## 5 Discussion  
In the present work we study the hierarchical pattern of drainages area distribution in Greece as extracted from an ASTER GDEM v2 Digital Elevation Model, offering a 30m resolution, enabling the creation of an accurate drainage basins' database within a GIS environment. The power law exponent \(\beta\) for classification of drainages based on a) the geotectonic pattern and b) on topographic characteristics are estimated. Our analysis, demonstrate that an empirical power law distribution could be used to describe, as a first approximation, the drainages' area distribution. We study the drainage basin area-frequency distribution in the set of Greek drainages along with the subsets constructed applied different geological or geomorphological criteria. The hierarchy pattern observed for the drainage's areas not only for all the Greece but also for the subsets of drainages in the internal and external Hellenides that are the main tectonic structures in Greece, and the drainages classified according to their elevation or the Topographic Position Index (TPI), was presented. Furthermore the feasibility of the non-extensive statistical physics applied to the size distribution of the drainages areas is demonstrated. The estimation of the entropic parameter q which is mainly in the range 1.45-1.55, indicates a sub-additive system, with significant long range interactions. Within the view of Tsallis entropy we extract as a first approximation of Q-exponential a power law of the form  
\[\text{N}(>\text{A})\ \sim\ \text{A}^{-\beta}\ \text{where}\ \beta=\frac{1}{ \text{Q}-1}\]  
with Q the non-extensive parameter extracted from the fitting of observations with a Q-exponential. The Q parameter is related with the q entropic Tsallis parameter introduced in the definition of Tsallis Entropy as  
\[\text{q}=2-\frac{1}{\text{Q}}.\]  
Following NESP, the Greek drainage system could be seen as a drainage set organized by the merging of different sub-systems, according to the classification used. In view of Tsallis statistics formulation for non-extensive systems composed of non-extensive subsystems having different Q's it is proposed in [62] and references therein, that enables to estimate the behavior of a composite system containing subsystems, each having its own Q, using the expression [62]:  
\[\text{Q}=\frac{\sum\text{Q}_{\text{i}}\text{ln}\text{N}_{\text{i}}}{\sum\text {ln}\text{N}_{\text{i}}}\]  
where Q\({}_{\text{i}}\) the Q-exponential parameter and Ni the number of elements (drainages) in each of the sub-sets formed the composite one. In this frame the Greek drainage system could be viewed as containing two Q-exponential subsystems with different Q's as the external and internal Hellenides drainage sets are. Since \(\mathrm{Q}_{\mathrm{int}}<Q<\mathrm{Q}_{\mathrm{ext}}\) an estimation of Q is given  
\[\mathrm{Q}=\frac{\mathrm{Q}_{\mathrm{int}}\mathrm{In}\mathrm{N}_{\mathrm{int}}+ \mathrm{Q}_{\mathrm{ext}}\mathrm{In}\mathrm{N}_{\mathrm{ext}}}{\mathrm{In} \mathrm{N}_{\mathrm{int}}+\mathrm{In}\mathrm{N}_{\mathrm{ext}}},\text{ where }\mathrm{N}_{\mathrm{int}}=11042\text{ and }\mathrm{N}_{\mathrm{ext}}=7076,\]  
the number of drainages in each subset. Substituting the values of \(\mathrm{Q}_{\mathrm{int}}\) and \(\mathrm{Q}_{\mathrm{ext}}\) we lead to Q = 2.30 in a good agreement with Q \(\approx\) 2.41 obtained with fitting a Q-exponential for all the Greek drainage system. We now discuss the possible origin of the deviation from the power law observed in a number of cases for large values of drainage area A, where an exponential tail (i.e., q = 1) observed.