Smooth vector fields defined on a spherical domain are principal objects of study inmany branches of science. Many geophysical and environmental processes occur as vector fields on a sphere, and usually have some important features, such as being tangential to a sphere, or derived from the gradient of harmonic scalar potentials, while exhibit different scale of variations and local features. Existing literature usually focuses on using large spatial scale component basis functions (spherical harmonics), which is not suitable for capturing local and non-Gaussian features. In this dissertation, we propose a new representation of functions on a sphere, localized in both space and frequency domain, and a Bayesian sparse regression framework for vector field mean estimation. The model is fitted efficiently by a Markov Chain Monte Carlo (MCMC) scheme employing Gibbs Sampling algorithm, and provides uncertainty quantication of the fitted field as a by-product. The validity of the framework and model fitting procedure are investigated by an extensive simulation study. We demonstrate practical utility of our method through applications to synthetic data generated from the known crustal field models (CHAOS-6) and satellite survey (CHAMP) data to reconstruct lithospheric magnetic field.