A generalized, compactly supported correlation function for data assimilation applications

Abstract

This work introduces a new, compactly supported correlation function that can be inhomogeneous over Euclidean three‐space, anisotropic when restricted to the sphere, and compactly supported on regions other than spheres of fixed radius. This function, which we call the Generalized Gaspari–Cohn (GenGC) correlation function, is a generalization of the compactly supported, piecewise rational approximation to a Gaussian introduced by Gaspari and Cohn in 1999 and its subsequent extension by Gaspari et al in 2006. The GenGC correlation function is a parametric correlation function that allows two parameters and to vary, as functions, over space, whereas the earlier formulations either keep both and fixed or only allow to vary. Like these earlier formulations, GenGC is a sixth‐order piecewise rational function (fifth‐order near the origin), while the coefficients now depend explicitly on the values of both and at each pair of points being correlated. We show that, by allowing both and to vary, the correlation length of GenGC also varies over space and introduces inhomogeneous and anisotropic features that may be useful in data assimilation applications. Covariances produced using GenGC are computationally tractable due to their compact support and have the added flexibility of generating compact support regions that adapt to the input field. These features can be useful for covariance modeling and covariance tapering applications in data assimilation. We derive the GenGC correlation function using convolutions, discuss continuity properties relating to and and its correlation length, and provide one‐ and two‐dimensional examples that highlight its anisotropy and variable regions of compact support.