## Appendix A Derivation of Equations  
(1) Apparent resistivity conversion by surface wave method  
The surface wave method mainly utilizes [PERSON]'s formula for the conversion of shear wave velocity to apparent resistivity. The wave speed in [PERSON]'s formula is primarily the acoustic wave speed. When using the surface wave method for geophysical sounding, the acoustic velocities extracted from the logs do not coincide with the seismic shear wave velocities. The relationship between the two data needs to be fitted by a function to derive the conversion relationship between apparent resistivity and seismic shear wave velocity to obtain apparent resistivity data for the surface wave method. [PERSON]'s formula is an empirical formula proposed by [PERSON] to describe the correlation between the resistivity of the subsurface medium and the acoustic wave velocity, which is applicable to the subsurface medium with a good statistical relationship between the acoustic wave velocity and resistivity. [PERSON]'s formula is shown in Equation (A1). \[V=KHC^{dR_{t}}\] (A1)  
where \(K\), \(C\), and \(d\) indicate stratigraphic parameters of different areas, \(H\) indicates depth, m, \(V\) indicates acoustic velocity, m/s, and \(R_{t}\) indicates resistivity, \(\Omega\)-m.  
Most of the dam materials used in earth and rock embankment projects are clay or loam materials. Its main lithologic properties are dominated by mudstones, which have a good statistical relationship between resistivity curves and acoustic curves. Therefore, [PERSON]'s formula can be used to express the relationship between resistivity and acoustic wave velocity. Considering that most of the earth and rock dam projects do not have acoustic logging conditions and cannot utilize the \(R_{t}\) and AC curves in the geologic data to fit the values of the stratigraphic parameters \(K\), \(C\), and \(d\), an improved method using [PERSON]'s formula is proposed. [PERSON]'s formula is mainly used to carry out the mutual conversion of acoustic velocity and density. The mathematical expression is as follows. \[\rho=kV^{m}\] (A2)  
where \(\rho\) indicates the density, g/cm\({}^{3}\), \(V\) indicates the acoustic wave velocity, m/s, \(k\) indicates the scale factor, taking the value of 0.31, and \(m\) indicates a constant, taking the value of 0.25. Substituting [PERSON]'s formula into [PERSON]'s formula for conversion. The improved [PERSON] formula is obtained by making \(K=a\) and \(C^{d}=b\) as shown in Equation (A3). \[\rho=0.31\left(ab^{R_{t}}H\right)^{0.25}\] (A3)  
Seismic exploration and sonic logging are two extremely important geophysical detection techniques in the detection of leakage hazards in earth and rock dams. The acoustic velocities extracted from logging do not coincide with the seismic shear wave velocities. [PERSON] [108] carried out a study on the function mapping relationship between sonic logging velocity and seismic wave velocity. The power function model was chosen as the final function mapping relation as shown in Equation (A4). \[V=0.65V_{P}^{1.05}\] (A4)where \(V\) indicates the actual sonic logging velocity, m/s, and \(V_{P}\) indicates the layer velocity in the depth domain of the seismic channel, m/s. Combining the function mapping relationship between sonic logging velocity and seismic velocity, and [PERSON]'s formula and the wave velocity conversion formula, the expression for the apparent resistivity of the surface wave method can be derived as shown in Equation (A6). \[V_{P}=\sqrt{\frac{2(1-v)}{1-2v}}V_{S}\] (A5)  
\[R_{t}=\frac{ln\frac{\left[0.65\left(\sqrt{\frac{2(1-v)}{1-2v}}V_{S}\right)^{1 05}\right]}{KH}}{lnc^{d}}\] (A6)  
where \(K\), \(C\), and \(d\) indicate the stratigraphic parameters, and \(v\) indicates the [PERSON]'s ratio, taking the value of 0.35.