  We investigate the feasibility of representing a structured multi-dimensional
stellar atmosphere with a single one-dimensional average stratification for the
purpose of spectral diagnosis of the atmosphere's average spectrum. In
particular we construct four different one-dimensional stratifications from a
single snapshot of a magneto-hydrodynamic simulation of solar convection: one
by averaging its properties over surfaces of constant height, and three
different ones by averaging over surfaces of constant optical depth at 500 nm.
Using these models we calculate continuum, and atomic and molecular line
intensities and their center-to-limb variations. From analysis of the emerging
spectra we identify three main reasons why these average representations are
inadequate for accurate determination of stellar atmospheric properties through
spectroscopic analysis. These reasons are: non-linearity in the Planck function
with temperature, which raises the average emergent intensity of an
inhomogeneous atmosphere above that of an average-property atmosphere, even if
their temperature-optical depth stratification is identical; non-linearities in
molecular formation with temperature and density, which raise the abundance of
molecules of an inhomogeneous atmosphere over that in a one-dimensional model
with the same average properties; the anisotropy of convective motions, which
strongly affects the center-to-limb variation of line-core intensities. We
argue therefore that a one-dimensional atmospheric model that reproduces the
mean spectrum of an inhomogeneous atmosphere necessarily does not reflect the
average physical properties of that atmosphere, and are therefore inherently
unreliable.
