therefore, the age-ranked discharge s-function, in this case depends only on the travel time, nor on the injection time,  neither on the clock time alone while the dependance of the injection time is retained only in the intensity of precipitation.  Analogously   \(t\ >\ t_{in}+\Delta t\) becomes \(T\ >\ \Delta t\)  and  then \(x=t_{in}\ \to z=t_{ }-t_{in}\ =T\) and \(x=t_{in}+\Delta t\ \to z=t-t_{in}-\Delta t\ =\ T\ -\ \Delta t\) while \(dz\ =\ -dx\) such that: