  The observed presence of water molecules in the dayside lunar regolith was an
unexpected discovery and remains poorly understood. Standard thermophysical
models predict temperatures that are too high for adsorbed water to be stable.
We propose that this problem can be caused by the assumption of locally flat
surfaces that is common in such models. Here we apply a model that explicitly
considers surface roughness, and accounts for solar illumination, shadows cast
by topography, self-heating, thermal reradiation, and heat conduction. We
couple the thermophysical model to a model of first-order desorption of lunar
surface water and demonstrate that surface roughness substantially increases
the capability of the Moon to retain water on its sunlit hemisphere at any
latitude, and within $45^{\circ}$ of the poles, at any time of the lunar day.
Hence, we show that lunar surface roughness has a strong influence on lunar
water adsorption and desorption. Therefore, it is of critical importance to
take account of surface roughness to get an accurate picture of the amount of
water on the Moon's surface and in its exosphere.
