Least squares spherical harmonics approximation on the Cubed Sphere

Abstract

The Cubed Sphere grid is an important tool to approximate functions or data on the sphere. We introduce an approximation framework on this grid based on least squares and on a suitably chosen subspace of spherical harmonics. The main claim is that for the equiangular Cubed Sphere with step size π/(2N), a relevant spherical harmonics subspace is the one of all SH of degree less than 2N. This choice, which matches the Nyquist’s cutoff frequency along the equatorial great circle, provides an approximation problem both well-posed and well-conditioned. A series of theoretical and numerical results supporting this fact are presented.