When attempting to predict array signal-to-noise gain in terms of correlation functions, a first approximation is obtained by assuming ideal propagation, zero time delay, and two-dimensional or three-dimensional isotropic background noise. These assumptions lead to the familiar noise correlation functions J0(X) and sin X/X. In this paper, noise correlation functions are given for anisotropic background noise fields and isotropic fields are treated as special cases. It is shown that only for a special orientation of elements does the noise correlation function reduce to J0(X) in a two-dimensional isotropic field. For a noise field whose power in a given direction is proportional to the radius vector of an oblate spheroid it is shown that when the elements are horizontal the noise correlation function departs from sin X/X toward J0(X) as the noise field changes from three-dimensional isotropic to two-dimensional isotropic. However, when the elements are vertical, the noise correlation function is shown to depart from sin X/X toward unity. Quantitative results are given for noise fields of different anistropy.