Context. With the advent of next generation high resolution telescopes, our understanding of how the magnetic field is organized in the internetwork (IN) photosphere is likely to advance significantly.
Aims: We aim to evaluate the extent to which we can retrieve accurate information about the magnetic vector in the IN photosphere using inversion techniques.
Methods: We use a snapshot produced from high resolution three-dimensional magnetohydrodynamic (MHD) simulations and employ the Stokes Inversions based on Response functions (SIR) code to produce synthetic observables in the same near infrared spectral window as observed by the GREGOR Infrared Spectrograph (GRIS), which contains the highly magnetically sensitive photospheric Fe I line pair at 15 648.52 Å and 15 652.87 Å. We then use a parallelized wrapper to SIR to perform nearly 14 million inversions of the synthetic spectra to test how well the `true' MHD atmospheric parameters can be constrained statistically. Finally, we degrade the synthetic Stokes vector spectrally and spatially to GREGOR resolutions and examine how this influences real observations, considering the impact of stray light, spatial resolution and signal-to-noise (S/N) in particular.
Results: We find that the depth-averaged parameters can be recovered by the inversions of the undegraded profiles, and by adding simple gradients to magnetic field strength, inclination, and line of sight velocity we show that an improvement in the χ2 value is achieved. We also evaluate the extent to which we can constrain these parameters at various optical depths, with the kinematic and thermodynamic parameters sensitive deeper in the atmosphere than the magnetic parameters. We find the S/N and spatial resolution both play a significant role in determining how the degraded atmosphere appears. At the same time, we find that the magnetic and kinematic parameters are invariant upon inclusion of an unpolarized stray light. We compare our results to recent IN observations obtained by GREGOR. We studied a linear polarization feature which resembles those recently observed by GRIS in terms of appearing as `loop-like' structures and exhibiting very similar magnetic flux density. Thus, we demonstrate that realistic MHD simulations are capable of showing close agreement with real observations, and the symbiosis between them and observations continues to prove essential. We finally discuss the considerations that must be made for DKIST-era observations.