A novel technique is presented that aids the reconstruction of solar coronal magnetic fields by augmenting the visible solar surface boundary conditions with an estimate of the simultaneous conditions on the far side of the Sun. In converting from line-of-sight measurements, two alternative assumptions are considered: (1) ignore the plasma beta differences between corona and photosphere and so find a radial field component that is consistent with an overlying potential (or force free) field corona; and (2) characterize the change in beta by a boundary layer, matching the coronal field to the photosphere where the field, seen in projection, is taken to be wholly radial. The radial magnetic field (or radial field component) over the unseen hemisphere of the Sun is deduced from information held within a time series of the Solar and Heliospheric Observatory Michelson Doppler imager photospheric magnetograms centered on the time of interest and combined with a full disc of visible data for that time. Comparison is made with the more usual synoptic map boundary conditions to test the time sensitivity of the problem. Several methods of extrapolation to the far side are assessed using data centered on an observation at the end of August 1996, when a large equatorial coronal hole was present on one side of the Sun. The corresponding magnetic field equilibrium in the corona is found, assuming a simple potential approximation, which employs an outer source surface in addition to the derived inner boundary condition. Together with the inherent assumption that the evolution of the global field is slow, the validity of the technique is confirmed by the self-consistency of the results. ¿ 1999 American Geophysical Union