Stability and error estimates for vector field interpolation and decomposition on the sphere with RBFS

A new numerical technique based on radial basis functions (RBFs) is presented for .tting a vector .eld tangent to the sphere, S2, from samples of the .eld at "scattered" locations on S2. The method naturally provides a way to decompose the reconstructed .eld into its individual Helmholtz-Hodge components, i.e., into divergence-free and curl-free parts, which is useful in many applications from the atmospheric and oceanic sciences (e.g., in diagnosing the horizontal wind and ocean currents). Several approximation results for the method will be derived. In particular, Sobolevtype error estimates are obtained for both the interpolant and its decomposition. Optimal stability estimates for the associated interpolation matrices are also presented. Finally, numerical validation of the theoretical results is given for vector .elds with characteristics similar to those of atmospheric wind .elds.