## 2 Method
### The Four-Dimensional Tracking Model
#### 2.3.2 The Lateral Offset Case  
However, the normal case is an ideal situation. Considering that the pipeline is underground and invisible, the receivers may be laterally offset from the pipeline when it is placed above the ground. In this lateral offset case, the dual tracking receivers are located above the pipeline, while the axes of the receivers are offset from the transmitter in the \(x\)-axis direction, as shown in Figure 2(b). Furthermore, the pipeline is parallel to the ground. In the figure, \(w\) is the distance between the PIG transmitter and the tracking receivers in the \(x\)-axis direction. Referring to (1) and (2), the vertical component of the radiation magnetic field of the PIG transmitter in the lateral offset case can be expressed as (9):  
\[\dot{\mathbf{H}}=\frac{NI\mathbf{S}}{4\pi}3 yze^{-\sqrt{\pi f\
u\omega}h}e^{- \mathrm{j}\beta h}(\frac{1}{{(x^{2}+y^{2}+z^{2})}^{2.5}}+\frac{\mathrm{j}k}{{(x ^{2}+y^{2}+z^{2})}^{2}})\mathbf{e}\mathbf{y}, \tag{9}\]The difference in the amplitude of the magnetic field intensity is calculated, marked as \(H_{\text{d}}(z)\):  
\[H_{d}(z)=\frac{N\text{IS}}{4\pi}e^{-\sqrt{\pi f\
u}n}\left(\frac{3y}{\left(x^{2 }+y^{2}+z^{2}\right)^{25}}-\frac{15 yz^{2}}{\left(x^{2}+y^{2}+z^{2}\right)^{35} }\right), \tag{10}\]  
Referring to (9), maximum and minimum values of the magnetic field intensity can be obtained by taking \(H_{d}(z)=0\). It can be found that the positions corresponding to the maximum and minimum values, the horizontal offset and the vertical distance meet a position-square relationship, shown as (11). \[z=\pm\frac{\sqrt{x^{2}+y^{2}}}{2}. \tag{11}\]  
Therefore, the lateral offset \(w\) can be obtained from the dual receivers tracking system, which can obtain two position-square relationships, while \(y=d_{1}\) and \(y=d_{2}\). Combining the (11) and (4), the lateral offset \(w\) can be expressed as (12):  
\[w_{0}=\frac{\sqrt{-r^{4}-{z_{1}}^{4}-{z_{2}}^{4}+2r^{2}{z_{1}}^{2}+2r^{2}{z_{2 }}^{2}+2{z_{1}}^{2}{z_{2}}^{2}}}{2r}, \tag{12}\]  
where \((*,*,z_{1})\) and \((*,*,z_{2})\) are the positions corresponding to the maximum and minimum values of the magnetic field intensity under the distance \(d_{1}\) and \(d_{2}\), respectively. \(w_{0}\) is the initial calculated value distance between the PIG transmitter and the tracking receivers in the \(x\)-axis direction. Furthermore, the nomenclature of the variables in the equations is listed in the Appendix A. In the case of known lateral offset, the method of the maximum value of MFI can also be used to accurately calculate the vertical distance. The magnetic field measured by tracking receivers in Figure 3b can be considered as the combination of the radial and circumferential components of the original magnetic field in a cylindrical coordinate system, \(H_{\text{rd}}\) and \(H_{\text{pqd}}\), as shown in (13). \[\left( \begin{array}{cc}H_{\text{d1}}&*\\ *&H_{\text{d2}}\end{array} \right)=\left( \begin{array}{cc}H_{\text{rd1}}&H_{ \varphi_{1}}\\ H_{\text{rd2}}&H_{\varphi_{2}}\end{array} \right)\left( \begin{array}{cc}\sin \varphi_{1}&\sin\varphi_{2}\\ \cos\varphi_{1}&\cos\varphi_{2}\end{array} \right), \tag{13}\]  
where \(\varphi_{1}\) and \(\varphi_{2}\) are circumferential angles, which are the angles between the \(x\)-axis and the line of the transmitter and receivers. Combining the (11) and (12), \(\varphi_{1}\) and \(\varphi_{2}\) in Figure 3b can be expressed as \(\sin\varphi_{1}=\frac{d_{1}}{z_{1}}\) and \(\sin\varphi_{2}=\frac{d_{2}}{z_{2}}\). \(H_{\text{d1}}\) and \(H_{\text{d2}}\) comprise the magnetic field measured by dual tracking receivers. Referring to Figure 2, it can be found that the \(\varepsilon_{\varphi}\) component of the magnetic field is 0, \(H_{\text{{{{\it{gi}}}}}}=0\), \(i=1,2\). Furthermore, the nomenclature of the variables in the equations is listed in the Appendix A. Therefore, the radial components of the original magnetic field in a cylindrical coordinate system, \(H_{\text{rd1}}\) and \(H_{\text{rd2}}\), can be solved inversely.