## 2 Method
### The Four-Dimensional Tracking Model
#### 2.3.1 The Normal Case  
The normal case assumes that the tracking receivers are located directly above the pipeline, and the pipeline is parallel to the ground, as shown in Figure 3a. References that consider only one-dimensional tracking all assume that AGMs work in the normal case [26; 27; 28]. It is obvious that distances \(r\), \(d_{1}\) and \(d_{2}\) meet the following relationship:  
\[d_{2}=d_{1}+r, \tag{4}\]  
Figure 3: Possible relative position cases between the PIG transmitter and receivers: (**a**) the normal case; (**b**) the lateral offset case; (**c**) the pitch angle case.  
Referring to (1) and (2) and taking \(x=0\), the radiation magnetic field of PIG transmitter in the normal case can be expressed as (5):  
\[\dot{\mathbf{H}}=\frac{3 NI\mathbf{S}yz}{4\pi}(\frac{1}{{(y^{2}+z^{2})}^{2.5}}+ \frac{\mathrm{j}k}{{(y^{2}+z^{2})}^{2}})e^{-(\beta+\mathrm{j}\alpha)h}\mathbf{ e}\mathbf{y}. \tag{5}\]  
The integration of the amplitude of the magnetic field intensity in (5) is calculaed with respect to the pipe axis direction and marked as \(H_{\mathrm{i}}(z)\):  
\[H_{\mathrm{i}}(z)=-\frac{NI\mathbf{S}}{4\pi}\frac{y}{{(y^{2}+z^{2})}^{1.5}}e^{ -\sqrt{\pi f\
u\omega}h}. \tag{6}\]  
Referring to (5), the amplitude of the magnetic field intensity follows the characteristics of origin symmetry, that is, \(H(z)=-H(-z)\) and \(H(0)=0\). Therefore, \(H_{\mathrm{i}}(z)\) reaches a maximum value at \(z=0\) shown as (7). The position corresponding to \(z=0\) is where the transmitter is located.  
\[H_{\mathrm{max}}(y)=-\frac{NI\mathbf{S}}{4\pi}\frac{1}{y^{2}}e^{-\sqrt{\pi f \
u\omega}h}. \tag{7}\]  
To conclude, the maximum value of the magnetic field integration (MFI) in (6) can be used to track the DOF \(z\) of the PIG. The method of locating PIG using the maximum value of MFI will be more reliable than using the zero magnetic field point. Because of the existence of noise and the limited accuracy of the sensor, it is difficult to accurately find the zero value of the magnetic field signal.  
Furthermore, the maximum value of MFI can also be used to calculate accurately the distance between the transmitter and receivers along the \(y\)-axis, that is, the buried depth of the pipeline. Referring to (7), the dual receivers tracking system can obtain two maximum values of MFI, \(H_{\mathrm{max}}(d_{1})\) and \(H_{\mathrm{max}}(d_{2})\). Combining (7) and (4), the distance between the PIG transmitter and the tracking receiver 1 and receivers 2 can be calculated as (8):  
\[\left\{\begin{array}{l}d_{1}=\frac{\sqrt{H_{\mathrm{i}max}(d_{1})H_{\mathrm{ i}max}(d_{2})+H_{\mathrm{i}max}(d_{2})}}{H_{\mathrm{i}max}(d_{1})-H_{\mathrm{i}max}(d_{ 2})}r,\\ d_{2}=\frac{\sqrt{H_{\mathrm{i}max}(d_{1})H_{\mathrm{i}max}(d_{2})+H_{\mathrm{ i}max}(d_{1})}}{H_{\mathrm{i}max}(d_{1})-H_{\mathrm{i}max}(d_{2})}r.\end{array}\right. \tag{8}\]  
The magnetic ratio in (8) has its own characteristics of filtering common-mode noise. As an example, for dual tracking receivers, the effects of pipeline wall and soil on the attenuation of ELF magnetic fields are the same, which can be automatically filtered out by the MFI method.