# RF Exposure Assessment by Drone-Based Technology
## 2 Materials and Methods
### Heat Maps  
From the discrete electric-field data obtained in the aforementioned samplings, we generated continuous distributions to visualize the electric field in different frequency bands using heat maps. We employed the geostatistical method known as kriging, a stochastic technique that uses a linear combination of weights at known points to estimate values at unknown locations [14].  
Geostatistics is a powerful tool for analyzing spatially distributed data and assumes that the variable under study is not entirely random but possesses a certain spatial structure. This spatial structure is typically analyzed by fitting a theoretical model to the experimental variogram. The variogram measures variability in terms of distance. A variogram, such as the one shown in Figure 4, is a graphical representation of variance as a function of the distance between points. In our case, if \(E(r_{i})\) and \(E(r_{i}+h)\) represent the electric fields measured at two locations separated by a distance \(h\), the variance \(\gamma(h)\) is defined by Equation (1) [14].  
\[\gamma(h)=\frac{1}{2\ N(h)}\sum_{i=1}^{N(h)}\left[E(r_{i})-E(r_{i}+h)\right]^ {2} \tag{1}\]  
where \(N(h)\) is the number of data pairs \([E(r_{i}),E(r_{i}+h)]\) separated by a distance \(h\). When \(h=0\), the variance is zero because there are no differences between points compared with themselves. As \(h\) increases, the points are compared with others that are farther away, so the variance increases. At a certain distance, called the range (\(R\)), the variance \(\gamma(h)\) reaches a value \(\gamma(R)\) known as the sill, which is equivalent to the sample variance. For values of \(h\) greater than the range, the variance does not increase further, and the data can be considered independent. Theoretically, the variogram \(\gamma(h)\) should be zero when \(h=0\), but experimental variograms often present a positive value of \(\gamma(0)\), giving rise to the so-called nugget effect (see Figure 4). This effect can be caused by (i) microvariance or  
Figure 3: Data collection points during walking sampling (**a**) and drone sampling (**b**).  
spatial variability at a scale smaller than that used for data collection, and (ii) experimental or sampling errors.  
Fitting a theoretical model to the experimental variogram allows us to estimate the parameters \(R\), \(\gamma(0)\), and \(\gamma(R)\), which can then be used to calculate electric-field values at unsampled points using kriging interpolation. This technique uses a weighted average of experimental data, where the weights depend not only on distance (as in distance-weighted averages) but also on the geometry of the sample locations, considering the spatial correlation structure deduced from the variogram analysis. The interpolated value, \(E_{interp}(r)\), at point \(r\) where no data were collected, is obtained using the linear estimator given by Equation (2).  
\[E_{interp}(r)=\sum_{i=1}^{N}\lambda_{i}E(r_{i}) \tag{2}\]  
where \(E(r_{1})\) and \(E(r_{N})\) are the known values at \(N\) locations and the weights \(\lambda_{i}\) are determined by minimizing the variance of the estimate [14]. The end result of kriging is a map including interpolated values of the variable.