# Propagation Attenuation Maps Based on Parabolic Equation Method +
## 5 PAM Accuracy Evaluation versus Angular Resolution of Terrain Profiles  
In this section, we present a preliminary assessment of the accuracy of the proposed algorithm versus the angular resolution of the terrain profiles. This assessment was made for the selected TX location with the coordinates (49.28\({}^{\circ}\)N, 19.96\({}^{\circ}\)E) and \(R_{0}=1\) km. The terrain around TX is shown in Figure 11a, while Figure 11b illustrates the PAM obtained for the angular resolution of the terrain profiles equal to \(\Delta\alpha=1^{\circ}\). In our analysis, this PAM is used as a reference to determine the algorithm errors at another angular resolution of the terrain profiles. Figure 12 shows exemplary PAMs for the analyzed area and two different angular resolutions equal to \(2^{\circ}\) and \(10^{\circ}\). Figure 10: Normalized number of points where calculations were made exactly \(r\) times for TX-RX distance equals to 1 km. Figure 9: (**a**) Absolute and (**b**) normalized number of all calculations \(CN\) for different distances between TX and RX. In the following, for every map point, \(\mathbf{PAM}(j,k)\), we calculate the attenuation error  
\[\Delta L(j,k)=\mathbf{PAM}_{\Delta\alpha}(j,k)-\mathbf{PAM}_{\Delta\alpha=1^{ \circ}}(j,k), \tag{8}\]  
where \(\mathbf{PAM}_{\Delta\alpha}(j,k)\) and \(\mathbf{PAM}_{\Delta\alpha=1^{\circ}}(j,k)\) are the PAM values (i.e., attenuations) obtained for the analyzed \(\Delta\alpha\) and reference \(\Delta\alpha=1^{\circ}\) angular resolutions of terrain profiles. In this way, the error maps, i.e., attenuation error matrices, are determined. Figure 13 shows sample error maps that were obtained for the two PAMs illustrated in Figure 12.  
These error maps are the basis for determining the empirical cumulative distribution function (CDF), \(\mathrm{CDF}(\Delta L)\), and root-mean-square error (RMSE), \(\mathrm{RMSE}(\Delta L)\), of the attenuation error. Exemplary CDFs for the two analyzed error maps (see Figure 13) are presented in Figure 14a. Generally, we can judge that the obtained functions are symmetrical about the sign of \(\Delta L\). This means that the attenuation errors in the PAM with \(\Delta\alpha\) oscillate around the attenuations for the reference PAM. Hence, we also computed the CDFs for the module of the attenuation error, \(\mathrm{CDF}(|\Delta L|)\), which are depicted in Figure 14b. Figure 11: (**a**) DTED-based elevation map for selected TX location, (**b**) reference PAM for angular resolution \(\Delta\alpha=1^{\circ}\). The CDFs analysis shows that increasing \(\Delta\alpha\) causes a significant increase in the attenuation errors in determined PAMs. This is due to the smaller number of terrain profiles included in the calculations. For \(\Delta\alpha=2^{\circ}\), 80% of the absolute attenuation errors are less than 3 dB, while for \(\Delta\alpha=10^{\circ}\), it is 7 dB. For \(\mathrm{CDF}(|\Delta L|)=0.9\), \(|\Delta L|=5\,\mathrm{dB}\) and \(|\Delta L|=10\,\mathrm{dB}\) for the angular resolutions equal to \(2^{\circ}\) and \(10^{\circ}\), respectively. Therefore, we may assume that for small Da, the number and time of calculations can be reduced at the expense of the accuracy of the PAMs obtained. This approach can be used in the preliminary stage of PAM determination, which requires initial visualization of the results. In the next step, with the platform computing resources available, the exact PAMs can be determined for the lower angular resolution of terrain profiles. Similar conclusions might be drawn by analyzing the RMSE scalar measure. Figure 15 shows the RMSE of attenuation error for selected angular resolution. Figure 14: CDFs of (**a**) attenuation error and (**b**) attenuation error module for selected angular resolution of terrain profiles. Figure 13: Error maps for selected angular resolutions (**a**) \(\Delta\alpha=2^{\circ}\) and (**b**) \(\Delta\alpha=10^{\circ}\). For the angular resolutions equal to \(2^{\circ}\) and \(10^{\circ}\), we obtained \(\text{RMSE}(\Delta L)=4\) dB and \(\text{RMSE}(\Delta L)=7\) dB, respectively.