Data generation for axially symmetric processes on the sphere

Abstract

In this article, we propose an algorithm that generates data on the entire sphere following the given axially symmetric covariance. We demonstrate that the axially symmetric data can be decomposed as Fourier series on circles, where the random Fourier coefficients can be expressed as circularly-symmetric complex random vectors. Our algorithm has the advantages of efficiency and scalability comparing with the classical method using the given covariance function directly. The data validation through the classical variogram estimation method demonstrates that our algorithm produces the data with smaller biases in recovering the given covariance function while maintaining the comparable mean squared errors.