Computation of isotropic random fields on spheres via needlet decomposition

Abstract

Isotropic random fields on the sphere have applications in environmental models and astrophysics. The classic Karhunen-Lo\`{e}ve expansion in terms of spherical harmonics has a drawback of requiring full observation of the random field over the sphere. Attempting to solve this difficulty, we study decomposition by needlets --- a highly localised basis --- of an isotropic random field on the sphere. We prove the $L_2$ convergence of the needlet  decomposition of a two-weakly isotropic random field. We use a quadrature rule to construct fully discrete needlets for computation and give the truncation error of the discrete needlet approximation for random fields.