Construction of correlation functions in two and three dimensions

Abstract

This article focuses on the construction, directly in physical space, of simply parameterized 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 R 3 . Among these are classes of compactly supported functions that restrict to covariance and cross-covariance functions on the unit sphere S 2 , and that vanish identically on subset...