## 5 Measurements Processing & Results  
The measurement campaign corresponds to the sampling of a spatial function. Moving from space discrete samples to a continuous surface is necessary in order to develop a Radio Environment Map (REM) and extract area-related metrics, such as the coverage area, or to visualize the transmission speed over the area of interest. This process, the estimation of new data points based on the range of a discrete set of known data points, is called interpolation. References [32; 33; 34; 35; 36; 37] outline some of the most common methods used to perform spatial interpolation for the purpose of developing a REM and provide information about the performance of the various techniques and their applicability. In this work, we select a well-known and broadly accepted spatial interpolation method, Ordinary Kriging \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Standard** & **Band** & **Bandwidth** & **Guard Interval/Spreading Factor** & **Number of Measurements** \\ \hline
802.11 ax & 5 GHz & 20 MHz & 3.2 μs & 1832 \\
802.11 ac & 5 GHz & 20 MHz & 0.8 μs & 1595 \\
802.11 ax & 2.4 GHz & 20 MHz & 3.2 μs & 2405 \\
802.11 n & 2.4 GHz & 20 MHz & 0.8 μs & 1939 \\ LoRaWAN & 868 MHz & 125 KHz & SF7 & 3802 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters and number of measurements for the different radio technologies and standards evaluated. Figure 6: (**a**) Satellite view of the area selected for the measurement campaign. The antenna mast was set up at the location marked with the red dot. (**b**) Elevated view of the same location from a UAV. The red arrow points to the location where the system was set up. with detrending [38; 39]. Despite being one of the most competent interpolation methods for the purpose of building REMs, we found the performance of other methods such as Radial Base Functions (RBFs) or Inverse Distance Weighting (IDW) to be similar due to the dense sampling that was carried out in this measurement campaign. Ordinary Kriging estimates the value \(\hat{Z}\) of a wide sense stationary random field at an unsampled location \(\mathbf{x}\) through the use of a linear combination of known sample values. \[\hat{Z}(\mathbf{x})=\sum_{i}\lambda_{i}\cdot Z(\mathbf{x}_{i}) \tag{1}\]  
The selection of the coefficients is based on spatial dependence. Samples closer to the unknown location have a greater impact on the predicted value. This dependence is captured by the variogram: a function describing the degree of spatial dependence between observations of the spatial random field \(Z(\mathbf{x})\) at two locations, given by the following equation. \[2\gamma(\mathbf{x}_{1}-\mathbf{x}_{2})=Var[Z(\mathbf{x}_{1})-Z(\mathbf{x}_{2} )]=E[(Z(\mathbf{x}_{1})-Z(\mathbf{x}_{2}))^{2}]=2\gamma(\mathbf{h}) \tag{2}\]  
where \(\mathbf{h}\) is the distance between the two positions, \(\|\mathbf{x}_{1}-\mathbf{x}_{2}\|\). Ordinary Kriging is a Best Linear Unbiased Estimator (BLUE), which means that the mean error must be zero for any value of the mean \(m\),  
\[E[\hat{Z}(\mathbf{x}_{0})-Z(\mathbf{x}_{0})]=\sum_{i}\lambda_{i}\cdot m-m=( \sum_{i}\lambda_{i}-1)\cdot m=0\Rightarrow\sum_{i}\lambda_{i}=1 \tag{3}\]  
and the estimation variance (mean square estimation error) \(\sigma_{\mathrm{E}}{}^{2}\),  
\[\sigma_{\mathrm{E}}{}^{2}(\mathbf{x}_{0})=-\gamma(\|\mathbf{x}_{0}-\mathbf{x }_{0}\|)-\sum_{i}\sum_{j}\lambda_{i}\lambda_{j}\gamma(\|\mathbf{x}_{1}-\mathbf{ x}_{j}\|)+2\sum_{i}\lambda_{i}\gamma(\|\mathbf{x}_{1}-\mathbf{x}_{0}\|) \tag{4}\]  
should be minimized. Equations (3) and (4) constitute a constrained optimization problem. Through the use of Lagrange multipliers, the following linear system of equations is derived, which can be used to calculate the coefficients \(\lambda_{i}\) for predicting the value of the unknown function at location \(\mathbf{x}_{0}\). \[