## 2 System and Channel Model
### Total Attenuation  
The total channel coefficient for each THz hop can be written as  
\[h_{n}=h_{l,n}h_{p,n}h_{a,n}, \tag{2}\]  
where \(h_{l,n}\), \(h_{p,n}\) and \(h_{a,n}\) denote the THz signal intensity due to the deterministic total attenuation, generalized pointing errors and atmospheric turbulence, respectively [21; 24; 41].  
The above deterministic total attenuation factor can be expressed in turn as  
\[h_{l,n}=h_{f,l,n}h_{u,l,n}h_{wl,n}, \tag{3}\]  
where \(h_{f,l,n}\), \(h_{u,l,n}\) and \(h_{wl,n}\) denote the attenuation due to FSPL, the molecular attenuation due to the water vapor and the attenuation caused by the weather effects, respectively. It should be clarified here that the latter attenuation factor obtains different values in case of fog and rain as it will be explained below. Additionally, it should be noted that \(h_{wl,n}=1\) in case of clear weather conditions. Indeed, in clear weather conditions, Equation (3) reduces to \(h_{l,n}=h_{f,l,n}h_{u,l,n}\) as it was reported in [19; 27; 41], where the presence of adverse weather effects was not taken into consideration.  
In more detail, the deterministic attenuation factor due to FSPL is obtained according to Friis equation as seen in [45]  
\[h_{fl,n}=c\sqrt{G_{t}G_{r}}(4\pi f_{n}z_{n})^{-1}, \tag{4}\]  
where \(c\) is the speed of the light in the free space, \(f_{n}\) represents the frequency of the information-bearing THz carrier wave which propagates along the \(n\)-th link, \(z_{n}\) denotes the \(n\)-th link length and \(G_{t}\), \(G_{r}\) are standing for the transmission and reception antenna gains of the corresponding nodes, respectively.  
Additionally, the molecular attenuation term due to the propagating signal's absorption along the \(n\)-th hop can be obtained according to the well-known Beer-Lambert Law as [11]:  
\[h_{vl,n}=exp(-a_{v,n}z_{n}), \tag{5}\]  
with \(a_{v,n}\) being the attenuation coefficient in m\({}^{-1}\) while the propagation distance along the \(n\)-th THz hop, \(z_{n}\), is expressed in meters. Bearing in mind that water vapor dominates in terms of THz attenuation over gaseous oxygen and any other species of gases as well as those at lower atmospheric altitudes where THz links are located, water vapor remains a major limiting factor in determining the maximum link distance, the maximum THz frequency of the link and the link performance [8; 10; 59], we have, without loss of generality, set \(a_{v,n}\) as the attenuation coefficient due to water vapor which at \(T_{0}=20\) degC surface temperature and for \(f\leq 350\) GHz is obtained as [60]:  
\[a_{v,l}\left(\frac{\mathrm{dB}}{\mathrm{km}}\right)= \tag{6}\] \[\left[0.067+\frac{2.4}{\left(f_{n}-2.3\right)^{2}+6.6}+\frac{7.33 }{\left(f_{n}-183.5\right)^{2}+5}+\frac{4.4}{\left(f_{n}-3238\right)^{2}+10} \right]f_{n}^{2}\rho_{n}\times 10^{-4}\]  
where the operation frequency \(f_{n}\) is now expressed in GHz, while \(\rho_{n}\) represents the water vapor concentration in g/m\({}^{3}\) along the corresponding channel. It should be mentioned here that as the surface temperature decreases, the attenuation coefficient is expected to increase by about 1% per degree Celsius, and vice versa [60].  
Similarly, the attenuation factor due to weather conditions prevailing along the \(n\)-th hop can be obtained as  
\[h_{ul,n}=exp(-a_{w,n}z_{n}), \tag{7}\]  
with \(a_{w,n}\) being the attenuation coefficient for the specific weather effect which is usually experimentally measured in km\({}^{-1}\), while the propagation distance along the \(n\)-th THz hop, \(z_{n}\), is expressed in km. Additionally, it should be clarified that the attenuation due to any weather condition such as rain or fog can be added to both attenuation due to FSPL and the molecular attenuation due to water vapor [19]. Consequently, in this work the total attenuation due to water vapor, rain or fog and FSPL will be jointly considered by properly utilizing the above attenuation expressions.