An Approach to Producing Space–Time Covariance Functions on Spheres

Abstract

For space–time processes on global or large scales, it is critical to use models that respect the Earth's spherical shape. The covariance functions of such processes should be not only positive definite on sphere × time, but also capable of capturing the dynamics of the processes well. We develop space–time covariance functions on sphere × time that are flexible in producing space–time interactions, especially space–time asymmetries. Our idea is to consider a sum of independent processes in which each process is obtained by applying a first-order differential operator to a fully symmetric process on sphere × time. The resulting covariance functions can produce various types of space–time interactions and give different covariance structures along different latitudes. Our approach yields explicit expressions for the covariance functions, which has great advantages in computation. Moreover, it applies equally well to generating asymmetric space–time covariance functions on flat or other spatial domains. We study various characteristics of our new covariance functions, focusing on their space–time interactions. We apply our model to a dataset of total column ozone levels in the Northern hemisphere.