- \(\Theta\left(t_{in}\right)\) is a coefficient of partition that identifies which portion of the rain input (at time \(t_{in}\) goes into discharge). \(\Theta\left(t_{in}\right)\le1\) depending if other outputs (as evapotranspiration) are present, but in the present case, for simplicity, will assume \(\Theta\left(t_{in}\right)=1\)
For simplicity, we will assume in the following that \(A\ \) is included in \(f\) definition, which now become \(\left[m^2s^{-1}\right]\). The notation used above is congruent with the one used in \cite{Rigon_2016} where, however, more general cases are treated.
According to the definition of the Heaviside step function the integral giving the age-ranked discharge can be considered for various time steps.
- For \(t\ <t_{in}\) is clearly:
\(q\left(t,t_{in}\right)=0\)
- For \(t_{in\ }<\ t\ \le t_{in}+\Delta t\):
\(q\left(t,t_{in}\right)\ =\ p_{t_{in}}\int_{_{t_{in}}}^tf\left(t-x\right)\ dx\)
because \(H\left(x\right)H\left(t-x\right)\ \equiv1\)
- For \(t\ >t_{in}+\ \Delta t\):
\(q\left(t,t_{in}\right)\ =\ p_{t_{in}}\int_{_{t_{in}}}^{t_{in}+\Delta t}f\left(t-x\right)\ dx\)
because \(H\left(x\right)H\left(t-x\right)\ \equiv1\) for \(t_{in}<\ x\le t_{in}+\Delta t\ \) and \(H\left(x\right)H\left(t-x\right)\ \equiv0\ \) for \(t\ >\ t_{in}+\ \Delta t\).
Let's now define the so called S-function (which, as it is known, correspond to the probability or residence/travel times \cite{Rigon_2016}, in this case multiplied by the catchment area \(A\):