  The physical conditions of the solar photosphere change on very small spatial
scales both horizontally and vertically. Such a complexity may pose a serious
obstacle to the accurate determination of solar magnetic fields. We examine the
applicability of Milne-Eddington (ME) inversions to high spatial resolution
observations of the quiet Sun. Our aim is to understand the connection between
the ME inferences and the actual stratifications of the atmospheric parameters.
We use magnetoconvection simulations of the solar surface to synthesize
asymmetric Stokes profiles such as those observed in the quiet Sun. We then
invert the profiles with the ME approximation. We perform an empirical analysis
of the heights of formation of ME measurements and analyze the uncertainties
brought about by the ME approximation. We also investigate the quality of the
fits and their relationship with the model stratifications. The atmospheric
parameters derived from ME inversions of high-spatial resolution profiles are
reasonably accurate and can be used for statistical analyses of solar magnetic
fields, even if the fit is not always good. We also show that the ME inferences
cannot be assigned to a specific atmospheric layer: different parameters sample
different ranges of optical depths, and even the same parameter may trace
different layers depending on the physical conditions of the atmosphere.
Despite this variability, ME inversions tend to probe deeper layers in granules
as compared with intergranular lanes.
