## 1 Introduction
Based on the \(x\)-axis and \(y\)-axis signals of the magnetic field of the transmitter, [PERSON] gives an orthogonal search coils model, which is applied to the tracking of high-speed PIGs through the least square criterion and ternary decision trees [28; 29]. There are three obvious problems with traditional ELF-AGMs. Firstly, the positioning of PIGs has not yet reached centimeter-level accuracy. The positioning error is 40 cm in [27] and 20 cm in [30]. Secondly, both [PERSON] [30] and [PERSON] [28] measure the ELF magnetic field in the axial and radial direction at the same time, which leads to signal redundancy. Thirdly, the traditional AGMs achieve only one-dimensional tracking for PIG (the axial direction of the pipe). However, the terrain above pipelines may change over time, and the actual location of pipelines may be inconsistent with the initial design [31]. Since the actual location of the underground pipeline is unknown, multiple dimensions tracking is required to accurately describe the location of the pipeline--including the relative position \((x,y,z)\) between the tracking receivers and the PIG, and the pitch angle between the pipeline and the ground plane--in the Cartesian coordinate system with the PIG transmitter as the origin and the direction along the pipeline as the \(z\)-axis. The coordinate \(z\) is used to track the PIG along the pipeline direction. The improvement of the tracking accuracy of the coordinate \(z\) can help to improve the accuracy of the defect location. The coordinate \(x\) is used to position the lateral offset distance between the PIG and the AGM, while the coordinate \(y\) is used to position the vertical distance between the PIG and the AGM, that is, the buried depth of the pipeline. The acquisition of the dimensions \(x\), \(y\), and pitch angle can quantify and compensate for their influence on the coordinate \(z\) tracking, which will help improve the robustness of the AGMs and improve the accuracy of defect location. In addition, the accurate lateral offset distance between the AGM and the pipeline obtained from the four-dimensional (4-D) tracking, the actual buried depth of the pipeline, etc., are helpful for the excavation and maintenance of the pipeline. At the same time, 4-D tracking of the pipeline is necessary correction information to compensate for the accumulated errors of the pipeline surveying system [18; 32]. Qi uses five receiving coils in a horizontal plane to form a receiver array, which has the function of tracking the PIG in multiple dimensions and further improves the tracking accuracy from 40 cm to 20 cm [30]. However, the tracking of multiple dimensions is not adequately realized in addition to the direction along the pipeline. Furthermore, the receiver array with the five coils on the same horizontal plane is too large to be convenient for field use, which can be seen from the experimental scene in Reference [30]. Therefore, the paper proposes a novel tracking receiver system with only two receiving coils distributed vertically and a corresponding 4-D tracking algorithm, which can achieve high-precision and 4-D tracking of the PIG. The dual receivers will detect the radiated magnetic field from transmitters at different heights. Using these dual magnetic signals and the distance between the dual receivers, it becomes possible to achieve 4-D tracking of the PIG. The tracking algorithm is mainly based on the origin symmetry feature of the radial component of the transmitter's magnetic field, which has the smallest error in various assumptions of the magnetic analysis. It determines that the tracking error of the algorithm proposed in this paper will be smaller than traditional ELF-AGMs. The algorithm is also proposed to solve the redundancy problem in the previous positioning system. First, the paper introduces the design of the dual receivers tracking system in detail. Then, based on the MDM, three possible cases are presented to discuss the 4-D tracking model and corresponding analytical solutions, which are verified by a finite element model (FEM).