A technique is described for predicting ground surface geomagnetic variations from measurements of the approaching interplanetary magnetic field (IMF) and solar wind. The method uses twin, empirical representations of the ionospheric electric and magnetic Euler potentials' response to the IMF drivers. The magnetic potential model, originally derived for mapping the large-scale field-aligned current structure, describes the curl-free component of the horizontal ionospheric current, also called the "potential current." Using approximations that the Hall and Pedersen conductances have a fixed ratio and that there are no conductivity gradients, then the Hall current is derived from the magnetic potentials. In this case the Hall current is the same as the divergence-free "equivalent current," which is used to derive the geomagnetic variations at the ground surface. The assumption that the ionospheric conductances have no gradients is avoided if the empirical model for the ionospheric electric potentials is used in addition to the magnetic potentials. In this second method the electric field provides additional information about the direction of the estimated equivalent current. Despite the approximation of a fixed conductance ratio, both calculation methods perform remarkably well for predicting the large-scale and long-period geomagnetic variations. The method that includes the electric fields has a slightly better performance, particularly in the polar cap. Corrections for the effects of currents induced underground were not applied for this demonstration. Such corrections could in principle improve the predictions, particularly for the short-period variations for which the effect is the greatest.