  A fraction of the electromagnetic radiation emitted from the surface of a
geometrically thin and optically thick accretion disk of a black hole returns
to the disk because of the strong light bending in the vicinity of the compact
object (returning radiation). While such radiation clearly affects the observed
spectrum of the source, it is often neglected in theoretical models. In the
present paper, we study the impact of the returning radiation on relativistic
reflection spectra. Assuming neutral material in the disk, we estimate the
systematic uncertainties on the measurement of the properties of the system
when we fit the data with a theoretical model that neglects the returning
radiation. Our NICER simulations show that the inclination angle of the disk
and the black hole spin parameter tend to be overestimated for low viewing
angles, while no clear bias is observed for high viewing angles. The iron
abundance of the disk is never overestimated. In the most extreme cases (in
particular, for maximally rotating black holes) the returning radiation
flattens the radial emissivity beyond a few gravitational radii. In such cases,
it also produces residuals that cannot be compensated by adjusting the
parameters of models neglecting the returning radiation. This may be an
important issue for interpretation of data from future X-ray missions (e.g.
Athena). When we simulate some observations with NuSTAR and we fit data above
10 keV, we find that some conclusions valid for the NICER simulations are not
true any longer (e.g., we can get a high iron abundance).
