Regularization of spherical cap harmonics

Spherical cap harmonic analysis has become a well-known technique for regional modelling of fields that can be expressed as the gradient of a scalar potential, such as for example the geomagnetic field and its secular variation. Up to now, the method has been regularized by a purely statistical technique: coefficients that are considered to be statistically insignificantly small are simply set to zero. This method lacks physical justification and ignores resolution; individual coefficients may be small but well-resolved, while coefficients with a large value may be poorly resolved. We implement the more physical regularization technique of minimizing a certain feature of the field, for example the field strength over the cap surface, analogous to the regularization techniques widely used in global spherical harmonic analysis. The mathematical difference between spherical cap harmonics (SCHA) and global spherical harmonics analyses (SHA) lies in the basis functions. While these are completely orthogonal in SHA, this is not the case in SCHA. This leads to the existence of certain linear combinations of coefficients that hardly contribute to the field, which makes the statistical rejection criterion meaningless. With the physical regularization the individual coefficients become meaningful, as we show by modelling the secular variation from a data set of 30 years of European observatory measurements, repeat station and ground vector surveys.