## 2 Method
### The Four-Dimensional Tracking Model
#### 2.3.3 The Pitch Angle Case  
If it is in the pitch angle case shown in Figure 3c, the receivers are placed perpendicular to the ground, but not perpendicular to the pipeline due to the change of terrain. In the figure, the tracking receiver is located directly above the pipeline with zero lateral offset in the \(x\)-axis, and \(\alpha\) is the pitch angle between the PIG transmitter axis and the ground plane. The angle between the transmitter and receivers is \((90^{\circ}-\alpha)\). The magnetic field measured by tracking receivers in this case is the combination of the \(y\)-axis and \(z\)-axis components of the original magnetic field in the Cartesian coordinate system, \(H_{\rm y}(z)\) and \(H_{\rm z}(z)\). Therefore, the amplitude of measured magnetic field intensity \(H(z)\) can be expressed as (16):  
\[H(z)=\left(\begin{array}{cc}H_{\rm y}(z)&H_{\rm z}(z)\end{array}\right) \left(\begin{array}{c}\cos\alpha\\ \sin\alpha\end{array}\right)=\frac{NIS}{4\pi}\frac{\left(3 yz\cos\alpha+\left(2 z^{2}-y^{2}\right)\sin\alpha\right)}{\left(y^{2}+z^{2}\right)^{2.5}}e^{-\sqrt{\pi f \mu\
u h}}. \tag{16}\]  
It is easy to find that the existence of the pitch angle causes the positions corresponding to the maximum values of MFI not to overlap on the \(z\)-axis. Furthermore, the distance difference of the positions corresponding to the maximum value can be defined as \(\Delta z\). Therefore, the pitch angle can be expressed as (17):  
\[\alpha=\arcsin\frac{\Delta z}{r}. \tag{17}\]  
Taking \(H(z)=0\) in (16), the positions corresponding to the maximum values of MFI can be calculated out:  
\[z=\left(-\frac{3}{4}\tan\alpha+\sqrt{\frac{1}{2}+\frac{9}{16}(\tan\alpha)^{2} }\right)y, \tag{18}\]  
where the pitch angle \(\alpha\) can be obtained from (17), \(y\) can be regarded as the initial value in \(y\)-axis and can obtained from (11). Therefore, after measuring the magnetic field radiated by the transmitter and calculating the position corresponding to the maximum values of MFI, the position of the transmitter can be located according to (18).  
To use the MFI method to locate the PIG in the pitch angle case, multiply (1) with the sign function, and it is easy to find that the \(y\)-axis component of the original magnetic field in the Cartesian coordinate system meets the characteristics of symmetry along the \(z\)-axis, \(H_{\rm y}(z){\rm sgn}(z)=H_{\rm y}(-z){\rm sgn}(-z)\), while the \(z\)-axis component meets the characteristics of origin symmetry, \(H_{\rm z}(z){\rm sgn}(z)=-H_{\rm z}(-z){\rm sgn}(-z)\). Therefore, one can integrate the product of measured magnetic field \(H(z)\) in (16) and sign functions to obtain \(H_{\rm Isgn}(y)\), as shown in (19). It can be found that the \(z\)-axis component of the magnetic field in (16) has been integrated to 0. Furthermore, the maximum value in the MFI method is half of the maximum in the magnetic field sign-integration (MFSI) method in (19).  
\[H_{\rm Isgn}(y)=\int H(z){\rm sgn}(z){\rm d}z, \tag{19}\]  
\[r_{\rm new}=\sqrt{(r^{2}-\Delta z^{2})}. \tag{20}\]  
To conclude, the MPSI method can position the dimension pitch angle of the pipeline by (17) and track the DOF \(z\) of the PIG by (18). By replacing the maximum value of MFI with \(\frac{1}{2}H_{\rm Isgn}(y)\) and updating the radial distance difference between the dual receivers cor responding with (20), the distance between the PIG transmitter and the tracking receivers in \(y\)-axis can be obtained using (8), which is also the buried depth of the pipeline. The MFSI method also works for the normal case and the lateral case, in which the maximum value of MFI can be replaced with \(\frac{1}{2}H_{\text{Isgn}}(y)\).