A new proposal for spherical cap harmonic modelling

The geomagnetic field above the surface of the Earth in the current-free region may be expressed as the gradient of a scalar potential solving Laplace's equation. For regions with a fairly dense coverage of data at different altitudes, a regional model ought to offer a better spatial resolution of the regional field over the volume under study than a global field expanded in spherical harmonics (SH). The spherical cap harmonics analysis (SCHA) is an attractive regional modelling tool having close relationship with global SH. With the SCHA adopted so far, difficulties arise in upward continuation and in establishing a relationship between global and local Gauss coefficients. Such a relationship would be useful, for instance, for introducing prior constraint on an inverse problem dealing with the estimation of local Gauss coefficients based upon a local data set. In this paper, we show that these difficulties are overcome if the SCHA modelling is formulated as a boundary value (BV) problem in a cone bounded radially by the surface of the Earth and an upper surface suitable for satellite data, and bounded laterally in order to encompass a specific region of study. Although the example worked out here applies only to a limited class of fields, which verifies some special flux condition, the ideas behind this formalism are quite general and should offer a new way of processing data in a bounded region of space.