Spatially restricted data distributions on the sphere: the method of orthonormalized functions and applications

In many geoscientific applications data are irregularly distributed and not globally available, e.g. caps around the poles which are uncovered due to nonpolar satellite orbits, or signals being defined solely on bounded regions on the globe. Starting from a sequence of base functions with global support. which in the present case is composed of spherical harmonics being initially non-orthogonal on a bounded subdomain, a set of functions is generated that constitutes an orthonormal basis. Different approaches to realize this transformation are studied and compared with respect to numerical stability and computational effort, and the corresponding effects on the coefficient recovery are investigated. A number of synthetic tests demonstrate the applicability, the benefit, but also the limitations, of this method.