  Usage of secondary cosmic muons to image the geological structures density
distribution significantly developed during the past ten years. Recent
applications demonstrate the method interest to monitor magma ascent and
volcanic gas movements inside volcanoes. Muon radiography could be used to
monitor density variations in aquifers and the critical zone in the near
surface. However, the time resolution achievable by muon radiography monitoring
remains poorly studied. It is biased by fluctuation sources exterior to the
target, and statistically affected by the limited number of particles detected
during the experiment. The present study documents these two issues within a
simple and well constrained experimental context: a water tower. We use the
data to discuss the influence of atmospheric variability that perturbs the
signal, and propose correction formulas to extract the muon flux variations
related to the water level changes. Statistical developments establish the
feasibility domain of muon radiography monitoring as a function of target
thickness (i.e. opacity). Objects with a thickness comprised between $\simeq$
50 $\pm$ 30m water equivalent correspond to the best time resolution. Thinner
objects have a degraded time resolution that strongly depends on the zenith
angle, whereas thicker objects (like volcanoes) time resolution does not.
