## 2 Materials and Methods
### Distributed Hydrological-Hydraulic Modelling
#### 2.2.2 Design Storm  
To represent the distribution of rainfall intensity over time, parametrised by \(P(t)\) in (1), we analysed the extreme values of available rainfall data since 1950. The main objective was to calibrate the rainfall rate \(\bar{P}\) and duration \(D\) of the step hyteograph:  
\[P(t)=\begin{cases}\bar{P}&\text{if }0\leq t\leq D\\ 0&\text{if }t>D\end{cases} \tag{4}\]  
Figure 3a shows the accumulated precipitation for 24 h (referred to as daily rainfall depth from now on) measured by a rain gauge located in the Guadalen Reservoir at a distance of 4 km downstream of the town of Arquillos. The maximum value was 95 mm on 15 August 1996 due to a sudden storm in the night. It caused the most catastrophic flood reported in the town of Arquillos. Real-time hydrologic monitoring has provided hourly average values since 2010. Figure 3b depicts the accumulated precipitation for 1 h (referred to as hourly rainfall depth), with a peak value of 30 mm\(\cdot\)h\({}^{-1}\) on 8 June 2011 at 7:00 a.m. There are other extreme events with hourly rainfall depths close to the peak value, for instance, 27.9 mm\(\cdot\)h\({}^{-1}\) on 21 October 2012 at 4:00 a.m. and 28.9 mm\(\cdot\)h\({}^{-1}\) on 7 September 2016 at 8:00 a.m. Usually, the storm lasts for three hours and leads to daily rainfall depths of 80-95 mm (Figure 3a). Subsequently, we set the design storm parameters given by \(\bar{P}=30\) mm\(\cdot\)h\({}^{-1}\) and \(D=3\) h that lead to daily/hourly rainfall depths similar to the highest values measured since 1950.  
The dangerous nature of short-rain floods in small-area catchments is related to the short period of concentration \(T\) required to achieve the peak streamflow \(Q_{\text{ss}}\), given in the absence of infiltration by [PERSON] [60]:  
\[Q_{\text{ss}}=A_{\text{drain}}\bar{P}\,, \tag{5}\]  
\[T=\frac{L}{U_{P}}\,, \tag{6}\]  
with the runoff velocity \(U_{P}\):  
\[U_{P}=(\bar{P}\,L)^{\frac{2}{3}}\cdot S_{0}^{\frac{3}{30}}\,n^{-\frac{3}{5}}\,. \tag{7}\]  
In (6) and (7), \(L\) is the characteristic length of the catchment (e.g., \(L=A_{\text{drain}}^{1/2}\) in a nearly square basin), \(S_{0}\) is the mean basin slope and \(n\) corresponds to [PERSON]'s roughness coefficient. Setting \(n=0.02\) s\(\cdot\)m\({}^{-1/3}\) as for similar basins [1; 6], we get \(U_{P}=1.3\) m\(\cdot\)s\({}^{-1}\), \(T=33\) min and \(Q_{\text{ss}}=57.4\) m\({}^{3}\cdot\)s\({}^{-1}\). The theoretical value of \(T\) can be interpreted as the minimum duration of the rain event \(T\) needed to provoke \(Q_{\text{ss}}\).  
Figure 3: Maximum values of daily (**a**) and hourly (**b**) precipitation depth recorded 1950–2022 and 2010–2022, respectively. Data source: [[http://www.chguadalquivires/aih/](http://www.chguadalquivires/aih/)]([http://www.chguadalquivires/aih/](http://www.chguadalquivires/aih/)), accessed on 1 July 2022.  
Such an extreme value of the peak water discharge was not observed before 1996. Conversely, it has become usual in the last decade because of sudden storms with a duration shorter than one day and longer than the concentration time \(T\), referred to as a short-rain flood [61]. Short-rain floods have acquired relevancy in small catchments under the actual climate change scenario not only in the Upper Guadalquivir River [1; 6] but also in the Mediterranean [2; 62]. Furthermore, the reported increase in the frequency and the magnitude of short-rain inundations corroborates the predictions of the IPCC [3] in the current study site.