Determination of equivalent current sources from spherical cap harmonic models of geomagnetic field variations

The intrinsic ability of the method of spherical cap harmonic analysis to separate external and internal sources allows the calculation of equivalent ionospheric and induced currents that are able to explain variations of the geomagnetic field over a portion of the earth's surface. Formulations for current densities and current functions are derived and found to be analogous to those derived for the global case from conventional spherical harmonic analysis. Although spherical cap formulations for current density have been given by another worker, they were incorrect because of an error in defining the equivalent current. An example of the use of current functions is given by modelling variations from hourly mean values recorded at 40 geomagnetic observatories over Europe during a very quiet day in 1978. The modelling can be done spatially for each of the 24 hours separately, or spatially and temporally either by expressing each spatial coefficient as a Fourier series or by smoothing the spatial coefficients obtained from the separate hourly models.