## 4 Propagation Model
### Blockage by Clouds  
The next component to determine is the attenuation by the cloud \(L_{C}(f,d)\). Recall that the attenuation caused by a single layer of clouds has been calculated in Section 2. However, depending on weather conditions, a propagation path may or may not be occluded by a cloud at each layer. Thus, the overall attenuation caused by clouds can be approximated by  
\[L_{C}(f,d)=E\left[\sum_{k=1}^{K}p_{k}L_{k}\right], \tag{13}\]  
where \(p_{k}\) is the probability that clouds at layer \(k\) occludes the propagation path, \(L_{k}\) is attenuation induced by cloud blockage at layer \(k\) computed in Section 2. We now proceed with deriving \(p_{k}\) by using the tools of integral geometry. Observe that, to determine \(p_{k}\) in (13), we basically need to determine the probability that a point randomly dropped on the plane in \(\Re^{2}\) and representing the intersection of the propagation path and clouds at layer \(k\) crosses the cloud and actually induced attenuation in (13). This task can naturally be solved using integral geometry [53] as recently demonstrated in applied studies [38; 54]. To proceed, we need two definitions  
**Definition 1** (**Kinematic density**). : _Let \(K\) be the group of motions of the set \(A\) in the plane. The kinematic density of the set of points \(K\) is calculated as_  
\[dA=dx\cap dy\cap d\phi, \tag{14}\]  
_which must be invariant under the group of motions of the plane._  
**Definition 2** (**Kinematic measure**). : _The kinematic measure \(m\) of a set of group motions \(K\) on the plane is defined as the integral of the kinematic density \(dA\) over \(K\), i.e.,_  
\[m_{A}=\int_{K}dA=\int_{K}dx\cap dy\cap d\phi. \tag{15}\]  
Consider one THz BS located in the region \(A\) of interest, and let \(B\) denote the cloud in this zone, see Figure 4, where multiple clouds \(B_{i}\) shown. First, we are interested in the probability \(p_{C}\) that a ground-to-aircraft transmission path crossing this cloud layer at a randomly chosen point \(P\) in \(A\) actually crosses the cloud, i.e., \(P\) is in \(B\). Using the definition of conditional probability, we have  
\[p_{C}=\frac{Pr\{P\in A\cap B\}}{Pr\{A\cap B\
eq 0\}}, \tag{16}\]  
where the probability that a ground-to-aircraft transmission path crossing this cloud layer at a randomly chosen point \(P\) belongs to the intersection of two sets, \(A\) and \(B\), is in the numerator, and the probability that these sets intersect is in the denominator. Using the definition of a kinematic measure, we obtain  
\[Pr\{P\in A\cap B\} =m(A:P\in A\cap B), \tag{17}\] \[Pr\{A\cap B\
eq 0\} =m(A:A\cap B\
eq 0),\]  
where the first expression is the kinematic measure of the set of motions \(A\) such that \(P\in A\), and the second gives the measure of all motions \(A\) for which the intersection between \(A\) and \(B\) is not equal to zero. Following further calculations from [53], the measures will be equal to  
\[m_{j}\{P\in A\cap B\} =\int_{P\in B}f(x,y)dx\cap dy\cap d\phi, \tag{18}\] \[m_{j}\{A\cap B\
eq 0\} =\int_{A\cap B\
eq 0}f(x,y)dx\cap dy\cap d\phi,\]  
where \(f(x,y)\) is the distribution of the point \(P\) in the set \(A\). Figure 4: Illustration of cloud blockage at a certain cloud layer. Finally, the implied probability is defined as  
\[p_{C}=\frac{\int_{P\in B}f(x,y)dx\cap dy\cap d\phi}{\int_{A\cap B\
eq 0}f(x,y)dx \cap dy\cap d\phi}, \tag{19}\]  
and can be computed for a particular shape of \(A\), \(B\) and \(f(x,y)\). The numerator can be calculated as  
\[dA=dx\cap dy\cap d\phi, \tag{14}\]0 \[dA=dx\cap dy\cap d\phi, \tag{14}\]1  
where \(E[S_{B}]\) is the average area of a cloud. The measure of motions \(A\) at \(A\cap B\
eq 0\) is expressed as  
\[dA=dx\cap dy\cap d\phi, \tag{14}\]2 \[dA=dx\cap dy\cap d\phi, \tag{14}\]3  
where \(E[L_{B}]\) is the perimeter of a cloud, \(S_{A}\) and \(L_{A}\) are the area and perimeter of \(A\).