Axially symmetric models for global data: A journey between geostatistics and stochastic generators

Decades of research in spatial statistics have prompted the development of a wide variety of models and methods whose primary goal is optimal linear interpolation (kriging), as well as sound assessment of the associated uncertainty (kriging variance). While kriging is of paramount importance for scientific investigations requiring high‐resolution maps, spatial statistics can be used for other classes of applications as well. Indeed, new areas are emerging where the main goal is to simulate from a statistical model whose parameters have been estimated from the data. This paper focuses on two different ways to model global data with axially symmetric Gaussian processes, for which the covariance function is nonstationary over latitudes and stationary over longitudes. Both strategies are illustrated through a global data set on surface temperatures generated by the National Center for Atmospheric Research (NCAR). On the one hand, we downscale surface temperatures through a classical geostatistical approach. We exploit Gaussianity assumption to focus on the second‐order structure, and we develop a novel class of axially symmetric models inspired from currently available isotropic models. We also propose a new covariance model that is axially symmetric. Covariance‐based approaches are notorious for their computational burden, and a considerable amount of recent literature has been devoted to overcome this problem. We propose a simulation‐based approach that works for processes defined on a lattice only. For such an approach, kriging cannot be performed as there is an underlying continuous process. At the same time, inference can be performed exactly on extremely large data sets.