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.

Return Period

Some people, I realised, have problems with the concept of return period. This is the definition in wikipedia (accessed October 25th, 2017):
A return period, also known as a recurrence interval (sometimes repeat interval) is an estimate of the likelihood of an event, such as an earthquake, flood[1], landslide[2], or a river discharge flow to occur.

It is a statistical measurement typically based on historic data denoting the average recurrence interval over an extended period of time, and is usually used for risk analysis (e.g. to decide whether a project should be allowed to go forward in a zone of a certain risk, or to design structures to withstand an event with a certain return period). The following analysis assumes that the probability of the event occurring does not vary over time and is independent of past events.

Something that wikipedia does not include is rainfall intensity. The first paragraph, should be then something like:

"A return period of x time units, also known as a recurrence interval (sometimes repeat interval) is an estimate of the likelihood of an event, such as an earthquake, flood[1], landslide[2], rainfall intensity, a river discharge flow or any observable, to occur (or be overcome) on average every x time units."

Return period clearly involves a statistical concept, which is traced back to a probability, and a time concept, that is the sampling time.
Let us assume we have a sequence of data, for which, at moment, the sampling time is unknown, composed by a discrete number, $n$, of data.
The empirical cumulative distribution function (ECDF) of the data is a representation of the empirical statistics for those data. Let $ECDFc$ be the complementary empirical cumulative distribution function, meaning $ECDFc(h) \equiv 1 - ECDF(h)$.
Let h* be one of the possible values of these data (not necessarily present in the sequence but included in the range of experimental values). We are interested in the probability of $h^*$ being overcome. If $m$ is the number of time $h^*$ is matched or overcome, then
$$ ECDFc(h^*)= m/n $$
$$ECDF(h^*) = 1 - m/n$$
We can, at this point assume that ECDF resembles some probability function, but this is a further topic we do not want to talk about here. What we want to stress i s that ECDFs (probabilities) are not automatically associated to a time. All the data in the sequence refers to different picks of a random variable, and these picks are not necessarily time-ordered or can be happened all at the same time. So the “frequencies" that can be associated to the above events are not time frequencies.
Now introduce time by saying that, for instance, each datum was sampled at regular time step $\Delta t$, what before I called “time units”, and, for practical reasons we are not interested to the ECDF of the data but to know how frequently (in clock time sense) it is repeated. So, we can say that the total time of our record is
$$T = n\, \Delta t$$
and in this time span, the number of time, h* is overcome is (by construction)
$$m=ECDFc(h^*)*n$$
On average, along the record obtained, the time frequency on which values greater than $h^*$ are obtained is the empirical return period:
$$T_r:=\frac{T}{m} =\frac{n *\Delta t}{ECDFc(h^*)*n} = \frac{\Delta t}{ECDF(h^*)}$$
So, the empirical return period of $h^*$ is inversely proportional to the complementary ECDF($h^*$) but, properly there is a “$\Delta t$” to remind that it is given in time units. One basic assumption in our definition is that the underneath probability is well defined, which is not if climate change is in action. This is a delicate and well discussed topic*, but again, not the core of this page.

There is a crucial initial step, the sampling of data which affects the final result. If the data in the sequence are, for instance annual maxima of precipitation, then the return period is given in years. If the data were daily precipitation totals, then the return period is given in days. And so on. Because usually the time unit has value “1” (but dimension of a time), the numeric value of the return period is just the inverse of the ECDFc. We should not forgot, however, that the equation contains a mute dimension. We are talking about times, not dimensionless numbers (probabilities).

Being Bayesian, probably you can introduce this in a different way. I let you as an exercise to do it.

** On the topic of stationarity, please give a look to:

Milly, P. C. D., Betancourt, J., Falkenmark, M., Hirsch, R. M., Kundzewicz, Lettenmaier, D. P., & Stouffer, R. J. (2008). Stationarity Is Dead: Whither Water Management? Science, 319, 1–2.

Montanari, A, and Koutsoyiannis, D,  Modeling and mitigating natural hazards: Stationarity is immortal!, Water Resources Research, 50 (12), 9748–9756, doi:10.1002/2014WR016092, 2014.

Serinaldi, F., & Kilsby, C. G. (2015). Stationarity is undead: Uncertainty dominates the distribution of extremes. Advances in Water Resources, 77(C), 17–36. http://doi.org/10.1016/j.advwatres.2014.12.013

A simple and trivial consideration about rainfall return period in relation with a spatial analysis, and discharges

Statistics at station, i.e. evaluated at ground measurements stations do not reflect the statistics of the areal event. This is obvious. However, even today, I read a report where the guys:

1. estimated the return period of rainfall in several point 
2. used the estimated depths to infer the spatially varying rainfall intensities
3. use these rainfalls as inputs of a rainfall-runoff model to obtain extreme discharges

In between there are a lot of technicalities, often useless. The point is, which is clear to the most, I hope, that, when moving from point 1 to point 2, one assumes that all the measured events are isochronous, which is not (otherwise we could not have let say 200 hundred return period events each year ia a space-wide area).  The above operation actually correspond to consider a precipitation with a higher return period than established (usually following some design criterion) and therefore maximise excessively the discharges.

What one should do is instead:

1. studying the spatial statistics of precipitation to enable a stochastic weather generator^1^2
2. run in continuos time her/his rainfall-runoff model for a long period, say 20 years if one wants to get some statistics a return period of 4 years,
3. analyse the results and extracting the statistics of the discharges (i.e. their return period), eventually extracting the extreme events

The validity of each component of the modelling chain should have been tested against available data (of the basin) independently.  Traditionally engineers do not like to simulate events at continuos time, and prefer to model events. This latter approach, however, has several drawbacks, and especially:

• one has to determine the initial conditions (which also introduce a bias in the return period) of the catchments (models that do not have this problem cannot be good models)
• fall back into the issue of determining a spatially distributed rainfall with a certain return period

Engineers usually also neglect the role of snow in producing discharges. This cannot be neglect, except than in particular climatic conditions. Using continuos time simulations also implies the use of some parameterisation of evapotranspiration (and therefore requires a model like JGrass-NewAGE).

^1 - Remarkably using a weather generator can also allows the inclusion of foreseen trends (either in precipitation characteristics, as depth, interstorm inter-arrival time, or evapotranspiration or radiation).

^2 -  Usually these models are site depedent. Therefore, waiting for a spatial stochastic weather generator, one should run several copies of the weather generator, each one for each sites where there can be information to drive it, and subsequently, using the spatial data, spatially interpolate at each time-step the desired quantity.