Magnetograms underestimate even unipolar magnetic flux nearly everywhere on the solar disk

We aim to test the reliability of determining the line-of-sight magnetic field from a 3D MHD simulation of a unipolar region. In contrast to earlier similar studies, we consider the full solar disk, i.e. considering the full centre-to-limb variation, as well as regions with different averaged field strengths. We synthesised Stokes profiles from MURaM MHD simulations of unipolar regions with varying mean vertical magnetic flux densities, ranging from quiet Sun to active region plage. We did this for a comprehensive range of heliocentric angles: from $\mu=1$ to $\mu=0.15$, and for two commonly used photospheric spectral lines: Fe I $6173.3$ and Fe I $5250.2${\AA}. The line-of-sight magnetic field was derived with a Milne-Eddington Inversion as well as with other commonly used methods. The inferred spatially averaged $\langle B_{LOS}\rangle$ is always lower than that present in the MHD simulations, with the exception of $\mu\approx 1$ and sufficiently high spatial resolution. It is also generally inconsistent with a linear dependence on $\mu$. Above $\mu=0.5$ the spatial resolution greatly impacts the retrieved line-of-sight magnetic field. For $\mu\leq0.5$ the retrieved $B_{LOS}$ is nearly independent of resolution, but is always lower than expected from the simulation. These trends persist regardless of the mean vertical magnetic field in the MHD simulations and are independent of the $B_{LOS}$ retrieval method. For $\mu\leq0.5$, a larger $\langle B_{LOS}\rangle$ is inferred for the $5250.2${\AA} spectral line than $6173.3${\AA}, but the converse is true at higher $\mu$. The results found here raise some doubts of the reliability of determining the radial field by dividing the line-of-sight field by $\mu$ and are of considerable importance for deducing the total magnetic flux of the Sun. They may also contribute to the resolution of the open flux problem.