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 ($ u_tau$) enters the Earth it produces $tau$ leptons as a result of nuclear charged current interactions. In some cases, this process results in a $tau$ lepton exiting the surface of the Earth, which can subsequently decay in the atmosphere and produce an extensive air shower. These upward-going air showers can be detected via fluorescence, optical Cherenkov, or geomagnetic radio emission. Several experiments have been proposed to detect these signals. We present a comprehensive simulation of the production of $tau$ leptons by $ u_tau$s propagating through Earth to aid the design of future experiments. These simulations for $ u_tau$s and leptons in the energy range from $10^{15}$ eV to $10^{21}$ eV treat the full range of incidence angles from Earth-skimming to diametrically-traversing. Propagation of $ u_tau$s and leptons include the effects of rock and an ocean or ice layer of various thicknesses. The interaction models include $ u_tau$ regeneration and account for uncertainties in the Standard Model neutrino cross-section and in the photo-nuclear contribution to the $tau$ energy loss rate.
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