The last decades have seen an unprecedented increase in the availability ofdata sets that are inherently global and temporally evolving, from remotelysensed networks to climate model ensembles. This paper provides a view ofstatistical modeling techniques for space-time processes, where space is thesphere representing our planet. In particular, we make a distintion between (a)second order-based, and (b) practical approaches to model temporally evolvingglobal processes. The former are based on the specification of a class ofspace-time covariance functions, with space being the two-dimensional sphere.The latter are based on explicit description of the dynamics of the space-timeprocess, i.e., by specifying its evolution as a function of its past historywith added spatially dependent noise. We especially focus on approach (a), where the literature has been sparse. Weprovide new models of space-time covariance functions for random fields definedon spheres cross time. Practical approaches, (b), are also discussed, withspecial emphasis on models built directly on the sphere, without projecting thespherical coordinate on the plane. We present a case study focused on the analysis of air pollution from the2015 wildfires in Equatorial Asia, an event which was classified as the year'sworst environmental disaster. The paper finishes with a list of the maintheoretical and applied research problems in the area, where we expect thestatistical community to engage over the next decade.
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