  We perform a new, detailed calculation of the flux and energy spectrum of
Earth-emerging $\tau$-leptons generated from the interactions of tau neutrinos
and antineutrinos in the Earth. A layered model of the Earth is used to
describe the variable density profile of the Earth. Different assumptions
regarding the neutrino charged- and neutral-current cross sections as well as
the $\tau$-lepton energy loss models are used to quantify the systematic
uncertainty from these on the results. A baseline simulation is then used to
generate the optical Cherenkov signal from upward-moving extensive air showers
generated by the $\tau$-lepton decay in the atmosphere, applicable to a range
of space-based instruments. We use this simulation to determine the neutrino
sensitivity for $E_\nu>$ 10 PeV for a space-based experiment with performance
similar to that for the Probe of Extreme MultiMessenger Astrophysics (POEMMA)
mission currently under study.
