Time Variables and Their Definitions
The foundation of travel time theory rests on two fundamental time variables: $T$ (transit time) and $t$ (actual or clock time). Transit time is defined as the time a water parcel takes to cross a control volume or domain, expressed mathematically as $T = t_{ex} - t_i$, where $t_{ex}$ is the exit time and $t_i$ is the injection (entry) time.
If we observe the system at time $t$ and have measured or estimated the transit time $T$, we are effectively positioned at the outlet of the domain, considering water parcels exiting at that moment. These exiting parcels entered the domain at various previous times $t_i$, extending theoretically back to $-\infty$, meaning $T$ ranges from 0 (for parcels entering at time $t$ and exiting instantaneously) to $\infty$ (for parcels that entered in the distant past). While this represents an idealization—since the distant past is practically unknowable—it provides a useful theoretical framework.
The Transit Time Distribution
Transit time follows a probability distribution that can be conceptualized as the distribution of a random variable $T$. This distribution is time-dependent and conditional on the observation time $t$. The proper notation is $p_Q(T|t)$, though the literature often uses $p_Q(T,t)$, which can be misleading as it suggests a bivariate distribution when it's actually a conditional one. Some authors use $\overset{\leftarrow}{p}_Q(T,t)$ or place a backward arrow above $p$ to indicate that this probability refers to past events—hence the term "backward probability" or "backward transit time distribution."
As noted in Benettin et al. (2022)'s comprehensive review on transit time estimation and your previous AboutHydrology posts on residence time approaches, this backward-looking perspective is crucial for understanding catchment memory and the age composition of streamflow.
The Complexity of Transit Time as a Variable
Using $T$ as the primary variable introduces conceptual complexity. Since $T = t - t_i$ with $t$ being the observation time (the conditioning variable), $T$ inherently depends on both the current time and the injection time $t_i$. The injection time $t_i$ is the true independent variable in this framework.
This distinction has important mathematical consequences: when expressed in terms of $t_i$, the mass conservation law remains an ordinary differential equation, but when formulated using $T$, it becomes a partial differential equation—a transformation that significantly complicates the mathematical treatment, as discussed in Botter et al.'s (2011) work on the master equation.
Residence Time Distribution
A distinct but related concept is the residence time distribution, denoted as $p_S(T_r|t)$, where the residence time at observation time $t$ is $T_r = t - t_i$. Unlike transit time, residence time considers all water parcels currently within the domain, not just those exiting. We assume we can label parcels by their age (their injection time $t_i$).
This distribution also looks backward from the current time $t$ and is often represented in literature as $p_S(T,t)$ or $\overset{\leftarrow}{p}_S(T,t)$, potentially causing confusion with transit time notation. Crucially, this distribution characterizes the age composition of water currently stored in the domain at time $t$—it makes no predictions about the future but provides a snapshot of the past up to the present moment.
The Link Between Distributions: StorAge Selection Functions
While $p_Q$ refers to water exiting the domain and $p_S$ refers to water stored within it, these distributions are naturally related—what exits must have been stored. The relationship between these probabilities is mediated by the StorAge Selection (SAS) function, denoted as $\omega(T,t)$, which describes how water of different ages is preferentially selected for discharge.
The SAS framework, as elaborated in recent work including studies on the contribution of groundwater to catchment travel time distributions, provides:
$$p_Q(T|t) = \omega(T,t) \cdot p_S(T|t)$$The Special Case of Complete Mixing
There exists a special case where residence time and transit time distributions coincide: when parcels exiting the domain are uniformly sampled from the population of ages within the domain. This represents the "complete mixing" or "random sampling" assumption, where $\omega(T,t) = 1$ for all $T$.
Under this condition:
- Transit time and residence time distributions become equal: $p_Q(T|t) = p_S(T|t)$
- All mass conservation equations simplify and become linear in the probability distributions
- The system behavior resembles that of a well-mixed reactor
As discussed in your posts about celerity versus velocity and the travel time problem, this simplification, while mathematically convenient, rarely holds in real catchments where preferential flow paths and incomplete mixing dominate the hydrological response.
Here's the complete reference list including your own AboutHydrology posts on the topic:
References
Published Literature
Benettin, P., Rodriguez, N. B., Sprenger, M., Kim, M., Klaus, J., Harman, C. J., van der Velde, Y., et al. (2022). Transit Time Estimation in Catchments: Recent Developments and Future Directions. Water Resources Research, 58(11). https://doi.org/10.1029/2022wr033096
Botter, G., Bertuzzo, E., & Rinaldo, A. (2010). Transport in the hydrologic response: Travel time distributions, soil moisture dynamics, and the old water paradox. Water Resources Research, 46, W03514. doi:10.1029/2009WR008371
Botter, G., Bertuzzo, E., & Rinaldo, A. (2011). Catchment residence and travel time distributions: The master equation. Geophysical Research Letters, 38, L11403. doi:10.1029/2011GL047666
Botter, G. (2012). Catchment mixing processes and travel time distributions. Water Resources Research, 48, W05545. doi:10.1029/2011WR011160
Comola, F., Schaefli, B., Rinaldo, A., & Lehning, M. (2015). Thermodynamics in the hydrologic response: Travel time formulation and application to Alpine catchments. Water Resources Research, 51(2), 1671-1687. doi:10.1002/2014WR016228
Cornaton, F., & Perrochet, P. (2006). Groundwater age, life expectancy and transit time distributions in advective-dispersive systems: 1. Generalized reservoir theory. Advances in Water Resources, 29(9), 1267-1291. doi:10.1016/j.advwatres.2005.10.009
McDonnell, J. J., et al. (2010). How old is the water? Open questions in catchment transit time conceptualization, modelling and analysis. Hydrological Processes, 24(12), 1745-1754.
McGuire, K. J., & McDonnell, J. J. (2006). A review and evaluation of catchment transit time modelling. Journal of Hydrology, 330, 543-563.
Niemi, A. J. (1977). Residence time distribution of variable flow processes. International Journal of Applied Radiation and Isotopes, 28, 855-860.
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 20 (12): 4929–47. https://doi.org/10.5194/hess-20-4929-2016.
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.
Rinaldo, A., & Rodriguez-Iturbe, I. (1996). Geomorphological theory of the hydrologic response. Hydrological Processes, 10, 803-829.
Rinaldo, A., Beven, K. J., Bertuzzo, E., Nicotina, L., Davies, J., Fiori, A., Russo, D., & Botter, G. (2011). Catchment travel time distributions and water flow in soils. Water Resources Research, 47, W07537. doi:10.1029/2011WR010478
van der Velde, Y., Torfs, P. J. J. F., van der Zee, S. E. A. T. M., & Uijlenhoet, R. (2012). Quantifying catchment-scale mixing and its effect on time-varying travel time distributions. Water Resources Research, 48, W06536. doi:10.1029/2011WR011310
AboutHydrology Blog Posts
Rigon, R. (2014). Residence time approaches to the hydrological budgets. AboutHydrology Blog. http://abouthydrology.blogspot.com/2014/06/residence-time-approaches-to.html
Rigon, R. (2016). Celerity versus velocity and the travel time problem. AboutHydrology Blog. http://abouthydrology.blogspot.com/2016/06/celerity-vs-velocity.html
Rigon, R. (2023). A Commentary on transit (travel) times theory. AboutHydrology Blog. http://abouthydrology.blogspot.com/2023/01/a-commentary-on-transit-travel-times.html
Rigon, R. (2025). Transit Time and Residence Time Distributions: Fundamental Time Variables in Catchment Hydrology. AboutHydrology Blog. [This post]
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