## 2 Materials and Methods
### A Brief Description of the Theoretical Principles of Drone-Based GPR Measurements  
The width of the explored area depends on the radiation pattern of the transmitting antenna, the EM wave transmission (soil properties), and the drone's height, namely on the radius of the First Fresnel Zone (FFZ) [57; 58]. In short, Fresnel zones are determined as concentric ellipses with the transmitter (Tx) and receiver (Rx) as the two focal points. The innermost ellipse, the First Fresnel Zone (FFZ), transfers over 70% of the signal energy to the receiver [59]. The problem of electromagnetic wave penetration in soils caused by a GPR antenna, as explained by Fresnel theory, has been extensively analyzed in [7; 60]. Here, we present a simplified approach to estimate the GPR survey parameters. The study area features dense tree growth and fences on the surface (Figure 3). With trees estimated to be about 4 m tall, the drone flight altitude (\(H\)) was set at 5 m. Three additional input parameters for the calculation are the target depth (\(D\)), the soil's cross-sectional content (soil dielectric permittivity), and the center frequency \(f\) of the GPR antenna. Figure 3: The community center construction site (dashed red line) is enclosed by a white fence (Figure 2). The drone’s precise flight path begins and ends at the west corner of the hilltop (purple circle), capturing changes in the natural surface under the distributed clayey soil. Obstacles like streetlights, trees (marked white), and a metal fence (marked blue) are visible on the map. The numbers show the straight section of the drone flight. The yellow point in the center marks the drilling test. As mentioned earlier (Section 2.1), the study's target depth is 15 m, and the subsurface is composed of clayey soil with a dielectric permittivity \(\varepsilon\) of approximately 15 [44]. Our previous study [45] demonstrated that conducting a GPR survey using antennas with 300-500 MHz center frequencies at a flight altitude exceeding 3 m does not yield high-quality data. The other three antennas included in the unmanned GPR kit have frequencies of 50, 100, and 150 MHz, which were considered for survey planning (see below). Using these three parameters, a six-step procedure can be used to determine the main exploration parameters for the GPR survey. The calculation scheme is shown in Figure 4.  
In Equations (1)-(3), \(R1\) is a function of the drone's altitude (\(H\) = 5), \(V\) is the speed of the electromagnetic wave in the material, and \(f\) is the radar center frequency [45; 46; 47; 48; 49]. \[V=300\Big{(}\frac{\text{mm}}{\text{ns}}\Big{)}/\sqrt{\varepsilon}, \tag{1}\]  
\[\lambda=V/f, \tag{2}\]  
\[R1=\frac{\lambda}{4}+\frac{\sqrt{H\cdot\lambda}}{2}, \tag{3}\]  
where \(\lambda\) is the wavelength and \(\varepsilon\) is the value of material dielectric permittivity [55; 56]. Considering the dielectric permittivity of air (\(\varepsilon_{a}\)-1), Equation (1) returns _Vair_ of 300 mm/ns. Based on Equation (2) and frequency \(f\) ranging 50-150 MHz, the \(\lambda\) value is 2-6 m (Table 1). Hence, considering the flight altitude \(H\) = 5 m, the value of \(R1\) (for the frequency range 50-150 MHz) equals \(\sim\)2-4.2 m.  
To estimate the radius \(R2\) of the maximum exposure area under the drone at a depth under the earth's surface (Figure 4) and additional parameters such as vertical resolution \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Parameter** & **Eq.** & \multicolumn{3}{c}{**Frequency, MHz**} \\ \hline  & & & **50** & **100** & **150** \\ \hline
1 & Wavelength in air, m & 2 & 6 & 3 & 2 \\
2 & Speed velocity in clay, mm/ns & 1 & & 77.5 & \\
3 & Wavelength in clay, m & 2 & 1.55 & 0.78 & 0.52 \\
4 & Footprint at Earth’s surface \(R1\), m & 3 & 4.24 & 2.7 & 2.1 \\
5 & Footprint at target depth R2, m & 4 & 6.7 & 4.4 & 3.5 \\
6 & Vertical resolution, m & 5 & 0.77 & 0.39 & 0.26 \\
7 & Minimum size of a detectable object, m & 6 & 2 & 2 & 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The list of calculated GPR parameters.