## 3 Verification of Theoretical Analysis and Numerical Simulation  
### Homogeneous Half-Space Medium  
According to the principle of the transient electromagnetic method, the secondary magnetic field of the homogeneous half-space medium is shown in Formula (1). In this case, the nonlinear features of the media are not considered in order to verify the feasibility of the detection method. \[H_{\theta}=\frac{2M}{4\pi R^{3}}[\varphi(u)-\sqrt{\frac{2}{\pi}}u(1+u^{2})e^{ -(u^{2}/2)}]\sin\theta \tag{1}\]  
where \(H_{\theta}\) is the secondary magnetic field component in the direction of the spherical coordinate system (A/m), \(M\) is the magnetic dipole moment (A\(\cdot\)m\({}^{2}\)), \(R\) and \(\theta\) are the components of the spherical coordinate system, \(\varphi(u)\) is the probability integral and its  
Figure 6: Principle and process of vehicle-mounted transient electromagnetic method: (**a**) basic principle; (**b**) detection process. expression is \(\varphi(u)=\sqrt{\frac{2}{\pi}}\int_{0}^{u}e^{-(t^{2}/2)}dt\), \(t\) is the electromagnetic field time (s), \(u\) is the judgment coefficient of time or distance of electromagnetic field and its expression is \(u=2\pi R/\sqrt{2\pi\rho t\times 10^{7}}\), and \(\rho\) is the resistivity of the medium (\(\Omega\cdot\)m). Two methods are used to verify the results of the electromagnetic field in a three-dimensional homogeneous half-space model. The first method is an analytical solution based on Formula (1), which is self-programming. The other method is the simulation solution, which uses electromagnetic field common software. In the analytical solution, the fast Hankel transform of 47 point filter coefficients can be used to solve the integral of the zero-order or first-order Bessel function in the transient electromagnetic field, and can be realized by MATLAB self-programming. In the process of calculation, the radius and turn of the transmitter coil are 20.0 m and 1, respectively. As for the ideal negative step, the maximum current of the transmitter coil is 1.0 A. The resistivity of the medium is 1000 \(\Omega\cdot\)m, and the radius and turn of the receiver coil is 0.01 m and 100, respectively. It should be noted that the radius of the transmitter coil in the analytical solution needs to be set large enough to ensure the high precision of the secondary magnetic field at the center of the coil. For the receiver coil, since the analytical calculation is linear, the coil form has no more limitations in the analytical calculation. In the simulation solution, a three-dimensional half-space model is adopted in the COMSOL software (Version 5.4), the center of the excitation coil is located at the center point of the three-dimensional model, and the normal direction of the transmitter coil is perpendicular to the X-Y plane of the Cartesian coordinate system [3]. The model is a cylindrical structure with a radius of 400 m and a height of 800 m. The upper part of the model is a surrounding rock medium with a resistivity of 1000 \(\Omega\cdot\)m, and the lower part is an air medium with a resistivity of 1,000,000 \(\Omega\cdot\)m. The infinite element is adopted at the boundary of the model, and its thickness is 20 m. The parameters of the coil in the simulation model are the same as the analytical solution method. The detailed model can be shown in Figure 7[3]. Through the calculation of the two methods, the numerical solution based on COMSOL and the analytical solution based on MATLAB can be obtained, and the results are shown in Figure 8. In order to compare the accuracy of the numerical solution and analytical solution, the relative error between them is also calculated, as shown in Figure 8.  
Figure 7: Layout of finite element model. It can be seen from Figure 8 that the numerical solution curve of the transient electromagnetic field in the three-dimensional half-space medium is consistent with the analytical solution curve. In general, the response curve of the secondary magnetic field exhibits an obvious linear attenuation trend in the middle and late stages.