A semiparametric class of axially symmetric random fields on the sphere

The paper provides a way to model axially symmetric random fields defined over the two-dimensional unit sphere embedded in the three-dimensional Euclidean space. Specifically, our strategy is to integrate an isotropic random field on the sphere over longitudinal arcs with a given central angle. The resulting random field is shown to be axially symmetric and to have the arc central angle as a tuning parameter that allows for isotropy as well as for longitudinal independence as limit cases. We then consider multivariate longitudinally integrated random fields, having the same properties of axial symmetry and a tuning parameter (arc central angle) proper to each random field component. This construction allows for a unified framework for vector-valued random fields that can be geodesically isotropic, axially symmetric, or longitudinally independent. Additionally, all the components of the vector random field are allowed to be cross-correlated. We finally show how to simulate the proposed axially symmetric scalar and vector random fields through a computationally efficient algorithm that exactly reproduces the desired covariance structure and provides approximately Gaussian finite-dimensional distributions.