## 2 Diffusion Characteristics and Proportional Coefficients of Space States  
### Transient Electromagnetic Theory and Space Environment  
[PERSON]'s equations are the basic laws for the propagation of macroscopic electromagnetic fields in a vacuum or medium [18; 19]. The equations are a set of partial differential equations used to describe the relationship between electric field, magnetic field, electric charge density, and current density [18; 20]. The VMTEM in operational tunnels in the medium can be described by [PERSON]'s equations. The equations reflect four basic physical laws, as shown in Formulae (1)-(4):  
\[\
abla\times E=-\frac{\partial B}{\partial t}\left(\text{Faraday's Law}\right) \tag{1}\]  
\[\
abla\times H=J+\frac{\partial D}{\partial t}\left(\text{Ampere}-\text{ Maxwell's Law}\right) \tag{2}\]  
\[\
abla\cdot D=\rho\left(\text{Gauss's Law of Electric Field}\right) \tag{3}\]  
\[\
abla\cdot B=0\left(\text{Gauss's Law of Magnetic Field}\right) \tag{4}\]  
where \(E\) represents the electric field intensity (V/m), \(H\) indicates the magnetic field intensity (A/m), \(D\) is the electric displacement vector (C/m\({}^{2}\)), \(B\) is the magnetic flux density (Wb/m\({}^{2}\)), \(J\) denotes the current density (A/m\({}^{2}\)), \(\rho\) represents the electric charge density (C/m\({}^{3}\)), and \(t\) is the time of the electromagnetic field (s). As a complete set of equations to determine the motion of the electromagnetic field and charge and the current system, the following constitutive relations of the electromagnetic field exist between electric field intensity \(E\) and electric displacement vector \(D\), magnetic field intensity \(H\) and magnetic induction intensity \(B\), and electric field intensity \(E\) and current density \(J\), as shown in Formula (5):  
\[D=\varepsilon E,\ B=\mu H,\ J=\sigma E \tag{5}\]  
where \(\varepsilon\), \(\mu\), and \(\sigma\) are the dielectric constant (F/m), magnetic permeability (H/m), and electric conductivity (S/m) of the medium, respectively. Assuming that the research problem of this paper is free space, \(\varepsilon=\epsilon_{0}=8.854\times 10^{-12}\) F/m, and \(\mu=\mu_{0}=4\pi\times 10^{-7}\) H/m, where \(\epsilon_{0}\) and \(\mu_{0}\) are the vacuum dielectric constant (F/m) and vacuum magnetic permeability (H/m), respectively. Introducing the vector magnetic potential \(A\), defined as \(B=\
abla\times A\), the formulae between (1) and (5) can be rewritten, as shown in Formula (6):  
\[\sigma\cdot\frac{\partial}{\partial t}A+\
abla\times H=0 \tag{6}\]  
The expression of the electric field vector can be obtained by Fourier transform, as shown in Formula (7):  
\[E=\text{i}\omega\eta\left[A+\frac{1}{k^{2}}\
abla(\
abla\cdot A)\right] \tag{7}\]  
where \(k\) represents the wave number or propagation constant, \(\omega\) indicates the electromagnetic frequency (rad/s), and i denotes the virtual number unit. When the transient electromagnetic method is applied to detect tunnel defects in operational tunnels, the specific space state and background field exist, which is different from the conventional application field. The detection process is a moving process from the entrance to the exit of operational tunnels. The space state of the VMTEM changes with the variation of the position of the vehicle-mounted detection system, as shown in Figure 1. As for ideal conditions, without the low-resistivity of a water body and uneven distribution, when the vehicle-mounted detection system is located above the infinite stratum, the lower part of the coil system is the uniform surrounding rock medium, and the air medium is above the coil system. The coil system can receive electromagnetic induction signals generated by the surrounding rock below, namely, the half space effect. When the vehicle-mounted detection system is located at the entrance of the tunnel, the lower part of the coil system is the uniform surrounding rock medium, the upper-left is the air medium, and the upper-right is the uniform surrounding rock medium.