AboutHydrology: Transit time, Residence time, Response time, Life expectancy

Saturday, June 10, 2023

Transit time, Residence time, Response time, Life expectancy

Just to help someone, a few definitions:

Travel Time (a.k.a. Transit Time), T: It is the time a parcel of water stays inside a control volume. If $t_{in}$ is the time it entered the control volume and $$t_{ex}$$ the travel time it exits, then $$ T= t_{ex}-t_{in} $$ For an observer placed at the outlet(s) of the control volume, since their actual (clock) time coincides with $$t_{ex}$$, i.e $$t_{ex}=t$$ it is $$ T= t-t_{in} $$. The actual variable in this definition is $$t_{in}$$

Residence time is $$T_R = t-t_{in}$$
Life expectancy is $$ T_L = T_{ex} - t$$ so, it is also: $$ T = T_R+T_L $$
Response time is $$R = t_{ex}-t_{in}$$ but only restricted to all the parcels injected at the same $$t_{in}$$ estimated at $$t=t_{in}$$ and is their life expectancy at $$t=t_{in}$$. The actual variable here is $$t_{ex}$$.

All of these definitions are given in statistical sense, meaning that they are stochastic variables described trough their distributions. Looking at the above definitions and if we do not read them very carefully, it looks like that transit time and response time are the same thing and we though for decades it was. Instead they are not since transit time distribution is conditional to the clock time, while response time distribution is conditional to the injection time. The first is a distribution in $$t_{in}$$, the second in $$t_{ex}$$.

To learn more, you can get further details at:

Rigon, Riccardo, Marialaura Bancheri, and Timothy R. Green. 2016. “Age-Ranked Hydrological Budgets and a Travel Time Description of Catchment Hydrology.” Hydrology and Earth System Sciences Discussions, May, 1–22. https://doi.org/10.5194/hess-2016-210. (This paper has a little weird description of life expectancy though)

Rigon, Riccardo, and Marialaura Bancheri. 2021. “On the Relations between the Hydrological Dynamical Systems of Water Budget, Travel Time, Response Time and Tracer Concentrations.” Hydrological Processes 35 (1). https://doi.org/10.1002/hyp.14007. (ad especially give a look to the supplemental material)

See also a concise summary unpublished paper available here.

Video lectures in the topic can be found in this blog. The most recent ones are:

Travel Time, Residence Time and Response Time

Here below we started a little series of lectures about a statistical way of seeing water movements in catchments. This view has a long history but recently had a closure with the work of Rinaldo, Botter and coworkers. Here it is presented an alternative vie to their concepts. Some passages could be of some difficulty but the gain in understanding the processes of fluxes formation at catchment scale is, in my view, of great value and deserves some effort. The way of thinking is the following: a) the overall catchments fluxes are the sum of the movements of many small water volumes (molecules); b) the water of molecules can be seen through 3 distributions: the travel time distribution, the residence time distribution and the response time distributions; c) the relationships between these distributions are revealed; d) the relation of these distributions with the the treatment of the catchments made through ordinary differential equations is obtained through the definition of age ranked distributions; e) The theory this developed is a generalizations of the unit hydrograph theory.