Construction of correlation functions in two and three dimensions

Abstract

This article focuses on the construction, directly in physical space, of simply parametrized covariance functions for data‐assimilation applications. A self‐contained, rigorous mathematical summary of relevant topics from correlation theory is provided as a foundation for this construction. Covariance and correlation functions are defined, and common notions of homogeneity and isotropy are clarified. Classical results are stated, and proven where instructive. Included are smoothness properties relevant to multivariate statistical‐analysis algorithms where wind/wind and wind/mass correlation models are obtained by differentiating the correlation model of a mass variable. the Convolution Theorem is introduced as the primary tool used to construct classes of covariance and cross‐covariance functions on three‐dimensional Euclidean space R3. Among these are classes of compactly supported functions that restrict to covariance and cross‐covariance functions on the unit sphere S2, and that vanish identically on subsets of positive measure on S2. It is shown that these covariance and cross‐covariance functions on S2, referred to as being space‐limited, cannot be obtained using truncated spectral expansions. Compactly supported and space‐limited covariance functions determine sparse covariance matrices when evaluated on a grid, thereby easing computational burdens in atmospheric data‐analysis algorithms. Convolution integrals leading to practical examples of compactly supported covariance and cross‐covariance functions on R3 are reduced and evaluated. More specifically, suppose that gi and gj are radially symmetric functions defined on R3 such that gi(x) = 0 for |x| > di and gj(x) = 0 for |xv > dj, O < di,dj ≦, where |. | denotes Euclidean distance in R3. the parameters di and dj are ‘cut‐off’ distances. Closed‐form expressions are determined for classes of convolution cross‐covariance functions Cij(x,y) := (gi * gj)(x‐y), i ≠ j, and convolution covariance functions Cii(x,y) := (gi * gi)(x‐y), vanishing for |x ‐ y| > di + dj and |x ‐ y| > 2 di, respectively, Additional covariance functions on R3 are constructed using convolutions over the real numbers R, rather than R3. Families of compactly supported approximants to standard second‐ and third‐order autoregressive functions are constructed as illustrative examples. Compactly supported covariance functions of the form C(x,y) := Co(|x ‐ y|), x,y ∈ R3, where the functions Co(r) for r ∈ R are 5th‐order piecewise rational functions, are also constructed. These functions are used to develop space‐limited product covariance functions B(x, y) C(x, y), x, y ∈ S2, approximating given covariance functions B(x, y) supported on all of S2 × S2.