Derivation and Application of a Scaling Between Hinode/SP and SDO/HMI Vector Magnetic Fields to Improve Magnetic Field Extrapolations

Full-disk measurements of the solar magnetic field by the Helioseismic and Magnetic Imager (HMI) are often used for magnetic field extrapolations, but its limited spatial and spectral resolution can lead to significant errors. We compare HMI data with observations of NOAA 12104 by the Hinode Spectropolarimeter (SP) to derive a scaling curve for the magnetic field strength, $B$. The SP data in the \ion{Fe}{i} lines at 630\,nm were inverted with the SIR code. We find that the Milne-Eddington inversion of HMI underestimates $B$ and the line-of-sight flux, $\Phi$, in all granulation surroundings by an average factor of 4.5 in plage and 9.2 in the quiet Sun in comparison to the SP. The deviation is inversely proportional to the magnetic fill factor, $f$, in the SP results. We derived a correction curve to match the HMI $B$ with the effective flux $B\,f$ in the SP data that scaled HMI $B$ up by 1.3 on average. A comparison of non-force-free field extrapolations over a larger field of view without and with the correction revealed minor changes in connectivity and a proportional scaling of electric currents and Lorentz force ($\propto B \sim 1.3$) and free energy ($\propto B^2\sim 2$). Magnetic field extrapolations of HMI vector data with large areas of plage and quiet Sun will underestimate the photospheric magnetic field strength by a factor of 5--10 and the coronal magnetic flux by at least 2. An HMI inversion including a fill factor would mitigate the problem.