Spherical cap harmonics revisited and their relationship to ordinary spherical harmonics

Abstract

The “global” representation of the geomagnetic field in terms of ordinary spherical harmonics (SHs) and its corresponding set {g,h} of coefficients has been studied extensively, but the “local” representation in terms of spherical cap harmonics (SCHs) and its corresponding set {G,H} of coefficients is not yet well understood. This paper clarifies some of the main properties of the SCHs and their proper use along with their relationship with the SHs. In particular, it shows that for the spherical cap part of a global field specified by spherical harmonics there is a strict relation between the ordinary Legendre functions of the global representation and the fractional functions of the local expansion; hence we can express the set of coefficients {G,H} in terms of the set {g,h}. Finally, some attention will be given to the role of the leading (n = 0, m = 0) term of the SCH expansion.