The understating of the dynamics of tracers and water transit times at catchment scale has increasingly grown in the last decades, becoming a consolidated approach in the field of hydrological and ecohydrological research. Recently, a benchmark contribution has been given in the work by Benettin et al. (2022), which reviews the state of art on the topic, also addressing present and future challenge, pointing out some open questions in transit time research. This commentary tries to contextualize the above article, highlighting the most focal points and relating it to a broader context in the field. A brief overview on the main concepts of backward transit times, StorAge selection functions and forward transit time distributions is given in a logical-historical order, giving to the reader the primary instruments for a later comprehensive understating of the Transit Time Theory. Eventually, a numerical example helps to clarify the above concepts in a very simple and effective way.

We present, by using previous results on extended Petri Nets, the relations of various hydrological dynamical systems ($\mathtt{HDSys}$) derived from the water budget ($\mathtt{DynWB}$). Once $\mathtt{DynWB}$ has been implemented, there exist a consistent way to get the equations for backward travel time distributions ($\mathtt{DynTT}$), for the forward response time distribution ($\mathtt{DynRTD}$) and for the concentration for a solute or a tracer ($\mathtt{DynC}$). We show that the $\mathtt{DynWB}$ has a correspondence one to many with the $\mathtt{DynTT}$. In fact to any one of the $\mathtt{DynWB}$ equation correspond as many equation as the input precipitation events times. The $\mathtt{DynTT}$ is related to $\mathtt{DynRTD}$ by the Niemi’s relationship and, in presence of multiple, $n$ outputs, by the specification of $n-1$ partition functions, which determine which fraction of water volume, injected in the control volume at a specific time $t_{in}$, goes asymptotically into a specific output. The $\mathtt{DynC}$, given $\mathtt{DynTT}$, depends further on the solute/tracer concentration in inputs. The paper clarifies the complicate set of relations above by using an example from literature. Upon the introduction of the appropriate information, it is also shown how these ($\mathtt{HDSys}$) can be solved simultaneously without duplicating calculations. It is also shown that these systems can be solved exactly, under the hypothesis of uniform mixing of water ages inside each reservoir within the system.

This is research material under development. If used, it should be cited properly. Citations to  published papers is scanty and it should be fixed along the way. To account for water movements in a catchment or any other control volume, let us assume that we can record the movements of small group of water molecules, called parcels, inside the control volume (cv) and at its boundaries. We do some assumptions first: