  The structure of the solar chromosphere is believed to be governed by
magnetic fields, even in quiet-Sun regions that have a relatively weak
photospheric field. During the past decade inversion methods have emerged as
powerful tools for analyzing the chromosphere of active regions. The
applicability of inversions to infer the stratification of the physical
conditions in a dynamic 3D solar chromosphere has not yet been studied in
detail.
  This study aims to establish the diagnostic capabilities of non-local
thermodynamical equilibrium (NLTE) inversion techniques of Stokes profiles
induced by the Zeeman effect in the Ca II 8542 line.
  We computed the Ca II atomic level populations in a snapshot from a 3D
radiation-MHD simulation of the quiet solar atmosphere in non-LTE using the 3D
radiative transfer code Multi3d. These populations were used to compute
synthetic full-Stokes profiles in the Ca II 8542 line using 1.5D radiative
transfer and the inversion code Nicole. The profiles were then spectrally
degraded to account for finite filter width and Gaussian noise was added to
account for finite photon flux. These profiles were inverted using Nicole and
the results were compared with the original model atmosphere.
  Our NLTE inversions applied to quiet-Sun synthetic observations provide
reasonably good estimates of the chromospheric magnetic field, line-of-sight
velocities and somewhat less accurate, but still very useful, estimates of the
temperature. Three dimensional scattering of photons cause cool pockets in the
chromosphere to be invisible in the line profile and consequently they are also
not recovered by the inversions. To successfully detect Stokes linear
polarization in this quiet snapshot, a noise level below 10^{-3.5} is
necessary.
