Some observations about long rainfall and the generated discharges

 In well-known hydrologic response theories like the IUH, it has been established that for a specific catchment and a constant rainfall, there exists a 'critical rainfall duration' resulting in the maximum discharge for that catchment, which is usually known as concentration time. 

The next step is to associate a return period with the constant rainfall. This allows us to demonstrate that given a precipitation with an assigned return period, there is a critical rainfall duration that yields the highest possible discharge in that river section.This is what has been accomplished in Rigon et al., 2011 (but the research dates back to early 00, which is another interesting story). BTW, In the paper, we have also shown that this time is less or equal to the concentration time. 

The above argument may lead to the misconception that the “maximum discharge” for the catchment cannot be exceeded (keep in mind that the concept of maximum discharge obtainable is incomplete when you do not mention a return period).  Consider doubling the duration of the rainfall while keeping the intensity fixed. The first impulse results in the highest discharge with the assigned return period. Yet, it also has a discharge tail that, depending on the catchment's features, can last quite long. When the second impulse of precipitation arrives with the same intensity, it adds to the recession of the first impulse, usually increasing the discharge beyond the maximum discharge obtained with a single impulse.

In certain cases, like in the kinematic hydrograph model (uniform IUH) the rise of the new impulse discharge may precisely compensate for the decreasing recession of the older impulse, resulting in a constant discharge. However, this is not the general scenario, as simple calculations can show and sticking with this idea can be erroneous. Typically in fact, and especially when there is a marked contrast between the response time of the surface and subsurface storm flow waves, the recession discharge generated of the first impulse decreases more slowly than the increase in the new impulse discharge, effectively acting as additional rainfall. This effect is equivalent to increase the intensity of the effective rainfall to a return period which can be estimated through inverse modelling. In other words, two subsequent rainfall impulses, each with an assigned return period, are equivalent to a precipitation event with a higher return period. While the IUH theory establishes a precise equality between the return period of rainfall and discharge for a single impulse, the two return periods of discharges and rainfall become decoupled when multiple rainfall impulses occur.  

Although real-world precipitations are not constant and uniform, and the response of the catchment may not be time-invariant,  the main qualitative findings described above remain statistically valid and could be tested by generating ensembles of time-variable precipitations with numerical models. Besides, there are additional factors like sediment and vegetation transport that can add volume to the water (see for instance these posts),  increasing more than linearly the return period of discharge with increasing rainfall intensities. 

References 

Rigon, R., P. D’Odorico, and G. Bertoldi. 2011. “The Geomorphic Structure of the Runoff Peak.” Hydrology and Earth System Sciences 15 (6): 1853–63. https://doi.org/10.5194/hess-15-1853-2011.

Concentration time, if existent, is a statistical concept

Among the various times we use in describing the catchment, concentration time is one of them. It is referred, in the old textbooks, as the largest travel time of water parcels (i.e. statistically significant amount of water molecules that are though to move together) in a catchment. Travel time, in turn is the time a parcel of water employs to across the catchment from its injection (as rainfall) to its exit (as part of discharge). The Figure 1 below illustrate two parcels with different travel times, with parcel 1 arriving faster to the outlet, for being close to it.

The concept of concentration time gained its importance since the Mulvaney theory of "the rational method" reported, for instance,  in K. Beven book (2012). For giving a meaning to it, we can assume that,  if parcels are though to move with constant velocity in a catchment, then, once their distance from the outlet along the drainage directions (see the width function concept) is known,  travel times is obtained by dividing  that distance by the parcels’ velocity.
Rigon et al., 2016 gives a review of this concept in the framework of the geomorphological unit hydrograph based on the width function (or WFIUH). The oldest hydrologists would also remind a simplified version of the story, where, essentially the catchment is seen as a rectangular planar hillslope and the flow is though to be parallel as in Figure 2 below.

Parcels move in essentially rectilinear paths, with constant velocity. Parcels like the no  2 are on the divide and parcel like the no 1 very close to the outlet, that is in Figure 1 a sort of trench. In this case, varying the duration of precipitations, we obtain a hydrograph  which is a triangle or a trapeze. It can be demonstrated that when a rain of constant fixed intensity falls on this catchment, we obtain the maximum discharge possibile when its duration equals the parcels no2 travel time, the largest one. Continuing to argue about models, not about what happens in reality, it can be seen also that, from the point of view of the instantaneous unit hydrograph theory (IUH),  concentration time is the extension of the domain of definition of the IUH distribution function ($t_c$  in Figure 3). 

Unfortunately, most of IUHs do not have a finite domain but an infinite one, the simplest being probably the exponential IUH $$IUH(t;\lambda) = \frac{1}{\lambda} e^{-t/\lambda}$$ (see also Rigon et al., 2011). This implies that for most IUHs, the concentration time does not exist as a rigorous concept.   Besides, the dynamics of water parcels as depicted in simplified theories was completely screwed up by tracers experiments that have determined that the age of water in floods is very much larger than believed, and usually what we see in rivers and torrents is old water not the one just fallen during the last precipitation (though undoubtedly was the rainfall to trigger it). 
The concept of concentration time,  resists in operational hydrology because there is a certain evidence that floods are generated by precipitations of increasing duration with increasing basins area,  and this correlates with the idea of concentration time exposed above for the planar hillslope.  However,  in complex catchments, it cannot be something different from a statistical concept. We already mentioned briefly that a catchment is not a huge planar hillslope and that water parcels move in complicate ways through it.  Moreover, the expansion of the river networks during storms (e.g. Durighetto et al.,  2020)  implies the necessity to add a further dynamic to concentration times perceptual model.

After all the above considerations, if something like the concentration time exists, it is a characteristic statistical time which identified the duration of the rainfalls that generate the  largest peak discharges. It should depend on catchment size and topology (besides on the rainfall). We believe that it increases with catchment size, but being any catchment different, it remains a slippery concept. A solid statistical study would be required to clarify, once for all, the issue.

References

• Beven, Keith J. 2012. Rainfall-Runoff Modelling: The Primer. The Primer. John Wiley & Sons.
• Durighetto, N., F. Vingiani, L. E. Bertassello, M. Camporese, and G. Botter. 2020. “Intraseasonal Drainage Network Dynamics in a Headwater Catchment of the Italian Alps.” Water Resources Research 56 (4): 33.
• Rigon, R., P. D’Odorico, and G. Bertoldi. 2011. “The Geomorphic Structure of the Runoff Peak.” Hydrology and Earth System Sciences 15 (6): 1853–63.
• 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.

The estimation of the discharge through the IUH explained

Time to time I go back to the estimation of the discharge through the Instantaneous Unit Hydrograph theory (IUH). It is interesting that this almost 90 year old concept is still alive and, being threathened by the studies on residence time from the physical point of view, is still valid when operational  activities are involved, and, in any case, as a benchmark tool.
IUH is at the core of our analysis of peak flows and of the code called PeakFlow in the Horton Machine, and it is known that the concept can include geomorphic characteristics. Many then can be interested in knowing how to estimate it and the manuscript you can access in Authorea, gives a definitive guide to do it.

The manuscript, can be found here.

La teoria dell'idrogramma Istantaneo Unitario

L'idrogramma instantaneo unitario è una teoria semplificata dell'aggregazione e propagazione dei deflussi. Ecco nel seguito, le mie lezioni (per il corso di Costruzioni Idrauliche per Ingegneri Civili a Trento).  In 2016 I restructured it in parts, and according to new pieces of theory that I developed with Marialaura Bancheri and finding new inspiration form works by Gianluca Botter, Enrico Bertuzzo and Andrea Rinaldo on travel times. A good review is certainly Rigon et al., 2015. Unfortunetely slides are (so far) in Italian.

The new slides:

1   - Introduction

2   - Preliminary notions on travel and residence times

3   - Alternative heuristics (demonstration of equivalence by travel time distribution -TTD- and IUH)

4   - Some mathematical aspects of IUH theory

5   - A couple of examples (uniform  and exponential TTD and their names in classical iterpretation)

6   - The geomorphologic instantaneous unit hydrograph GIUH

7   - The width function unit hydrograph WFIUH (for this see also this post on the width function)

8   - The problem of the peak flows and maximum discharge

9   - Geomorphological dispersion (coming soon)

10 - Hillslope and channel contributions (coming soon)

The old slides

Slides e Audio:

1 - Introduzione allo IUH. Audio 2014 (31.1 MB);  Audio 2015 (19.5 Mb);
2 - Alcune distribuzioni dei tempi di residenza (5.8 Mb); Audio 2015: Alcune distribuzione per tempi di residenza (7.2 Mb).
3 - Idrogramma Istantaneo unitario Geomorfologico. Audio 2014  (16.8 Mb). Audio 2015 (20.3 Mb)
4 - Portate massime. Audio 2014 (17.4 Mb).
5 - L'idrogramma istantaneo unitario basato sulla funzione di ampiezza.

The even older ones:

All the old slides together: La teoria dell'idrogramma istantaneo unitario e dell'idrogramma istantaneo unitario 

IUH and GIUH methods for modelling discharges (in Italian)

This contains the material, in Italian, that I use for my classes. It is old and the new material is here instead.

Please find below the old material with audio lectures.

Le slides sulla teoria dell'idrogramma istantaneo unitario e dell'idrogramma istantaneo unitario GEOmorfologico (credo un metodo non convenzionale di mostrare la materia per un corso di Costruzioni Idrauliche). Audio: IUH (31.1 Mb), Alcuni Tipici IUH (5.9 Mb), GIUH (16.8 Mb), sulle portate massime  (17.4 Mb)

Le note [pdf, 11 Mb] scritte sullo stesso tema e su altri di idrologia, funzionali al corso. Le note hanno ormai qualche anno e non sono completamente allineate con le slides. Tuttavia ritengo sia meglio averle che non averle.

Il tutorial di Peakflow (che è un modello per il calcolo delle onde di piena nei bacini naturali. La parte di teoria contiene una descrizione della teoria del calcolo delle portate massime - alle sezioni 5 e 6 - che è la stessa usata da Trento_p).

For the English an further references on GIUH see the English post (not yet ready).