An Improved Virial Estimate of Solar Active Region Energy
  The MHD virial theorem may be used to estimate the magnetic energy of active regions based on vector magnetic fields measured at the photosphere or chromosphere. However, the virial estimate depends on the measured vector magnetic field being force-free. Departure from force-freeness leads to an unknown systematic error in the virial energy estimate, and an origin dependence of the result. We present a method for estimating the systematic error by assuming that magnetic forces are confined to a thin layer near the photosphere. If vector magnetic field measurements are available at two levels in the low atmosphere (e.g. the photosphere and the chromosphere), the systematic error may be directly calculated using the observed horizontal and vertical field gradients, resulting in an energy estimate which is independent of the choice of origin. If (as is generally the case) measurements are available at only one level, the systematic error may be approximated using the observed horizontal field gradients together with a simple linear force-free model for the vertical field gradients. The resulting `improved' virial energy estimate is independent of the choice of origin, but depends on the choice of the model for the vertical field gradients, i.e. the value of the linear force-free parameter $\alpha$. This procedure is demonstrated for five vector magnetograms, including a chromospheric magnetogram. 
M.S. Wheatland and T.R. Metcalf