## 2 Exploratory Data Analysis in Spino d'Adda  
In this section, we use the 10-year 1 min rain rate data bank collected in Spino d'Adda to study how the experimental 1 min rain rate time series \(R_{1}(t)\) (mm/h) are connected with the corresponding 10 min time series \(R_{10}(t)\) (mm/h). The latter is generated by \(R_{1}(t)\)  
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline \multirow{2}{*}{**Site**} & \multirow{2}{*}{**Latitude N (\({}^{\circ}\))**} & \multirow{2}{*}{**Longitude E (\({}^{\circ}\))**} & \multirow{2}{*}{**Altitude \(H_{5}\) (m)**} & \multirow{2}{*}{**Continuous Observation Time (Years)**} & \multirow{2}{*}{**Wind Speed at \(\sim\)700 mb Height (m/s)**} \\  & & & & & **(Years)** \\ \hline Spino d’Adda (Italy) & 45.4 & 9.5 & 84 & 10 (1993–2002) & 10.6 \\ \hline Gera Lario (Italy) & 46.2 & 9.4 & 210 & 5 (1978–1982) & 8.2 \\ \hline Fucino (Italy) & 42.0 & 13.6 & 680 & 5 (1978–1982) & 10.4 \\ \hline Madrid (Spain) & 40.4 & 356.3 & 630 & 9 (2006–2014) & 10.9 \\ \hline Prague (Czech Republic) & 50.0 & 14.5 & 250 & 5 (1999–2003) & 12.6 \\ \hline Tampa (Florida) & 28.1 & 277.6 & 50 & 4 (1995–1998) & 9.2 \\ \hline White Sands (New Mexico) & 32.5 & 253.4 & 1463 & 5 (1994–1998) & 9.1 \\ \hline Vancouver (British Columbia) & 49.2 & 236.8 & 80 & 3 (1995, 1996, 1998) & 12.4 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Geographical coordinates, altitude, number of years of continuous rain rate time series measurements, and wind speed at \(\sim\)700 mb (this latter parameter is necessary for applying the SST locally) at the indicated sites.  
according to the following relationship, applied to disjoint and contiguous 10 min intervals during the rainfall event:  
\[R_{10}\ =\ \frac{1}{10}\sum\
olimits_{k\ =\ 1}^{10}R_{1}(k) \tag{1}\]  
Figure 1 shows an example of this linear transformation. We notice, of course, that \(R_{10}(t)\) is smoothed; therefore, any application using \(R_{10}(t)\) will underestimate the effect of the \"instantaneous\" largest rain rates whose intensity can be significantly reduced.  
Since we transform \(R_{10}(t)\) (mm/h) into an estimated/simulated \(R_{1}(t)\), the first step is to study the distribution of \(R_{1}\) within intervals of 10 min conditioned to \(R_{10}\). These conditional distributions will be useful to simulate \(R_{1}(t)\) from \(R  
Figure 4: Histograms of \(R_{1}(t)\) in the ranges of 8–10 mm/h of \(R_{10}(t)\) (**left panel**) and 10–15 mm/h (**right panel**) of \(R_{10}(t)\).  
Figure 5: Histograms of \(R_{1}(t)\) in the ranges of 15–20 mm/h of \(R_{10}(t)\) (**left panel**) and 20–30 mm/h (**right panel**) of \(R_{10}(t)\).  
Figure 3: Histograms of \(R_{1}(t)\) in the ranges of 4–6 mm/h of \(R_{10}(t)\) (**left panel**) and 6–8 mm/h (**right panel**) of \(R_{10}(t)\).  
Figure 6: Histograms of \(R_{1}(t)\) in the ranges of 30–40 mm/h of \(R_{10}(t)\) (**left panel**) and \(>40\) mm/h (**right panel**) of \(R_{10}(t)\).  
From these figures, we can reasonably assume that a log-normal probability density function (PDF) [9] can adequately model the central part of the experimental histograms--a model used since the years 1970's (e.g., [10])--therefore, we calculated the mean value and standard deviation of \(\log(R_{1})\)--the natural logarithm--and assumed a log-normal probability PDF characterized by the values reported in Table 2, together with the correlation coefficient between two successive \(\log(R_{1})\) samples within the same 10 min interval. These parameters are fundamental because they will be used in Section 3 to simulate \(R_{1}(t)\) from \(R_{10}(t)\).