A novel technique is presented which 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 assumptions 
could be made: to 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, or to characterise 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 photosphere is deduced from 
information held within a time-series of SOHO MDI 
magnetograms centred on the time of interest, and following 
appropriate rotations is combined with a full disc of 
visible data for that time. Comparison is made with the more 
usual synoptic map of observations to test the time 
sensitivity of the problem. Several methods of extrapolation 
to the far side are assessed using data centred 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.