## Appendix A Derivation of Equations
(2) Apparent resistivity conversion by ground-penetrating radar method  
The ground-penetrating radar method (GPR) involves transmitting electromagnetic waves through a high-frequency radar antenna into the subsurface medium of the detection area, and when the electromagnetic wave encounters underground anomalies or hidden parts, it will be reflected back to the ground to be absorbed by the electromagnetic wave receiving antenna and transmitted back to the radar mainframe. By processing and analyzing the electromagnetic wave signals acquired by the radar host, it is possible to judge the underground anomalies and hidden parts in the detection area. Commonly used electromagnetic wave velocity extraction methods mainly include the Hough transform method, frequency-wave number domain offset method, and double curve fitting method. These methods are based on trial-and-error operations on the hyperbolic vertices in the B-scan and the electromagnetic wave velocity, and the closest result is chosen as the estimate of the electromagnetic wave velocity. The geometric relationship between the target and the radar is shown in Figure A1. The corresponding echo delay at \(x_{0}\) above the target in the underground medium is \(t_{0}\). The target echo delay at \(x\) is \(t_{h}\), and the propagation speed of the electromagnetic wave in the underground medium is \(v\). The hyperbolic equation can be deduced. \[(x-x_{0})^{2}=(vt_{h}/2)^{2}-(vt_{0}/2)^{2}\] (A7)\[v(x)=2|x-x_{0}|/\sqrt{t_{h}^{2}-t_{0}^{2}} \tag{10}\]  
From the above equation, it can be seen that the propagation velocity \(v\) of the electromagnetic wave in the subsurface medium can be obtained by estimating the hyperbolic curvature in the B-scan. Inverse distance-weighted interpolation is performed to obtain the electromagnetic wave velocity distribution in the geological body profile. The velocity of propagation of electromagnetic waves in a subsurface medium is given by:  
\[v=\frac{1}{\sqrt{\varepsilon\mu}}=\frac{1}{\sqrt{\varepsilon_{0}\varepsilon_{ r}\mu_{0}\mu_{r}}} \tag{11}\]  
where \(\varepsilon\) indicates the dielectric constant of the medium, \(\varepsilon_{r}\) indicates the relative dielectric constant (\(\varepsilon_{r}=\varepsilon/\varepsilon_{0}\)), \(\mu\) indicates the magnetic permeability, and \(\mu_{r}\) indicates the relative magnetic permeability (\(\mu_{r}=\mu/\mu_{0}\)). The speed of propagation of electromagnetic waves in a vacuum is given by:  
\[c=\frac{1}{\sqrt{\varepsilon_{0}\mu_{0}}}=2.98\times 10^{8}\text{m}/\text{s} \tag{12}\]  
where \(\varepsilon_{0}\) indicates the dielectric constant in vacuum, and \(\mu_{0}\) indicates the magnetic permeability in vacuum. From the above equation, the formula for the propagation speed of an electromagnetic wave in a medium can be deduced. \[\rho=kV^{m}\]0  
When geophysical surveys are carried out using the geological radar method, changes in the magnetic permeability of the surveyed area are generally ignored. By making \(\mu_{r}\) in the formula one, the following general formula for the propagation velocity of electromagnetic waves in the detection region can be obtained. \[\rho=kV^{m}\]1  
where \(c\) indicates the propagation speed of the electromagnetic wave in vacuum (\(c=2.98\times 10^{8}\) m/s), and \(\varepsilon_{r}\) indicates the relative dielectric constant (\(\varepsilon_{r}=\varepsilon/\varepsilon_{0}\)). From the above equation, it can be seen that the relationship between resistivity and relative dielectric constant is a prerequisite for deriving the conversion equation of apparent resistivity by the geo-radar method. [PERSON] et al. (2019) studied the influencing factors of electromagnetic wave resistivity measurements, and fitted the conversion relationship between resistivity and relative dielectric constant as shown in Equation (13) through a large amount of experimental data of geotechnical bodies.