## 5 Discussion  
This paper's modeling and simulation results show the contrast between the simplest free space and a more realistic LoRa signal spatial distribution in a real-world cityscape. Despite its simplicity, the free-space path loss model Equation (8) provides some basic understanding of the propagating electromagnetic signals, specifically, the influence of frequency and transmitter gain in the signal strength attenuation over distance. The accurate prediction of the coverage range also depends on a correct receiver sensitivity calculation. In this work, the estimated value of \(-124\) dBm was obtained partly from actual implemented LoRa parameters and reference values for the signal-to-noise ratio and noise factor (Table 3), which introduces some uncertainty. The FSPL model predicts a maximum free-space coverage range of up to 435 km for the parameters considered. Indeed, previous works (using different implementations) have set the experimental world record at an astonishing 766 km [59]. Other works have reported ranges of 9.7, 6.9, and 4.6 km in rural, urban and suburban environments [60], respectively. Also 1.6 km [23] and 1.2 km [19] were reported in urban environments, and reaching down to about 100 m in high-density urban environments [19]. We show in Section 4.2 that the wave-propagation O-FDTD method can be used to simulate the propagation of LoRa signals in urban environments and generate high-resolution RSSI maps. The relative dielectric permittivity defines the propagation conditions in air and concrete building walls. The imaginary part of the permittivity describes the signal absorption by the walls, while the dielectric contrast at the air/wall interfaces leads to reflections. Further, O-FDTD provides exact modeling of wave-effects, such as diffraction, which is particularly relevant in urban environments and disregarded by many other models, such as analytical path loss models and line-of-sight ray-tracing [43]. Yet, at larger scales, the O-FDTD also reproduces ray-like reflections, as shown in Figure 4, which is vital for distance regimes where scattering starts to play a dominant role [22]. However, despite considering building wall arrangements, several factors are not considered, such as other scattering objects like cars and trees, variations of terrain, weather parameters, the possibility of interference, and the influence of RF noise. Also, the signal mapping is incomplete inside buildings, and variable construction materials are disregarded (a single permittivity). Nevertheless, it provides excellent coverage predictions with 84%. In the following, we address each of the identified error sources and discuss improvements possibilities:  
1. Implementation and impact of cars and trees: Apart from path loss models specifically developed to account for trees [25], which require knowledge of tree parameters such as spatial distribution, leaf area index, and trunk diameter, most approaches use semi-empirical methods to determine an average path loss characteristic of the city in question [20]. We do not do this in our approach but instead use the building walls arrangement of a real-world city map directly. We remark that trees are sparse in this particular area of the campus, and cars do not entirely block the signal propagation above 2 m from the ground. Nevertheless, future works beyond this proof-of-concept could, in areas containing denser distributions of such scattering objects, include a map of trees or additional scattering elements assigned to specific regions where cars or trees are known to be abundant. The accurate location of trees may, for example, be obtainable from satellite images from Google Maps using some dedicated image processing algorithm. Such additional elements would be treated in the O-FDTD method analogously to the way building walls were treated in this work. 2. Absence of terrain and weather data: As a 2D wave propagation method, O-FDTD does not trivially take terrain information into account.