Comparison of inversion codes for polarized line formation in MHD simulations. I. Milne-Eddington codes

Milne-Eddington (M-E) inversion codes for the radiative transfer equation are the most widely used tools to infer the magnetic field from observations of the polarization signals in photospheric and chromospheric spectral lines. Unfortunately, a comprehensive comparison between the different M-E codes available to the solar physics community is still missing, and so is a physical interpretation of their inferences. In this contribution we offer a comparison between three of those codes (VFISV, ASP/HAO, and HeLIx$^+$). These codes are used to invert synthetic Stokes profiles that were previously obtained from realistic non-grey three-dimensional magnetohydrodynamical (3D MHD) simulations. The results of the inversion are compared with each other and with those from the MHD simulations. In the first case, the M-E codes retrieve values for the magnetic field strength, inclination and line-of-sight velocity that agree with each other within $\sigma_B \leq 35$ (Gauss), $\sigma_\gamma \leq 1.2\deg$, and $\sigma_{\rm v} \leq 10$ ms$^{-1}$, respectively. Additionally, M-E inversion codes agree with the numerical simulations, when compared at a fixed optical depth, within $\sigma_B \leq 130$ (Gauss), $\sigma_\gamma \leq 5\deg$, and $\sigma_{\rm v} \leq 320$ ms$^{-1}$. Finally, we show that employing generalized response functions to determine the height at which M-E codes measure physical parameters is more meaningful than comparing at a fixed geometrical height or optical depth. In this case the differences between M-E inferences and the 3D MHD simulations decrease to $\sigma_B \leq 90$ (Gauss), $\sigma_\gamma \leq 3\deg$, and $\sigma_{\rm v} \leq 90$ ms$^{-1}$.