Modeling Helioseismic and Magnetic Imager observables for the study of solar oscillations

Helioseismology aims to infer the properties of the solar interior by analyzing observations of acoustic oscillations. Interpreting helioseismic data, however, is complicated by the non-trivial relationship between helioseismic observables and the physical perturbations associated with acoustic modes as well as by various instrumental effects. We aim to improve our understanding of the signature of acoustic modes measured in the Helioseismic and Magnetic Imager (HMI) continuum intensity and Doppler velocity observables by accounting for radiative transfer, solar background rotation, and spacecraft velocity. We started with a background model atmosphere that accurately reproduces solar limb darkening and the Fe I 6173$ Å$ spectral line profile. We employed first-order perturbation theory to model the effect of acoustic oscillations on inferred intensity and velocity. By solving the radiative transfer equation in the atmosphere, we synthesized the spectral line, convolved it with the six HMI spectral windows, and deduced the continuum intensity ( and Doppler velocity ( according to the HMI algorithm. We analytically derived the relationship between mode displacement in the atmosphere and the HMI observables and show that both the intensity and velocity deviate significantly from simple approximations. Specifically, the continuum intensity does not simply reflect the true continuum value, while the line-of-sight velocity does not correspond to a straightforward projection of the velocity at a fixed height in the atmosphere. Our results indicate that these deviations are substantial, with amplitudes of approximately 10% and phase shifts of around 10^̧irc across the detector for both observables. Moreover, these effects are highly dependent on the acoustic mode under consideration and the position on the solar disk. To achieve accurate modeling of the observables, it is important to account for the impact of radiative transfer on oscillation velocities and perturbations in atmospheric thermodynamic quantities, which influence the line profile. The combination of these effects leads to non-trivial systematic errors (in amplitude and phase) across the detector that must be taken into account to understand the observables. This framework can be used to study mode visibility across the solar disk and for asteroseismology applications.