# Propagation Attenuation Maps Based on Parabolic Equation Method +
## 3 Propagation Attenuation Map
### PAM Algorithm  
The proposed approach allows determining path loss values in all discrete points around a fixed TX position (marked in red in Figure 1). The indexes of individual pixels are correlated with geographic coordinates, and the distance between them depends on the available topography data, i.e., DTED resolution, \(\Delta R\). The analyzed area is contained in a square with a side length of \(2R_{0}\), where \(R_{0}\) means the radius of the analyzed terrain area for which the PAM is determined. The size of the created PAM matrix is \(K\times K\), where \(K=\left\lfloor 2R_{0}/\Delta R\right\rfloor+1\). Calculations according to the PEM method are performed on defined directions around the TX depending on the angular resolution \(\Delta\alpha\). An example procedure for determining attenuation on a selected azimuth direction \(\alpha_{n}=n\Delta\alpha\) is depicted in Figure 1 (marked in green). The first step is to compute the terrain profile along the \(\alpha_{n}\) direction. This profile contains the height of the terrain with \(\Delta z\) resolution for discrete distance steps (\(\Delta x\)) along \(R_{n}=R_{0}/AF(\alpha_{n})\), where \(AF(\alpha_{n})\) is the following angular factor: \(AF(\alpha_{n})=\sin\alpha_{n}\) for \(45^{\circ}\leq\alpha_{n}\leq 135^{\circ}\) or \(225^{\circ}\leq\alpha_{n}\leq 315^{\circ}\), and \(AF(\alpha_{n})=\cos\alpha_{n}\) for \(0^{\circ}\leq\alpha_{n}<45^{\circ}\), \(135^{\circ}<\alpha_{n}<225^{\circ}\), or \(315^{\circ}<\alpha_{n}<360^{\circ}\). PEM calculations based on the SSFT approach are performed in the next phase. Then, path loss vectors for a given height of the receiving antenna, \(h_{R}\), are recorded into the PAM matrix. The method of determining the final PAM is explained in detail by Algorithms 1 and 2 for the sparse and full PAM matrices, respectively. Algorithm 1 shows a determination way of the sparse PAM matrix based on PEM calculations. The procedure for determining the path loss values should be repeated for every defined azimuth and the corresponding terrain profile. Depending on the selected azimuth resolution, the number of loop executions (lines 4 to 10 in the algorithm) vary. ```
0: DTED, DTED resolution (\(\Delta R\)), radius of the analyzed area (\(R_{0}\)), TX location, angular resolution (\(\Delta\alpha\)), receiving antenna height (\(h_{R}\)). 1: Calculate \(K=\left[2R_{0}/\Delta R\right]+1\), i.e., dimension of PAM matrix. 2: Create zero matrix for PAM with dimension \(K\times K\). 3: Set \(n=0\). repeat
4: Set \(\alpha_{n}=n\Delta\alpha\). 5: Calculate \(R_{n}=R_{0}/AF(\alpha_{n})\). 6: Determine terrain profile \(\mathbf{TP}(x)\), vector of length \(R_{n}\), for TX location and azimuth \(\alpha_{n}\) based on DTED. 7: Calculate PEM matrix, \(\mathbf{PEM}(x,z)\), for terrain profile \(\mathbf{TP}(x)\), e.g., using SSFT method. 8: Read path loss profile \(\mathbf{L}(D)=\mathbf{PEM}(x=D,z=h_{R})\) from determined PEM matrix for receiving antenna height \(h_{R}\). 9: Save path loss profile \(\mathbf{L}(D)\) to appropriate cells (green ones in Figure 1) of PAM matrix, \(\mathbf{PAM}(j,k)=\mathbf{PAM}(\alpha_{n})=\mathbf{L}(D)\), corresponding to analyzed azimuth \(\alpha_{n}\), where \(j\), \(k=1\), \(2\),..., \(K\) are indexes of PAM matrix and TX is located in central cell of PAM matrix (red one in Figure 1). 10: Set \(n=n+1\). until\(n\Delta\alpha>360^{\circ}\). 11: Output PAM as sparse matrix based on PEM for selected terrain profiles. ```  
**Algorithm 1**(_Creating sparse PAM matrix based on PEM for selected terrain profiles)The full PAM matrix is obtained by launching Algorithm 2. The proposed approach allows determining empty spaces (i.e., zero value cells) in the sparse PAM matrix (white ones in Figure 1). For this aim, we use linear regression. The interpolation method for creating the full PAM is explained in Figure 2. The input for this procedure is a partially completed path loss matrix, i.e., the sparse PAM matrix.