.

The results from section 2 provide the foundation for building anisotropic correlation models with the diffusion equation. This is discussed in sections 3 and 4. The Daley tensor is introduced, which is defined as the negative inverse of the tensor of second derivatives of the correlation function evaluated at zero distance (the Hessian tensor). The Daley tensor is an anisotropic generalization of the Daley length-scale. Expressions relating the Daley tensor to the diffusion tensor of the diffusion models are given. Section 4 discusses techniques for estimating the Daley tensor from statistics of a sample of simulated errors such as those that would be available from an ensemble data assimilation system. Idealized experiments are then presented to compare the effectiveness of two of the estimation techniques. Conclusions are given in section 5. Appendix A provides a derivation of the relationship between the Daley and diffusion tensors for the correlation functions represented by the implicit diffusion equation in . Appendix B provides a derivation of the key formulae for estimating the Daley tensor.