# Propagation Attenuation Maps Based on Parabolic Equation Method +
## 3 Propagation Attenuation Map
### PAM Algorithm
Attenuation values in gray-marked points (see Figure 1 or Figure 2) are calculated according to Algorithm 1 described above. Zero cells (white ones in Figure 1) of the sparse PAM matrix are calculated based on linear regression and the two attenuation values located in the nearest non-zero cells. Depending on the coordinates of the considered point, i.e., indexes of matrix cell, there are two possible ways of linear interpolation:  
* in a column--for the azimuth directions: \(\bigcirc\) \(225^{\circ}\leq\alpha_{n}\leq 315^{\circ}\) (i.e., \(k\leq j\leq K-k+1\) for \(k=1,2,\ldots,\lfloor K/2\rfloor+1\)), \(\bigcirc\) \(45^{\circ}\leq\alpha_{n}\leq 135^{\circ}\) (i.e., \(K-k+1\leq j\leq k\) for \(k=\lfloor K/2\rfloor+2,\lfloor K/2\rfloor+3,\ldots,K\)),
* or in a row--for the remaining azimuth directions: \(\bigcirc\) \(315^{\circ}<\alpha_{n}<360^{\circ}\) or \(0^{\circ}\leq\alpha_{n}<45^{\circ}\) (i.e., \(j+1\leq k\leq K-1\) for \(j=1,2,\ldots,\lfloor K/2\rfloor+1\)), \(\bigcirc\) \(135^{\circ}<\alpha_{n}<225^{\circ}\) (i.e., \(K-j+2\leq k\leq j-1\) for \(j=\lfloor K/2\rfloor+2,\lfloor K/2\rfloor+3,\ldots,K\)). In Figure 2, we showed these selected interpolation cases by colored cells of the PAM matrix. For column interpolation, the blue-marked zero cell values are determined based on two yellow non-zero PAM cell values of the sparse PAM matrix, i.e., \(\mathbf{PAM}(g,k)\) and \(\mathbf{PAM}(h,k)\) obtained for the azimuths \(\alpha_{n}\) and \(\alpha_{n+1}\), respectively. For row interpolation, the purple-marked zero cell values are derived from the two orange non-zero cell values of the sparse PAM matrix, i.e., \(\mathbf{PAM}(j,p)\) and \(\mathbf{PAM}(j,q)\) obtained for the azimuths \(\alpha_{m+1}\) and \(\alpha_{m}\), respectively. In these cases, the linear regression is performed according to the following formulas:  
\[\mathbf{PAM}(j,k)=\tfrac{\mathbf{PAM}(h,k)-\mathbf{PAM}(g,k)}{h-g}(j-g)+ \mathbf{PAM}(g,k)\quad\text{for}\quad g<j<h, \tag{2}\]  
\[\mathbf{PAM}(j,k)=\tfrac{\mathbf{PAM}(j,p)-\mathbf{PAM}(j,q)}{q-p}(k-p)+ \mathbf{PAM}(j,p)\quad\text{for}\quad p<k<q. \tag{3}\]  
According to Algorithm 2 presented above, the interpolation procedure is repeated column-by-column and then row-by-row so as to determine the path loss corresponding to the receiving antenna height at all points (i.e., cells) of the PAM matrix. ```
0: sparse PAM matrix (PAM). Interpolation in columns: 1. Set \(k=1\). repeat
2. Find \(g\)th row (\(g\geq k\)) corresponding first non-zero cell in \(k\)th column (see one of yellow cells in Figure 2). repeat
3. Find \(h\)th row corresponding next non-zero cell in \(k\)th column (see one of yellow cells in Figure 2). 4. Calculate all zero cells in \(k\)th column between \(g\)th and \(h\)th rows, i.e., PAM(\(j\), \(k\)) for \(g<j\prec h\), based on Equation (2) (see blue cells in Figure 2). 5. Set \(g=h\). until\(h\leq\) K-\(k+1\). 6. Set \(k=k+1\). until\(k\leq\)\([K/2]+1\). repeat
7. Find \(g\)th row (\(g\geq K\)-\(k+1\)) corresponding first non-zero cell in \(k\)th column (see one of yellow cells in Figure 2). repeat
8. Find \(h\)th row corresponding next non-zero cell in \(k\)th column (see one of yellow cells in Figure 2). 9. Calculate all zero cells in \(k\)th column between \(g\)th and \(h\)th rows, i.e., PAM(\(j\), \(k\)) for \(g<j<h\), based on Equation (2) (see blue cells in Figure 2). 10. Set \(g=h\). until\(h\leq k\). 11. Set \(k=k+1\). until\(k\leq K\). ```  
_Interpolation in rows_:
12. Set \(j=1\). repeat
13. Find \(p\)th column (\(p\geq j+1\)) corresponding first non-zero cell in \(j\)th row (see one of orange cells in Figure 2). repeat
14. Find \(g\)th column corresponding next non-zero cell in \(j\)th row (see one of orange cells in Figure 2). 15. Calculate all zero cells in \(j\)th row between \(p\)th and \(g\)th columns, i.e., PAM(\(j\), \(k\)) for \(p<k\)\(<q\), based on Equation (3) (see purple cells in Figure 2). 16. Set \(p=q\). until\(q\leq K\)-\(j\). 17. Set \(j=j+1\). until\(j\leq[K/2]+1\). repeat
18.