## 4 Hierarchical Drainage Basins Analysis
In this study, TPI generated from 2000 m neighborhoods due to the extended spatial coverage of the study area. Positive TPI values represent areas that are higher than the average of their neighborhoods (ridges), while negative TPI values represent locations with less elevation than their neighborhood (valleys). TPI values near zero are characterized as flat areas or areas of constant slope. The mean TPI value was further calculated for each drainage basin unit. At the scale of 2000 m, TPI reflects the broader valley morphology and the relative relief of streams and their surrounding topography. Drainage basins with higher mean values have a high proportion of streams in relatively deeper and narrower drainages, with narrowness defined by the spatial scale of the index [55]. For the area distribution analysis we obtained 5 classes of drainage basins with different mean TPI values using the quantile classification method (Figure 4a,b). As a first step in our analysis we present the cumulative drainage area distribution for all the territory of Greece (Figure 5). As Figure 5 presents a power law scaling of the form [2]  
\[\text{N(>A)}\sim\text{A}^{-\beta}, \tag{6}\]  
where N(>A) is the number of drainages with area greater than A, is observed with \(\beta_{\text{all}}\)\(\approx\)0.67, with a deviation from power law to observed for large drainage areas. A power law (see Equation (6)) fits the data for both the data sets organized for the drainage areas in external and internal Hellenides, respectively, implying a hierarchical organization of drainage basins. For each one of the cases we have \(\beta_{\text{ext}}\approx\) 0.65 and \(\beta_{\text{int}}\approx\) 0.86 (see Figure 6a,b). For the case of External Hellenides a deviation of observation from power law is observed for large drainage areas, i.e., for A > 100 km\({}^{2}\). Figure 4: (**a**) TPI values in Greece at a scale of 2000m. Inset: Detail of the TPI map with drainage basins overlay, (**b**) Quantile classification of drainage basins according to their mean TPI value. **Inset**: detail of the map. In Table 1 we present the power law exponent \(\beta\) observed for each of the six elevation classes defined (Figure 7). Figure 5: The drainage area distribution for Greece. A power law fitting is presented with \(\beta_{\rm all}\)\(\approx\)0.67 (green line) along with the Q-exponential (red line) with Q\(\approx\) 2.41 (see text). Figure 6: The drainage distribution for Internal (**a**) and External (**b**) Hellerides, Greece. A power law fitting is presented with \(\beta_{\rm int}\)\(\approx\) 0.86, and \(\beta_{\rm ext}\)\(\approx\) 0.65 for the internal and external Hellerides, respectively (green line) along with the Q-exponential (red line) with Q\({}_{\rm int}\)\(\approx\) 2.16 and Q\({}_{\rm ext}\)\(\approx\) 2.45 for the internal and external Hellerides, respectively (see text). In Table 2 we present the power law exponent \(\beta\) observed for each of TPI defined classes (see Figure 8). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Mean Elevation** & \(\beta\) & **Q** & **q** & \(\beta^{\text{cal}}\) \\ \hline
**0–30 m** & 0.95 & 1.91 & 1.48 & 1.10 \\
**30–90 m** & 1.25 & 1.85 & 1.46 & 1.18 \\
**90-150 m** & 0.995 & 1.98 & 1.495 & 1.02 \\
**150–300 m** & 1.02 & 2.00 & 1.5 & 1.00 \\
**300–900 m** & 0.715 & 2.45 & 1.59 & 0.69 \\
**>900 m** & 0.29 & - & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The power law exponent \(\beta\), the Q and q non extensive parameters and the \(\beta^{\text{cal}}\) value as estimated for each of elevation class of Greece drainages. Figure 7: The drainage distribution for the six elevation classes defined in Greece. A power law fitting is presented along with the Q-exponentials. The scaling exponent \(\beta\) and the Q values as presented in Table 1 (see text). All the above mentioned distributions are analyzed in terms of Tsallis entropy estimating the q-entropic index characterizing the distribution.