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In this paper we present ab initio 3D Monte-Carlo simulations of Ganymede's surface sputtered and sublimated H₂O exosphere. As inputs, we include surface water content maps and temperature distribution maps based on Galileo and Very Large Telescope (VLT) observations. For plasma precipitation, we use hybrid model results for thermal H⁺ and O⁺, energetic H⁺, O⁺⁺, S⁺⁺⁺, and electrons, with unprecedented energy resolution. Our results show that up to a solar zenith angle of ∼60° and up to ∼600 km altitude, sublimated H₂O dominates the atmosphere by up to four orders of magnitudes in number density, while sputtering dominates elsewhere. Sputtering is mainly induced by the impinging O⁺, O⁺⁺, and S⁺⁺⁺ ions, while protons (H⁺) and electrons only add about 1% of the total sputtered H₂O molecules to the atmosphere. Electrons are thus not important for the generation of the atmosphere, but they are important for spectroscopic observability of the atmosphere since they are the main inducer of the Lyman-α and O I emission lines. The extended H₂O atmosphere at altitudes ≳1 Ganymede radius is mainly the result of sputtering by thermal O⁺ ions, which is the only ion species with substantial fluxes in the low-energy range (10 eV–10 keV), i.e., is the only species that efficiently induces nuclear sputtering. Most released H₂O molecules return to the surface where they immediately adsorb, not forming a thermalized atmosphere. The morphology of Ganymede's magnetosphere, and the resulting dichotomies in the surface fluxes of the precipitating magnetospheric particles (polar fluxes > equatorial fluxes and leading equatorial fluxes > trailing equatorial fluxes), are thus well discernible in the sputtered atmosphere, persisting up to altitudes of a few thousand kilometers. In-situ measurements, as they are planned for the upcoming JUpiter ICy Moons Explorer (JUICE) mission, will mainly probe this sputtered atmosphere, except for encounters with the near-surface atmosphere on Ganymede's day-side, where the sublimated atmosphere will be probed instead. Finally, we compare our model results to the first observational evidence for a sublimated H₂O atmosphere on Ganymede, and find a very good agreement.