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| > 2di, 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.

TOPAZ4: an ocean-sea ice data assimilation system for the North Atlantic and Arctic

“…In contrast to TOPAZ3, TOPAZ4 uses smooth localisation (rather than a box-car type localisation) that yields spatially continuous analyses. The smoothing is implemented by multiplying local ensemble anomalies, or perturbations, by a quasi-Gaussian, isotropic, distance dependent localisation function (Gaspari and Cohn, 1999). The localisation radius, beyond which the ensemble-based covariance between two points is artificially reduced to zero, is uniform in space and is set to 300 km.…”

An ETKF approach for initial state and parameter estimation in ice sheet modelling

“…To solve this, Hunt et al (2007) select a subset of observations and increase error variance for observations close to the boundary of the subset. It is equivalent for a given location to replace R with ρ −1 l •R, with ρ l some distance-based correlation matrix (such as the one suggested by Gaspari and Cohn, 1999), the origin being the given location and • denoting the elementwise (Hadamard or Schur) product. Note that ρ −1 l • R must be changed for each given location even if it uses the same observations as for another location.…”

“…The observation error covariance matrix R is modified into ρ −1 l • R, with ρ l some distance-based correlation matrix (Gaspari and Cohn, 1999). The distance l must be tuned manually in order to achieve good results.…”

Adjoint sensitivity of the model forecast to data assimilation system error covariance parameters

“…In practice, several simplifying assumptions are necessary to achieve a feasible implementation, and an increased amount of research in numerical weather prediction (NWP) is dedicated to observation-and background-error covariance modelling (Gaspari and Cohn, 1999;Hamill and Snyder, 2002;Lorenc, 2003;Buehner et al, 2005;Frehlich, 2006;Janjić and Cohn, 2006;Bannister, 2008a,b) and to the development of effective techniques for diagnosis, estimation, and tuning of unknown error covariance parameters in both variational and Kalman filter-based assimilation systems (Dee, 1995;Dee and Da Silva, 1999;Desroziers and Ivanov, 2001;Desroziers et al, 2005;Chapnik et al, 2006;Desroziers et al, 2009;Li et al, 2009).…”