Global-scale phenomena, of which there are a multitude of important applications ranging from climate science to epidemiology, can be viewed as random fields on a sphere. Huang et al. (2019) proposed a class of non-stationary random fields called the intrinsic random functions and studied their probabilistic properties. In this research, we present the statistical estimation of the degree of non-stationarity and the associated generalized covariance function. This paves the way for truly universal kriging on the sphere. This is in stark contrast with universal kriging in Euclidean space, where the variogram is often used in place of a generalized covariance function. We demonstrate our approach with simulation studies and evaluate its performance using cross validation applied to temperature anomaly data of Earth's troposphere.