Investigating magnetic field inference from the spectral region around the Mg I b2 line using the weak-field approximation

Context. The understanding of the magnetic field structure in the solar atmosphere is important in assessing both the dynamics and the energy balance of the solar atmosphere. Our knowledge about these magnetic fields comes predominantly from the interpretation of spectropolarimetric observations. Simpler approaches based on approximations such as the weak-field approximation (WFA) deserve special attention as key methods in the interpretation of large, high-resolution datasets. Aims. We investigate the applicability of the WFA for retrieving the depth-dependent line-of-sight (LOS) magnetic field from the spectral region containing the Mg I b(2) spectral line and two photospheric Ti I and Fe I lines in its wings. Methods. We constructed and applied a 12-level model for Mg I atom that realistically reproduces the b(2) line profile of the mean quiet Sun. We tested the applicability of the WFA to the spectra computed from the FAL C atmospheric model with different magnetic and velocity fields added on an ad hoc basis . Then we extended the analysis to the spectra computed from two 3D magneto-hydrodynamic (MHD) MURaM simulations of the solar atmosphere. The first MHD cube was used to estimate the Stokes V formation heights of each spectral line. These heights correspond to optical depths at which the standard deviation of the difference between the WFA-inferred magnetic field and the magnetic field in the MHD cube is minimal. The estimated formation heights were verified using the second MHD cube. Results. The LOS magnetic field retrieved by the WFA is reliable for the magnetic field strength up to 1.4 kG even when moderate velocity gradients are present. The exception is the Fe I line, for which we found a strong discrepancy in the WFA-inferred magnetic fields because of the line blend. We estimated the Stokes V formation heights of each spectral line to be: log tau(Fe) = -2.6, log tau(Mg) = -3.3, and log tau(Ti) = -1.8. We were able to estimate the LOS magnetic field from the MURaM cube at these heights with the uncertainty of 150 G for the Fe I and Ti I lines and only 40 G for the Mg I b(2) line. Conclusions. Using the WFA, we can quickly get a reliable estimate of the structure of the LOS magnetic field in the observed region. This offers a significant advantage in comparison with otherwise time-consuming classical spectropolarimetric inversions. We find that the Mg I b(2) line profile calculated from the quiet Sun MURaM simulation agrees very well with the observed mean spectrum of the quiet Sun.