## 2 Methods and Tools
### Data Acquisition  
#### 2.2.1 The Survey Parameters  
Designing an appropriate GPR survey grid pattern is crucial for data acquisition. Several factors come into play when deciding on the grid layout: spatial resolution, depth of penetration, the maximum exposed area, etc. Figure 3 shows the scheme for calculating the GPR parameters. Following [43; 44; 45; 46], the general expression for the radius \(R_{0}\) of the maximum exposure area under the GPR center on the Earth's surface can be calculated as a function of the GPR altitude above the Earth's surface H, the center frequency of GPR antenna \(f\), and dielectric permittivity of the medium \(\varepsilon_{a}\) as follows:  
\[R_{0}=\ \ \frac{0.075}{f\sqrt{\varepsilon_{a}}}+\sqrt{\frac{0.075H}{f\sqrt{ \varepsilon_{a}}}} \tag{1}\]  
The constant '0.075' units, \(H\), and \(f\), are m/ns, m, and GHz, respectively. Since the value of the dielectric permittivity of air is equal to 1, Equation (1) can be simplified as follows:  
\[R_{0}^{D}=\ \ \frac{0.075}{f}+\sqrt{\frac{0.075H}{f}} \tag{2}\]  
Figure 3: The scheme for calculating the GPR parameters: (**a**) drone-based GPR, and (**b**) land-based GPR. where \(R_{0}^{D}\) is the radius of the maximum exposure area under the GPR center for drone-based GPR on the Earth's surface. Equation (2), used for the radius calculation for drone-based GPR, can be simplified for the case of ground-based GPR, taking into account the fact that the height of the GPR antennae above the ground surface, in this case, is negligibly small (\(H\approx 0\)) and hence can be neglected:  
\[R_{0}^{G}=\ \ \ \frac{0.075}{f} \tag{3}\]  
Estimating \(R_{z}\), that is, the radius of the maximum exposure area under the GPR center at depth Z, based on Equation (2) yields:  
\[R_{Z}^{D}=\ \ \frac{0.075}{f}+\sqrt{\frac{0.075}{f}}\Bigg{(}\sqrt{H}+\sqrt{ \frac{Z}{\sqrt{\epsilon_{s}}}}\Bigg{)} \tag{4}\]  
And for the case of ground-based GPR based on Equation (3):  
\[R_{Z}^{G}=\ \ \ \frac{0.075}{f}+\sqrt{\frac{0.075}{f}}\sqrt{\frac{Z}{\sqrt{ \epsilon_{s}}}} \tag{5}\]  
The depth of the desirable object Z and the maximum vertical resolution of the survey \(D_{Z}\) are critical factors affecting the quality of data collected by GPR [41; 42; 43; 44]. Balancing the depth and resolution is an ongoing challenge, and it is essential to strike a harmonious equilibrium to obtain the most informative and high-quality data during the GPR survey. The vertical resolution (\(D_{Z}\)) and minimum size (A\({}_{\text{min}}\)) of detectable objects can be calculated as follows [43; 44; 45; 46]:  
\[D_{Z}=\frac{0.15}{f\sqrt{\epsilon_{s}}} \tag{6}\]  
where the unit of the coefficient '0.15' is m/ns, and  
\[A_{min}=\left\{ \begin{array}{rl}(Z+H)/10,&if(Z+H)/10>D_{Z}\\ &\frac{0.075}{f\sqrt{\epsilon_{s}}},&if\ (Z+H)/10\leq D_{Z}\end{array} \right. \tag{7}\]  
An additional necessary parameter for the linear objects' detection (e.g., a wall) is their diameter (wall width), which follows [41; 42; 43; 44], usually estimated as 0.25-0.5 of the wavelength in the soil or, in a conservative estimation, 0.4 of the soil wavelength:  
\[D=(0.25\div 0.5)\frac{0.3}{f\sqrt{\epsilon_{s}}}\approx 0.4\frac{0.3}{f\sqrt{ \epsilon_{s}}} \tag{8}\]  
As can be seen, the flight altitude affects the data accuracy (Equations (2), (4) and (7)) and hence must be optimized. Our experience shows that the flight altitude for small objects should be no more than 1 m above the Earth's surface. Similarly to ground-based GPR, while scanning for small objects, slow velocity scanning is preferable; therefore, the flight speed was set to 1 m/s. Considering the typical near-surface soil lithology (Section 1.3), the estimated value of dielectric permittivity \(\epsilon_{s}\) was \(\sim\)3. This value was checked using the common for GPR measurements hyperbolic method.