Magnetic Field Measurements in the Solar Chromosphere Using the H$_β$ 4861Å~Line I: Forward Modeling Based on 1D Models

The chromosphere is a complex solar atmosphere that hosts a variety of transients and transports significant free energy to heat the corona. However, due to the limited sensitivity of polarization measurement and the influence of spectral line broadening, the basic magnetic field configuration in the chromosphere has not yet been fully revealed to correspond to the observed phenomena. In this work, we investigated the validity and application of the magnetic field inversion method for the H$_β$~4861~Å spectral line with non-local thermodynamic equilibrium approximations. We generated synthetic spectra by incorporating magnetic fields into semi-empirical FAL models for quiet Sun and sunspots, and then performed inversions to obtain the magnetic fields, which were then compared with the magnetic fields in the models. In addition, we evaluated the accuracy of the magnetic fields obtained using the weak field approximations and the impact of using the WFA results as the initial guess model for non-LTE inversion on the final results. Our work validates the effectiveness of the inversion method for the measurement of line-of-sight magnetic field components, which significantly improved the accuracy in both weak field (0 -- 500~G) and strong field ($>$2000~G) regions, while maintaining accuracy in the intermediate field range of 500 -- 2000~G. This demonstrates that the inversion techniques we employed are capable of resolving Zeeman-sensitive spectral lines in the chromosphere, which can be applied to the H$_β$ observational data from the new generation Solar Full-disk Multi-layer Magnetograph at GanYu Solar Station to provide full disk chromospheric magnetic field information.