Oh Notation ! About notation in Transit Time literature

What written here is nothing new. Is just what already present in previous papers and post, just said in a different manner to put a bridge among what people write in literature and what we think should be written.

 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. Please also observe that the injection time $t_i$ is a discrete variable, since the rainfall happens at discrete time steps and the same happens to any transported material, even if it is usually approximated as a continuous one. (Note:We cannot  distinguish ages from data at a very fine resolution. Therefore blurring the concept could not be wrong).  

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. In literature $p_Q(T|t)$ is treated as a continuous variable, but for the reason $t_i$ is a discrete variable, it is a discrete time distribution.  

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 collapse in a unique variable and the 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 my 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]

And finally we talk about isotope concentration

 If I understand correctly, colleagues arre measuring the concentration of an isotope in stream or some other catchment outlet.  This is quite not a measure of the age of water but the latter has to be inferred from  a hypothesis the system behavior, and possibly the knowledge of the distribution of water ages within the control volume, obtained iteratively from an initial data (derived from the isotopic data of precipitation).

Therefore, assume you have measured in the outflow a concentration of 5 of solute X in appropriate units. You mai have  the knowledge of the concentration of the same solute in the water of a given age resident in the control volume. It would be a tuple like (7,3,6,2) where the number of concentration is the number of precipitation happened before the present instant. The 5 units out of the tuple can be taken all from the oldest (7) concentration, from the youngest (3,2) where the 3 are out of the 6 of the penultimate concentration.

Therefore a concentration of 5 can be fitted with a water age of 4 in the first case or (3*2+2*1)/5=1.8 in the second case.

In fact, the concentration doesn't identify  the exact age of the water but rather a range of ages spanning from the "first in first out" age to the "last in first out" age illustrated in the simple example.

Potentially the old water is infinitely old and, therefore, the lower bound could be very large, while the upper bound for the ages is, clearly, related to the last rainfall age.

So, the division between young water and old water is reasonable. Instead of trying to guess the distribution, you're trying to guess parts of its integral.

Then SAS (StorAge Selection functions, see this previous post for understanding what they are) are summarizing the mechanics of mixing inside the control volume. If the waters mix a lot the travel time and the residence time should coincides  because of the uniform selection of isotopes (or tracers) among the various populations. SAS have a non trivial form if the mixing of waters is incomplete.

Due to the localized nature of water physics, the actual travel times are intricately linked to the specific location from which water emerges or is extracted from the soil. Within the unsaturated zone, it's logical to posit that newly introduced water is the first to flow out for two primary reasons: it occupies the empty pores, and these pores are typically where water is loosely retained. However, when young water mixes with old water in the saturated zone, the aquifer's pressure rises, causing a pressure wave that pushes out the oldest water near the streams.

Consequently, in a typical zeroth-order catchment, runoff tends to adhere to a "last in, first out" (LIFO) logic, while subsurface stormwater tends to expel old water first, following a "first in, first out" (FIFO) logic. Consequently, the recession of the flood wave is expected to contain a higher proportion of old water than young water. In this hypothetical simplified model delineating two distinct dynamics, and drawing from the aforementioned insights, it becomes feasible to estimate the global effect of the system, as proposed by Rigon and Bancheri in 2022 and applying the recent methods of TT probability determination.

Vegetation withdrawal we can think to extract water from the places where active roots are, according to root density. Plants with shallow roots should mostly sample young water, while plants with deeper roots wold sample older water but maybe also being effective in mixing water of different ages because of exudation and because of preferential water infiltration close to the roots.

The presentation (under construction) is exemplifying these concepts.

A series of talks and material on Transit (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 that, while having a long history (e.g. Niemi, 1977, Rigon et al, 2016) has been largely renewed recently starting from Botter et al., 2010 and Botter et al., 2011. The material is the same prepared for the Hydrological Modelling class however grouped here separately for the readers convenience. 

An alternative perspective is presented here regarding their concepts. While certain passages may pose some challenges, the enhanced comprehension of flux formation processes at the catchment scale is, in my opinion, immensely valuable and well worth the effort. The proposed approach involves the following line of thinking: a) the collective fluxes within catchments result from the cumulative movements of numerous small water volumes (water parcels); b) parcels can be understood through three key distributions: the travel time distribution, the residence time distribution, and the response time distribution; c) the interrelations among these distributions are elucidated; d) linking these distributions to catchment processes is achieved through the formulation of age-ranked distributions within ordinary differential equations; e) the theory developed here represents a generalization of the unit hydrograph theory.

• The view of the catchment as the statistics of elementary water volumes moving stochastically, a storyboard
• Travel Time, Residence Times (Vimeo2024)
• A summary (Vimeo 2022)
• A short note about past and future (Vimeo2022)
• The Python Notebook that created the Figure in slides
• Vimeo 2021-Ita, Vimeo 2021-Eng, Vimeo2022
• Some discussion (In English)

• Previous lesson recap - Blackboard2024
• StorAge Selection Functions (Vimeo2024)
• A summary on SAS, Blackboard2024, 
• Some further considerations on our goals - Blackboard2024 
• A Python notebook where the age-ranked tables are created within a simple example
• Response time and Life Expectancy (Vimeo2024)
• A post on travel (transit)  time, residence time and response time definitions
• A Notebook Estimating the empirical response time probability from the age-ranked (He) table
• Niemi's identity (Vimeo2024)
• Multiple Reservoirs (Vimeo2024)
• Multiple Reservoirs treatment is not that complicate as it can be imagined from theformal mathematics - Blackboard2024
• Partitions (Vimeo2024)
• Pollutants and Tracers (Vimeo2024)

• Q&A - A student asks and I respond on travel times (in Italian)
• Q&A - Another session of explanations
• Klicker session on Travel times, Residence Time, etc. (List of questions and answers by students, Zoom2020)
• More material on travel time, residence time and response time on this blog. 
• Old material on the same topic
• StorAge Selection functions (Vimeo2022)
• Vimeo 2021-It
• Response Times  (Vimeo 2023)
• Vimeo2020
• Vimeo 2021-Eng, Vimeo 2021-It
• A little of discussion (in English)
• Pollutants, Tracers, Nutrients Transport (Vimeo2023)
• (Vimeo2022)
• Partitioning the outputs (Vimeo2023)
•  (Vimeo2022)

Some References

• Benettin, P., Soulsby, C., Birkel, C., Tetzlaff, D., , G. and Rinaldo, A. (2017) Using sas functions and high resolution isotope data to unravel travel time distributions in headwater catchments. Water Resources Research, 53, 1864–1878. URL: http: //doi.org/10.1002/2016WR020117. 
• Benettin, Paolo, and Enrico Bertuzzo. 2018. “Tran-SAS v1.0: A Numerical Model to Compute Catchment-Scale Hydrologic Transport Using StorAge Selection Functions.” Geoscientific Model Development Discussions, January, 1–19.
• Benettin, Paolo, Nicolas B. Rodriguez, Matthias Sprenger, Minseok Kim, Julian Klaus, Ciaran J. Harman, Ype van der Velde, 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, Gianluca, Enrico Bertuzzo, and Andrea Rinaldo. 2010. “Transport in the Hydrologic Response: Travel Time Distributions, Soil Moisture Dynamics, and the Old Water Paradox.” Water Resources Research 46 (3). http://doi.wiley.com/10.1029/2009WR008371.
• Botter, Gianluca, Enrico Bertuzzo, and Andrea Rinaldo. 2011. “Catchment Residence and Travel Time Distributions: The Master Equation.” Geophysical Research Letters 38 (11). http://doi.wiley.com/10.1029/2011GL047666.
• Drever, Mark C., and Markus Hrachowitz. 2017. “Migration as Flow: Using Hydrological Concepts to Estimate the Residence Time of Migrating Birds from the Daily Counts.” Methods in Ecology and Evolution / British Ecological Society 8 (9): 1146–57.
• Harman, Ciaran J. 2015. “Time-Variable Transit Time Distributions and Transport: Theory and Application to Storage-Dependent Transport of Chloride in a Watershed.” Water Resources Research 51 (1): 1–30.
• Harman, Ciaran J., and Esther Xu Fei. 2024. Mesas.py v1.0: A Flexible Python Package for Modeling Solute Transport and Transit Times Using StorAge Selection Functions.†Geoscientific Model Development 17 (2): 477–95. https://doi.org/10.5194/gmd-17-477-2024.
• Hrachowitz, M., Benettin, P., van Breukelen, B. M., Fovet, O., Howden, N. J. K., Ruiz, L., van der Velde, Y. and Wade, A. (2016) Transit times-the link between hydrology and water quality at the catchment scale: Linking hydrology and transit times. Wiley Interdisciplinary Reviews: Water, 3, 629–657. 
• McDonnell, Jeffrey J. 2014. The Two Water Worlds Hypothesis: Ecohydrological Separation of Water between Streams and Trees? Wiley Interdisciplinary Reviews: Water, April. http://doi.wiley.com/10.1002/wat2.1027.
• Niemi, Antti J. 1977. “Residence Time Distributions of Variable Flow Processes.” The International Journal of Applied Radiation and Isotopes 28 (10): 855–60.
• 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.
• Rigon, R., and M. Bancheri. “On the Relations between the Hydrological Dynamical Systems of Water Budget, Travel Time, Response Time and Tracer Concentrations.” http://abouthydrology.blogspot.com/2020/05/equivalences-and-differences-among.html.
• Sprenger, M., Stumpp, C., Weiler, M., Aeschbach, W., ST, A., Benettin, P., Dubbert, M., Hartmann, A., Hrachowitz, M., Kirchner, J., McDonnel, J., Orlowski, N., Penna, D., Pfahl, S., Rinderer, M., Rodriguez, N., Schmidt, M. and Werner, C. (2019) The demographics of water: A review of water ages in the critical zone. Rev. Geophys., 2018RG000633. 
• Schwemmle, Robin, and Markus Weiler. 2024. Consistent Modeling of Transport Processes and Travel Times: coupling Soil Hydrologic Processes with StorAge Selection Functions. Water Resources Research 60 (1). https://doi.org/10.1029/2023wr034441.
• Velde, Y. van der, P. J. J. F. Torfs, S. E. A. T. M. Van der Zee, and R. Uijlenhoet. 2012. “Quantifying Catchment-Scale Mixing and Its Effect on Time-Varying Travel Time Distributions.” Water Resources Research 48 (6): W06536–13.
• Velde, Ype van der, Ingo Heidbüchel, Steve W. Lyon, Lars Nyberg, Allan Rodhe, Kevin Bishop, and Peter A. Troch. 2014. “Consequences of Mixing Assumptions for Time-Variable Travel Time Distributions.” Hydrological Processes 29 (16): 3460–74.
• Wilusz, Daniel C., Ciaran J. Harman, and William P. Ball. 2017. “Sensitivity of Catchment Transit Times to Rainfall Variability Under Present and Future Climates.” Water Resources Research 53 (12): 10231–56.

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. 

• The view of the catchment as the statistics of elementary water volumes moving stochastically, a storyboard
• Travel Time, Residence Times (Vimeo2022)
• A summary (Vimeo 2022)
• A short note about past and future (Vimeo2022)
• Vimeo 2021-Ita, Vimeo 2021-Eng
• Some discussion (In English)

• StorAge Selection functions (Vimeo2022)
• Vimeo 2021-It

• Response Times  (Vimeo 2023)
•  Vime20220
• Vimeo 2021-Eng, Vimeo 2021-It
• A little of discussion (in English)
• Pollutants, Tracers, Nutrients Transport (Vimeo2023)
• (Vimeo2022)
• Partitioning the outputs (Vimeo2023)
•  (Vimeo2022)

• Q&A - A student asks and I respond on travel times (in Italian)
• Q&A - Another session of explanations
• Klicker session on Travel times, Residence Time, etc. (List of questions and answers by students, Zoom2020)

A Commentary on transit (travel) times theory

We were recently to  produce a commentary on a review paper by Benettin et al. 2022. The preprint of the paper is here below clicking on the Figure. 

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. 

References

Benettin, Paolo, Nicolas B. Rodriguez, Matthias Sprenger, Minseok Kim, Julian Klaus, Ciaran J. Harman, Ype van der Velde, et al. 2022. “Transit Time Estimation in Catchments: Recent Developments and Future Directions.” Water Resources Research 58 (11). https://doi.org/10.1029/2022wr033096.

Life Expectation & Response Times explained

This is the continuation of the presentations that try to explain what residence time, travel time and response time are. The first presentation, on travel and residence times can be found here.

This presentation regards response times and their relation with life expectancies, which is an aspect quite unresolved so far and that was clarified to our knowledge for the first time in the supplemental material of the paper "On the relations among HDSys" by Rigon and Bancheri. 

Travel times and Residence Times explained

Long time ago it was believed that Residence times and travel times were the same concept. This is not true. Long time ago it was also assumed that they could be chosen as time invariant. Recent literature showed that this is not usually the case.  All of this is explained, hopefully in a clearly definitive way in the following presentation.

Yo can find it by clicking on the figure above. This presentation uses drawings, plots, in a way that it cannot be done in a paper (but would be useful sometimes) and can be considered to be a companion of part of our 2016 paper below. Other presentations will follow on the topics of response time and its relation to life expectation. The talk I gave for the WATZON project is here below

 

 

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.

Equivalences and differences among various Hydrological Dynamical Systems

In this paper we want to show that once the topology of a Hydrological Dynamical System is determined, the structure of the equations of the Water Budget Dynamical System is determined but with it also the travel time and residence time distributions are. This is obvious since our paper on age-ranked functions but here it is rigorously stated and worked out.

Ideally this paper is also a continuation of the paper on the representation of Hydrological Dynamical systems with Petri Nets of which in some sense, it represents an extension. In a ideal menù, the reader should read first the paper on the historical-critical approach to the GIUH, I would say excluded its last section, then the Age-Ranked paper, then the Petri Net paper and finally this one. Clicking on the Figure you can have the preprint, or on the bibliography below you can have access to all the manuscripts. This is part of my research program on trying a statistical-mechanical approach to hydrological modeling which across all my research activity since the last (almost) thirty years. 

References

• Rigon, Riccardo, Marialaura Bancheri, Giuseppe Formetta, and Alban de Lavenne. 2016. “The Geomorphological Unit Hydrograph from a Historical-Critical Perspective.” Earth Surface Processes and Landforms, EGU Reprint Series, 41 (1): 27–37.
• 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.
• Bancheri, Marialaura, Francesco Serafin, and Riccardo Rigon. 2019. “The Representation of Hydrological Dynamical Systems Using Extended Petri Nets (EPN).” Water Resources Research 55 (11): 8895–8921.
• Marialaura, Bancheri, Francesco Serafin, and Riccardo Rigon. 2019. “Supporting Material for: The Representation of Hydrological Dynamical Systems Using Extended Petri Nets (EPN).” Water Resources Research.
• Rigon, Riccardo, and M. Bancheri, 2020, “Equivalences and Differences between the Hydrological Dynamical Systems of Water Budget, Travel Time and Tracers Concentration and the Legacy of Models’ Topology.”, submitted to Hydrol. Processes.
• Rigon, Riccardo, and Marialaura Bancheri, 2020. “Supplementary Material, Equivalences and Differences between the Hydrological Dynamical Systems of Water Budget, Travel Time, Response Time, and Tracer Concentrations and the Legacy of Models’ Topology.” submitted to Hydrol. Processes.

The daemons of water times

This that you find on Authorea is an elaboration of what I used with my students to teach them what residence times and travel times are in catchment hydrology. Probably I can expand it to be more comprehensive of all the aspects but for the moment it can be a starting point.

I introduced three fictional characters to talk about of the perspectives we can have on the kinematics of water parcels, the little daemons She, He and They. She records what happens inside the catchment. He look at the boundaries. They makes calculations. If you are interested in these stories, click on the figure.

Some about the legacy of model topology in estimating travel times

This talk is connected to the paper in preparation about the equivalence of hydrological dynamical systems (HDSys), and on the Extended Petri Nets. We discuss how to pass from the water budget to the related system of age-ranked functions and  show some simple solutions of the master equations of the related travel times.

Our thesis is that once you have the structure of the water budget model, you can get the equations for the age-ranked functions automatically. Under the hypothesis of uniform mixing, you also know the solutions of your system ....

Travel-Time Based Modelling of Transport in Hydrological Systems

This is one of the posters I will present next 5th of April at Hydromod2018.

It present some of our results in TT theory and conveys the fact that, given a set of equations, the travel time distribution is (obviously) set. There are no hidden degree of freedom. All of it is contained in the paper about age-ranked equations. Clicking on the Figure you have the high resolution pdf.

Marialaura Bancheri defense

The Ph.D. Thesis of Marialaura Bancheri is already available in a previous post. On december 14, she finally defended it. This is the video of her performance. Her topics are: research reproducibility, GEOFRAME, reservoir based modelling (or semidistributed modelling) of the hydrological cycle, travel times theory re-interpreted in the perspective of reservoirs modelling.

I have no doubt that it could be very useful to all who are interested in our recent work and to all those that try to interpret catchment scale behavior through travel times. Marialaura was an outstanding student, is an exceptional team manager, and she is looking for an appropriate post-doc position.

About Benettin et al. 2017, equation (1)

Gianluca (Botter) in his review of Marialalaura (Bancheri) Ph.D. Thesis brought to my attention the paper Benettin et al. 2017. A great paper indeed, where a couple of ideas are clearly explained:

• SAS functions can be derived from the knowledge of travel and residence times probability
• a virtual-experiment where they show that traditional pdfs (travel times pdf) can be seen an the ensamble of the actual time-varying travel times distributions.

The paper is obviously relevant also for the hydrological contents it explains, but it is not the latter point the one which I want  to argue a little. I want here just to argue about the way they present their first equation.

SAS stands for StorAge Selection functions and they are defined, for instance in Botter et al. 2011 (with a little difference in notation) as:
$$
\omega_x(t,t_{in}) = \frac{p_x(t-t_{in}|t)}{p_S(t-t_{in}|t)} \ \ \ (1)
$$
as the ratio between the travel time probability related to output $x$ (for instance discharge or evapotranspiration) and the residence time probability.
In the above equation (1)

•  $\omega_x$ is the symbol that identifies the SAS
• $t$ is the clock time
• $t_{in}$ is the injection time, i.e. the time when water has entered the control volume
• $p_x(t-t_{in}|t)$ with $x \in \{Q, ET, S\}$  is the probability that a molecule of water entered in the system at time $t_{in}$ is inside the control volume, $S$, revealed as discharges, $Q$, or evapotranspiration, $ET$

Equation (1) in Benettin et al. is therefore written as
$$
\frac{\partial S_T(T,t)}{\partial t} + \frac{\partial S_T(T,t)}{\partial T} = J(t) - Q(t) \Omega_Q(S_T(T,t),t)-ET(t) \Omega_{ET}(S_T(T,t),t) \ \ \ \ (2)
$$
Where:

• $T$ is residence time (they call  water age but this could be a little misleading because the water age of water in storage could be, by their own theory different in storage, discharge, evapotranspiration)
• $S_T$ is the age-ranked storage, i.e. “the cumulative volumes of water in storage as ranked by their age” (I presume the word “cumulative”  implies some integration. After thinking a while and looking around, also to paper van der Velde et Al. 2012, I presume the integration is over all the travel times up to $T$ which, because the variable of integration in my notation is $t_{in}$ means that $t_{in} \in [t,t-T]$  )
• $J(t)$ is  the precipitation rate at time $t$
• $Q(t)$ is the discharge rate at time $t$
• $\Omega_x$ are the integral of the integrated SAS function which are more extensively derived below.

In fact, this (2) should be just an integrated version (integrated over $t_i$) of equation (9) of Rigon et al., 2016:
$$
\frac{ds(t,t_{in})}{dt} = j(t,t_{in}) - q(t,t_{in}) -et(t,t_{in})
\ \ \ \ (3)
$$
where:

• $s(t,t_{in})$ is the water stored in the control volume at time $t$ that was injected at time $t_{in}$
• $j(t,t_{in})$ is the water input which can have age $T=t-t_i$
• $q(t,t_{in})$ is the discharge that exits the control volume at time $t$ and entered the control volume at time $t_{in}$
•  $et(t,t_{in})$ is the evapotranspiration that exits the control volume at time $t$ and entered the control volume at time $t_{in}$

In terms of the SAS and the formulation of the problem given in Rigon et al. (2016), the $\Omega$s can be defined as follows:
\begin{equation}
\Omega_x(T,t) \equiv \Omega_x(S_T(T,t),t) := \int_{t-T}^t \omega_x(t,t_i) p_S(t-t_i|t) dt_i = \int_0^{p_S(T|t)} \omega_x(P_S,t) dP_S
\end{equation}
Where the equality ":=" on the l.h.s is a definition, so the $\Omega$s ($\Omega_Q$ and $\Omega_{ET}$) are this type of object. The identity $\equiv$ stresses that the dependence on $t_in$ is mediated by a dependence on the cumulative storage $S_T$ and $T$ is the travel time. As soon as $T \to \infty$, $\Omega \to 1$ (which is what written in equation (2) of Benettin's paper). This is easily understood because by definition ${\omega_x(t,t_i) p_S(t-t_i|t)} \equiv {p_x(t-t_i|t)}$ are probabilities (as deduced from (1)).
An intermediate passage to derive (2) from (3) requires to make explicit the dependence of the age-ranked functions from the probabilities. From definitions, given in Rigon et al., 2016. It is
$$
\frac{d S(t) p_S(t-t_{in}|t)}{dt} = J(t) \delta (t-t_in) - Q(t) p_Q(t-t_{in}|t) - ET(t) p_{ET}(t-t_{in}|t)
$$
which  is Rigon et al. equation (14).
Now integration over $t_i \in [t-T, t]$ can be performed to obtain:
$$
S_T(t_{in},t):= \int_{t-T}^t s(t_{in},t) dt_{in}
$$
and, trivially,
$$
J(t) = J(t) \int_{t-T}^t \delta(t-t_{in}) dt_{in}
$$
while for the $\Omega$s I already said.
The final step is finally to make a change of variables that eliminate $t_{in}$ in favor of $T := t-t_{in}$. This actually implies the last transformation. In fact:
$$
\frac{dS(t,T(t_{in},t))}{dt} =\frac{\partial S(t,T(t_{in},t))}{\partial t} + \frac{\partial S(t,T(t_{in},t))}{ \partial T}\frac{\partial T}{ \partial t} = \frac{\partial S(t,T(t_{in},t))}{\partial t} + \frac{\partial S(t,T(t_{in},t))}{ \partial T}
$$
since $\partial T/\partial t$ =1. Assembling all the results, equation (2) is obtained.

Note:
Benettin et al., 2017 redefines the probability $p_S$ as “normalized rank storage … which is confined in [0,1]” which seems weird with respect to the Authors own literature. In previous papers this $p_S$ was called backward probability and  written as $\overleftarrow{p}_S(T,t)$. Now probably they have doubt we are talking about probability.  In any case, please read it again: "normalized rank storage … which is confined in [0,1]”. Does not sound unnatural is not a probability ? Especially when you repetitively estimate averages with it and comes out with “mean travel times”?Operationally, it IS a probability. Ontologically the discussion about if there is really random sampling or not because there is some kind of convoluted determinism in the travel times formation can be interesting but it brings to a dead end. On the same premised we should ban the word probability from the stochastic theory of water flow, that, since Dagan has been enormously fruitful.

This long circumlocution,  looks to me like the symbol below

or TAFKAP, which was used by The Artist Formerly Known As Prince when he had problems with his record company.
In any case, Authors should pay attention in this neverending tendency to redefine the problem rather beacause it can look what Fisher (attribution by Box, 1976) called mathemastry. This is fortunately not the case of the paper we are talking about. But then why not sticking with the assessed notation ?

The Authorea version of this blog post can be found here.

References 

• Benettin, P., Soulsby, C., Birkel, C., Tetzlaff, D., Botter, G., & Rinaldo, A. (2017). Using SAS functions and high-resolution isotope data to unravel travel time distributions in headwater catchments. Water Resources Research, 53(3), 1864–1878. http://doi.org/10.1002/2016WR020117
• Botter, G., Bertuzzo, E., & Rinaldo, A. (2011). Catchment residence and travel time distributions: The master equation. Geophysical Research Letters, 38(11), n/a–n/a. http://doi.org/10.1029/2011GL047666
• Rigon, R., Bancheri, M., & Green, T. R. (2016). Age-ranked hydrological budgets and a travel time description of catchment hydrology. Hydrology and Earth System Sciences, 20(12), 4929–4947. http://doi.org/10.5194/hess-20-4929-2016

The geomorphic unit hydrograph from a historical-critical perspective

I have just submitted to the review process a paper on the geomorphic unit hydrograph. (Now published here). It crosses close to four decades of progress the field of describing the treatment of the hydrologic response with the travel time concept, and hopefully was able to convey the core of the ideas behind this very nice (and effective) theory, and to open the way to new researches. The paper was accepted as a "State of  art paper" by Earth Surface Processes and Landforms.

Its abstract:
"In this paper we present a brief overview of Geomorphic Instanteneous Unit Hydro- graph (GIUH) theories and analyze their successful path without hiding their limitations. The history of the GIUH can be subdivided into three major chapters. The first chap- ter is based on the pioneering work by Rodríguez-Iturbe and Valdés (1979), and Gupta and Waymire (1983), which recognized that a treatment of water discharges with ”travel times” could give a rich interpretation of the theory of the Instantaneous Unit Hydrograph (IUH). We show how this was possible, what assumptions were made, which of these assumptions can be relaxed, and which have become obsolete and been discarded. The second chapter focuses on the Width Function Based IUH (WFIUH) approach and its achievements in assessing the interplay of the topology and geometry of the network with water dynamics. The limitations of the WFIUH approach are described, and a way to work around them is suggested. Finally, a new formal approach to estimating the water budget by ”travel times”, which derives from a suitable use of the water budget equation and some mixing hypotheses, has been disentangled and presented."

The little post on the width function can be considered a complementary reading too.  The pre-print of the paper is here.

Bibliography (including some more papers)


Band LE, 1986. Topographic Partition of Watersheds with Digital Elevation Models. Water Resources Research 22(1): 15–24. ISSN 1944-7973. doi:10.1029/WR022i001p00015.

Beven KJ, 2011. Rainfall-runoff modelling: the primer. John Wiley & Sons.

Botter G, Bertuzzo E, Rinaldo A, 2010. Transport in the hydrologic response: Travel time dis- tributions, soil moisture dynamics, and the old water paradox. Water Resources Research 46(3): n/a–n/a. ISSN 1944-7973. doi:10.1029/2009WR008371.

Botter G, Bertuzzo E, Rinaldo A, 2011. Catchment residence and travel time distributions: The master equation. Geophysical Research Letters 38(11): L11403. ISSN 1944-8007. doi:10.1029/2011GL047666.

Botter G, Rinaldo A, 2003. Scale effect on geomorphologic and kinematic dispersion. Water Resour. Res. 39: 10 PP. doi:200310.1029/2003WR002154.

Clark C, 1945. Storage and the unit hydrograph. Transactions of the American Society of Civil Engineers 110(1): 1419–1446.

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 : n/a–n/aISSN 1944-7973. doi:10.1002/2014WR016228.

Cudennec C, Fouad Y, Gatot IS, Duchesne J, 2004. A geomorphological explanation of the unit hydrograph concept. Hydrol. Process. 18(4): 603–621. ISSN 0885-6087. doi: 10.1002/hyp.1368.

Cudennec C, Slimani M, Le Goulven P, 2005. Accounting for sparsely observed rainfall space-time variability in a rainfall-runoff model of a semiarid Tunisian basin. Hydrolog. Sci. J. 50(4): 617–630. ISSN 0262-6667. doi:10.1623/hysj.2005.50.4.617.

Córdova J, Rodríguez-Iturbe I, 1983. Geomorphoclimatic estimation of extreme flow probabilities. Journal of Hydrology 65(1–3): 159 – 173. ISSN 0022-1694. doi: http://dx.doi.org/10.1016/0022-1694(83)90215-9. Scale Problems in Hydrology.

Dagan G, 1989. Flow and Transport in Porous Formations. Springer-Verlag Heidelberg Berlin New York. doi:10.1007/978-3-642-75015-1. 465 p.

de Lavenne A, Boudhraâ H, Cudennec C, 2013. Streamflow prediction in ungauged basins through geomorphology-based hydrograph transposition. Hydrology Research doi: 10.2166/nh.2013.099. In press.

de Lavenne A, Rigon R, Formetta G, Cudennec C, 2015. What is the best WFIUH transfer function? .

Dingman SL, 1994. Physical Hydrology. Macmillan, New York.

D’Odorico P, Rigon R, 2003. Hillslope and channel contributions to the hydrologic response.
Water Resour. Res. 39(5): 1113. ISSN 0043-1397. doi:10.1029/2002WR001708.

Dooge JCI, 1959. A general theory of the unit hydrograph. Journal of Geophysical Research
64(2): 241–256. ISSN 2156-2202. doi:10.1029/JZ064i002p00241.
Dooge JCI, 2003. Linear Theory of Hydrologic Systems, volume 1. EGU Reprint Series,
Katlenburg-Lindau, Germany.

Fairfield J, Leymarie P, 1991. Drainage networks from grid digital elevation models. Water Resources Research 27(5): 709–717. ISSN 1944-7973. doi:10.1029/90WR02658.

Fenicia F, Savenije HHG, Matgen P, Pfister L, 2008. Understanding catchment behavior through stepwise model concept improvement. Water Resour. Res. 44(1): W01402. ISSN 0043-1397. doi:10.1029/2006WR005563.

Formetta G, Mantilla R, Franceschi S, Antonello A, Rigon R, 2011. The JGrass-NewAge system for forecasting and managing the hydrological budgets at the basin scale: models of flow generation and propagation/routing. Geoscientific Model Development 4(4): 943– 955. doi:10.5194/gmd-4-943-2011.

Grimaldi S, Petroselli A, Alonso G, Nardi F, 2010. Flow time estimation with spatially variable hillslope velocity in ungauged basins. Adv. Water. Resour. 33(10, Sp. Iss. SI): 1216–1223. ISSN 0309-1708. doi:10.1016/j.advwatres.2010.06.003.

Grimaldi S, Petroselli A, Nardi F, 2012. A parsimonious geomorphological unit hydrograph for rainfall–runoff modelling in small ungauged basins. Hydrolog. Sci. J. 57(1): 73–83. doi:10.1080/02626667.2011.636045.

Gupta VK, Mesa OJ, 1988. Runoff generation and hydrologic response via channel network geomorphology — Recent progress and open problems. J. Hydrol. 102(1–4): 3–28. ISSN 0022-1694. doi:10.1016/0022-1694(88)90089-3.

Gupta VK, Waymire E, 1983. On the formulation of an analytical approach to hydrologic response and similarity at the basin scale. J. Hydrol. 65(1–3): 95–123. ISSN 0022-1694. doi:10.1016/0022-1694(83)90212-3. 

Gupta VK, Waymire E, Wang CT, 1980. A representation of an instantaneous unit hydrograph from geomorphology. Water Resour. Res. 16(5): 855–862. doi: 198010.1029/WR016i005p00855.

Hall MJ, Zaki AF, Shahin MMA, 2001. Regional analysis using the Geomorphoclimatic In- stantaneous Unit Hydrograph. Hydrol. Earth Syst. Sci. 5(1): 93–102. doi:10.5194/hess-5- 93-2001.

Henderson FM, 1966. Open channel flow. Macmillan, New York. ISBN 0023535105. 522 pp.

Kirkby MJ, 1976. Tests of the random network model, and its application to basin hydrology. Earth Surface Processes 1(3): 197–212. ISSN 1931-8065. doi:10.1002/esp.3290010302.

Leopold LB, Maddock T, 1953. The Hydraulic Geometry of Stream Channels and Some Physiographic Implications. US Geol. Surv. Prof. Pap. 252: 56.

McDonnell JJ, McGuire K, Aggarwal P, Beven KJ, Biondi D, Destouni G, Dunn S, James A, Kirchner J, Kraft P, Lyon S, Maloszewski P, Newman B, Pfister L, Rinaldo A, Rodhe A,

Sayama T, Seibert J, Solomon K, Soulsby C, Stewart M, Tetzlaff D, Tobin C, Troch P, Weiler M, Western A, Wörman A, Wrede S, 2010. How old is streamwater? Open ques- tions in catchment transit time conceptualization, modelling and analysis. Hydrological Processes 24(12): 1745–1754. ISSN 1099-1085. doi:10.1002/hyp.7796.

McGuire KJ, McDonnell JJ, 2006. A review and evaluation of catchment transit time modeling. Journal of Hydrology 330(3–4): 543 – 563. ISSN 0022-1694. doi: http://dx.doi.org/10.1016/j.jhydrol.2006.04.020.

Mesa O, Mifflin E, 1986. On the Relative Role of Hillslope and Network Geometry in Hy- drologic Response. In V Gupta, I Rodríguez-Iturbe, E Wood (eds.), Scale Problems in Hydrology, Springer Netherlands, volume 6 of Water Science and Technology Library. ISBN 978-94-010-8579-3, pages 1–17. doi:10.1007/978-94-009-4678-1_1.

Morse P, Feshbach H, 1953. Methods of theoretical physics. Cambridge University Press.

Naden P, Broadhurst P, Tauveron N, Walker A, 1999. River routing at the continental scale: use of globally-available data and an a priori method of parameter estimation. Hydrology and Earth System Sciences 3(1): 109–123. doi:10.5194/hess-3-109-1999.

Naden PS, 1992. Spatial variability in flood estimation for large catchments: the exploitation of channel network structure. Hydrolog. Sci. J. 37(1): 53–71. ISSN 0262-6667. doi: 10.1080/02626669209492561. 

Nicótina L, Alessi Celegon E, Rinaldo A, Marani M, 2008. On the impact of rainfall patterns on the hydrologic response. Water Resources Research 44(12): W12401. ISSN 1944-7973. doi:10.1029/2007WR006654.

Niemi AJ, 1977. Residence time distributions of variable flow processes. The International Journal of Applied Radiation and Isotopes 28(10–11): 855 – 860. ISSN 0020-708X. doi: http://dx.doi.org/10.1016/0020-708X(77)90026-6.

Pattison I, Lane SN, Hardy RJ, Reaney SM, 2014. The role of tributary relative timing and sequencing in controlling large floods. Water Resources Research 50(7): 5444–5458. ISSN 1944-7973. doi:10.1002/2013WR014067.

Rigon R, D’Odorico P, Bertoldi G, 2011. The geomorphic structure of the runoff peak. Hydrol. Earth Syst. Sci. 15(6): 1853–1863. doi:10.5194/hess-15-1853-2011.

Rinaldo A, Beven KJ, 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(7): n/a–n/a. ISSN 1944-7973. doi:10.1029/2011WR010478.

Rinaldo A, Marani A, 1987. Basin scale model of solute transport. Water Resources Research 23(11): 2107–2118. ISSN 1944-7973. doi:10.1029/WR023i011p02107.

Rinaldo A, Marani A, Rigon R, 1991. Geomorphological dispersion. Water Resour. Res.
27(4): 513–525. ISSN 0043-1397.

Rinaldo A, Rodríguez-Iturbe I, 1996. Geomorphological theory of the hydrological re- sponse. Hydrol. Process. 10(6): 803–829. ISSN 1099-1085. doi:10.1002/(SICI)1099- 1085(199606)10:6<803::AID-HYP373>3.0.CO;2-N.

Rinaldo A, Vogel GK, Rigon R, Rodriguez-Iturbe I, 1995. Can one gauge the shape of a basin? Water Resour. Res. 31(4): 1119–1127. ISSN 0043-1397. doi: 10.1029/94WR03290.

Robinson JS, Sivapalan M, Snell JD, 1995. On the relative roles of hillslope processes, chan- nel routing, and network geomorphology in the hydrologic response of natural catchments. Water Resour. Res. 31(12): 3089–3101. ISSN 0043-1397.

Rodríguez-Iturbe I, González-Sanabria M, Bras RL, 1982. A geomorphoclimatic theory of the instantaneous unit hydrograph. Water Resour. Res. 18(4): 877–886. doi: doi:10.1029/WR018i004p00877.

Rodríguez-Iturbe I, Rinaldo A, 1997. Fractal River Basins: Chance and Self-Organization. Cambridge University Press, Cambridge (UK). doi:10.2277/0521004055.

Rodríguez-Iturbe I, Valdés JB, 1979. The geomorphologic structure of hydrologic response. Water Resour. Res. 15(6): 1409–1420. ISSN 0043-1397.

Saco PM, Kumar P, 2002a. Kinematic dispersion in stream networks 1. Coupling hydraulic and network geometry. Water Resour. Res. 38: 14 PP. doi:200210.1029/2001WR000695.

Saco PM, Kumar P, 2002b. Kinematic dispersion in stream networks 2. Scale issues and self-similar network organization. Water Resour. Res. 38(11): 1245. ISSN 0043-1397. doi:10.1029/2001WR000694.

Sivapalan M, 2003. Process complexity at hillslope scale, process simplicity at the watershed scale: is there a connection? Hydrol. Process. 17(5): 1037–1041. ISSN 0885-6087. doi: 10.1002/hyp.5109.

Sivapalan M, Jothityangkoon C, Menabde M, 2002. Linearity and nonlinearity of basin re- sponse as a function of scale: Discussion of alternative definitions. Water Resour. Res. 38: 5. doi:200210.1029/2001WR000482.

Smith MB, Koren VI, Zhang Z, Reed SM, Pan JJ, Moreda F, 2004. Runoff response to spatial variability in precipitation: an analysis of observed data. Journal of Hydrology 298(1–4): 267 – 286. ISSN 0022-1694. doi:http://dx.doi.org/10.1016/j.jhydrol.2004.03.039. The Distributed Model Intercomparison Project (DMIP).

Snell JD, Sivapalan M, 1994. On geomorphological dispersion in natural catchments and the geomorphological unit hydrograph. Water Resour. Res. 30(7): 2311–2323. ISSN 0043-1397.

Tarboton DG, Bras RL, Rodriguez-Iturbe I, 1991. On the extraction of channel networks from digital elevation data. Hydrological Processes 5(1): 81–100. ISSN 1099-1085. doi: 10.1002/hyp.3360050107.

Valdés JB, Fiallo Y, Rodríguez-Iturbe I, 1979. A rainfall-runoff analysis of the geomor- phologic IUH. Water Resources Research 15(6): 1421–1434. ISSN 1944-7973. doi: 10.1029/WR015i006p01421.

van der Tak LD, Bras RL, 1990. Incorporating hillslope effects into the geomorphologic instantaneous unit hydrograph. Water Resources Research 26(10): 2393–2400. ISSN 1944-7973. doi:10.1029/WR026i010p02393.

Wang CT, Gupta VK, Waymire E, 1981. A geomorphologic synthesis of nonlinearity in sur- face runoff. Water Resour. Res. 17(3): 545–554. doi:198110.1029/WR017i003p00545.

Wilkinson L, Friendly M, 2009. The History of the Cluster Heat Map. The American Statistician 63(2): 179–184. doi:10.1198/tas.2009.0033.

Woods R, Sivapalan M, 1999. A synthesis of space-time variability in storm response: Rainfall, runoff generation, and routing. Water Resour. Res. 35(8): 2469–2485. doi: 199910.1029/1999WR900014.

Zoccatelli D, Borga M, Viglione A, Chirico GB, Blöschl G, 2011. Spatial moments of catch- ment rainfall: rainfall spatial organisation, basin morphology, and flood response. Hydrol- ogy and Earth System Sciences 15(12): 3767–3783. doi:10.5194/hess-15-3767-2011.


Zoch RT, 1937. On the relation between rainfall and stream flow. Monthly Weather Review 55: 135–147.