  We estimate the rate of observable Horizontal and Upward Tau Air-Showers
(HORTAUs, UPTAUS) considering both the Earth opacity and the finite size of the
terrestrial atmosphere. We calculate the effective target volumes and masses
for Tau air-showers emerging from the Earth. The resulting model-independent
masses for satellite experiments such as EUSO may encompass at E_nu_tau = 10^19
eV a very large volume, V= 1020 km^3. Adopting simple power law neutrino
fluxes, E^-2 and E^-1, calibrated to GZK-like and Z-Burst-like models, we
estimate that at E= 10^19 eV nearly half a dozen horizontal shower events
should be detected by EUSO in three years of data collection by the
"guaranteed" GZK neutrino flux. We also find that the equivalent mass for an
Earth outer layer made of rock is dominant compared to the water, contrary to
simplified all-rock/all-water Earth models and previous Montecarlo simulations.
Therefore we expect an enhancement of neutrino detection along continental
shelves nearby the highest mountain chains, also given the better geometrical
acceptance for Earth skimming neutrinos. The Auger experiment might reveal such
a signature at E_nu= 10^{18} eV (with 26 events in 3 yr) towards the Andes, if
the angular resolution at the horizon (both in azimuth and zenith) would reach
an accuracy of nearly one degree needed to disentangle tau air showers from
common UHECR. The number of events increases at lower energies; therefore we
suggest an extension of the EUSO and Auger sensitivity down to (or even below)
E_nu = 10^19 eV and E_nu = 10^18 eV respectively.
