The solution of Laplace's equation, in spherical coordinates, is developed for the boundary value problem appropriate to fitting the geomagnetic field over a spherical cap. The solution involves associated Legendre functions of integral order but nonintegral degree. The basis functions comprise two infinite sets, within each of which the functions are mutually orthogonal. The series for the expansion of the potential can by design be differentiated term by term to yield uniformly convergent series for the field components. The method is demonstrated by modeling the International Geomagnetic Reference Field 1980 at the earth's surface and upward continuing it to 300 and 600 km. The rate of convergence of the series is rapid, and standard errors of fit as low as the order of a nanotesla can be obtained with a reasonable number of coefficients. Upward continuation suffers from not considering data outside the cap, the deterioration being confined to the boundary at low continuation altitudes but spreading inward over the cap with increasing altitudes. At 600 km the standard error of upward continuation is about 5 times the standard error of fit. Both the fit and the upward continuation can be greatly improved at a given truncation level by subtraction of a known spherical harmonic potential determined from data from the whole earth.

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S U M M A R YThe 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.

Regional models of the earth's magnetic field have developed considerably since the days of hand contouring. They have incorporated varying levels of mathematical sophistication, from partial "mutual consistency" on a surface to full electromagnetic consistency in three-dimensional space. Each method has its advantages and its limitations. Some of the methods allow for radial variation so that data acquired at different altitudes can be analyzed directly, and so that fields from the resulting models can be calculated at any altitude. For other methods, altitude continuation is theoretically possible but numerically difficult and statistically unreliable. Some methods are limited, for numerical reasons, to small areas or are able to represent only fairly long wavelengths. The different methods also differ greatly with respect to their mathematical and computational complexity, the less complex methods also usually being more subjective.

[1] A spherical cap harmonic analysis (SCHA) technique is introduced for mapping the 2-D high-latitude ionospheric convection pattern based on Super Dual Auroral Radar Network (SuperDARN) velocity measurements. The current method for generating such maps is the FIT technique which generates global-scale maps over the entire convection region. This is accomplished by combining observations with a statistical model to prevent unphysical solutions in areas away from the observation points and by forcing the plasma flow to zero at the low-latitude boundary of the convection zone. Both constraints distort the mapped convection and require a preconception of where the plasma flow lines should close. By focusing on mapping the convection over a region well covered by velocity observations, the SCHA technique is freed of these constraints and more accurately reproduces local convection. We generate large-scale convection maps from SuperDARN data for various interplanetary magnetic field (IMF) conditions during periods of widespread radar coverage to show the patterns are consistent with expectations for various IMF configurations. We validate the SCHA maps by comparing them with the 2-D ion drifts measured by the DMSP satellites and with the 2-D convection vectors obtained by merging SuperDARN measurements at beam crossings. The SCHA technique is shown to perform comparably to the FIT technique over regions of good data coverage. For limited data coverage and over regions of highly variable flow, particularly near the equatorward edge of the mapping region, the SCHA technique provides a better solution for mapping ionospheric convection based on SuperDARN radar observations.