  We consider propagation of high energy earth-skimming taus produced in
interactions of astrophysical tau neutrinos. For astrophysical tau neutrinos we
take generic power-law flux, $E^{-2}$ and the cosmogenic flux initiated by the
protons. We calculate tau energy loss in several approaches, such as dipole
models and the phenomenological approach in which parameterization of the $F_2$
is used. We evaluate the tau neutrino charged-current cross section using the
same approaches for consistency. We find that uncertainty in the neutrino cross
section and in the tau energy loss partially compensate giving very small
theoretical uncertainty in the emerging tau flux for distances ranging from $2$
km to $100$ km and for the energy range between $10^6$ GeV and $10^{11}$ GeV,
focusing on energies above $10^8$ GeV. When we consider uncertainties in the
neutrino cross section, inelasticity in neutrino interactions and the tau
energy loss, which are not correlated, i.e. they are not all calculated in the
same approach, theoretical uncertainty ranges from about $30\%$ and $60 \%$ at
$10^8$ GeV to about factors of 3.3 and 3.8 at $10^{11}$ GeV for the $E^{-2}$
flux and the cosmogenic flux, respectively, for the distance of 10 km rock. The
spread in predictions significantly increases for much larger distances, e.g.,
$\sim 1,000$ km. Most of the uncertainty comes from the treatment of
photonuclear interactions of the tau in transit through large distances. We
also consider Monte Carlo calculation of the tau propagation and we find that
the result for the emerging tau flux is in agreement with the result obtained
using analytic approach. Our results are relevant to several experiments that
are looking for skimming astrophysical taus, such as the Pierre Auger
Observatory, HAWC and Ashra. We evaluate the aperture for the Auger and discuss
briefly application to the the other two experiments.
