Let's have a record of precipitation of duration \(\Delta t\ \ll\ t_c\) where \(t_c\) is the concentration time \cite{Rigon_2011} of the catchment we are analyzing  (\(t_c\) can be thought as the length of the domain over which the IUH is defined or a conventional time where the contribution to the discharge given by a particular rainfall record is not anymore detectable).  Let \(t\) be the clock time and \(t_{in}\) the time at which the precipitation record happens. Finally let \(p_{t_{in}}\) the precipitation intensity (for instance in mm/h, [L/T]) during the time interval \(\left(t_{in},t_{in}+\Delta t\right]\)