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There has been a recent surge in interest in the detection of $\tau$lepton-induced air showers from detectors at altitude. When a $\tau$ neutrino($\nu_\tau$) enters the Earth it produces $\tau$ leptons as a result of nuclearcharged current interactions. In some cases, this process results in a $\tau$lepton exiting the surface of the Earth, which can subsequently decay in theatmosphere and produce an extensive air shower. These upward-going air showerscan be detected via fluorescence, optical Cherenkov, or geomagnetic radioemission. Several experiments have been proposed to detect these signals. Wepresent a comprehensive simulation of the production of $\tau$ leptons by$\nu_\tau$'s propagating through Earth to aid the design of future experiments.These simulations for $\nu_\tau$'s and leptons in the energy range from$10^{15}$ eV to $10^{21}$ eV treat the full range of incidence angles fromEarth-skimming to diametrically-traversing. Propagation of $\nu_\tau$'s andleptons include the effects of rock and an ocean or ice layer of variousthicknesses. The interaction models include $\nu_\tau$ regeneration and accountfor uncertainties in the Standard Model neutrino cross-section and in thephoto-nuclear contribution to the $\tau$ energy loss rate.

A Comprehensive Approach to Tau-Lepton Production by High-Energy Tau Neutrinos Propagating Through Earth