## 3 Analysis of Spatial Transformation and Proposal of Critical Distance
In order to highlight the comparability of spatial transformation, the transient electromagnetic response amplitude of different offset distances was divided by the transient electromagnetic response amplitude of the full space state, and the proportional coefficient \(K\) under different offset distances was obtained. The calculation results are shown in Figure 7.  
As is exhibited from the results in Figure 7, the proportional coefficient curves of the 3/4 space and full space state showed a straight line. The proportional coefficient under different distances had obvious three-stage variation, namely, early, middle and late stages. The upper limit of the proportional coefficient curve was the amplitude of the proportional coefficient in the full space state, and the lower limit of the proportional coefficient curve was the amplitude of the proportional coefficient in the 3/4 space state. Overall, the proportional coefficients of different distance conditions were close to 1.0 in the early stage, and the proportional coefficients in the middle stage were gradually nonlinearly reduced from 1.0 to 0.568. The proportional coefficients in the late stage became gradually close to 0.568. When the offset distance was small, the early duration was relatively  
Figure 7: Space proportional coefficient under different distances. short, the proportional coefficient curve of the medium duration was relatively slow, the corresponding duration was relatively large, and the late duration was relatively large. When the offset distance was large, the early duration was relatively long, the proportional coefficient curve of the medium-term was relatively fast, and the corresponding duration was relatively small. It should be noted that the amplitude of the proportional coefficient in the early stage was slightly larger than the amplitude of the proportional coefficient in the full space state under different offset distances. The increase rate was not more than 5.19%, and the influence of the increase rate of this part could be ignored. Therefore, the offset distance determined the distribution of the induced current and the amplitude of the electromagnetic induction signal. The smaller the offset distance was, the closer it was to the 3/4 space state. The larger the offset distance was, the closer it was to the full space state. When the vehicle-mounted detection system enters and leaves the tunnel, there is an obvious spatial transformation process. Since the inversion of the vehicle-mounted detection system is based on the uniform medium of the full space state, and the spatial transformation process is obviously asymmetric, the spatial transformation process of the vehicle-mounted detection system needs to be corrected. Based on the relevant investigations [13; 16; 22], the space correction coefficient \(\Phi(t)\) is proposed for the spatial transformation process, as shown in Formula (16):  
\[\Phi(t)= \begin{cases}1&t\leq t_{1}\\ 1/\big{(}a_{0}+a_{1}\log(t)+a_{2}\log(t)^{2}+\cdots+a_{n}\log(t)^{n}\big{)}&t_ {1}<t\leq t_{2}\\ 1/0.568&t_{2}<t\end{cases}  \tag{16}\]  
where \(\Phi(t)\) is the correction coefficient of spatial transformation, \(t_{1}\) and \(t_{2}\) are the boundary points of the three stages, and their value is related to the distance between the vehicle-mounted detection system and the entrance and exit of the tunnel. The \(\mathrm{a}_{0}+\mathrm{a}_{1}\log(\mathrm{t})+\mathrm{a}_{2}\log(\mathrm{t})^ {2}+\cdots+\mathrm{a}_{n}\log(t)^{n}\) is the polynomial fitting formula of the proportional coefficient curve in the middle term under different offset distances, \(a_{0}\), \(a_{1}\), \(a_{2}\), and \(a_{n}\) are polynomial fitting coefficients, and \(n\) is the number of the polynomial fitting. The value \(t\) is the time of the vehicle-mounted detection system, and \(\log(t)\) represents the logarithm of time to the base 10. According to the relevant investigations [15], the average value of the characteristic parameters of the lifting time of the water-bearing abnormal body is \(-6.83\).