## 1 Introduction  
Pipelines are widely used in oil and natural gas transportation projects. Now, the total length of pipelines in a single country may have exceeded \(10^{5}\) km [1, 2]. However, because pipelines are buried underground for a long time, they may also be corroded [3], even with the typical corrosion prevention techniques (coating, cathodic protection, etc.) [4]. Under the effect of stress-strain, corrosion on the pipe wall gradually evolves into defects [5]. The further development of defects may lead to leakage incidents of the oil or gas, which may cause economic losses and environmental pollution [6]. Based on a series of nondestructive testing technologies such as magnetic flux leakage (MFL) detection [7], eddy current testing [8] and ultrasonic testing [9], the pipeline inspection gauge (PIG) can be developed to detect and evaluate pipeline defects. The two most important issues in the nondestructive inspection of pipelines are whether defects present and, if so, where they are located [10]. The defect localization is mainly divided into two types: active locating method and above-ground tracking method. In the active locating method, the pipeline inspection gauge (PIG) measures its own movement through the instruments it carries. Furthermore, there are two ways to achieve active locating method, the odometer positioning method [11] and the inertial measurement unit (IMU) method [12, 13]. However, due to the slip caused by paraffin oil or the mechanical error caused by wear [10], the positioning error of the odometer can be as high as 10% [14]. What is worse, owing to cumulative counting, there is a cumulative error in the odometer positioning method [15]. Similarly, IMU realizes positioning through the integration of acceleration and angular velocity, and inevitably also has an accumulated error, which is 20 cm per 100 m [16; 17]. The inspection mileage of hundreds of kilometers will accumulate positioning errors of hundreds of meters, which is unacceptable in engineering applications. Therefore, it is necessary to place above-ground markers (AGM) at regular intervals along the pipeline to communicate with the PIG and correct the accumulated error of the active locating method [18]. At present, the most commonly used AGMs are based on the magnetic flux leakage signal, which uses the magnetic field leaking from the permanent magnet carried by the MFL-PIG to achieve the positioning of the PIG [19]. Obviously, it is only applicable to MFL-PIG. The friction between PIG and the pipe wall will generate a self-excited vibration acoustic signal, while the collision between PIG and the pipe weld will generate a transient shock vibration acoustic signal [20]. Therefore, the MEMS acoustic vector sensors can be used in AGMs to track the PIG [21; 22]. However, the amplitude of the friction sound is small, which results in a limited propagation distance and a weak anti-interference ability. The amplitude of the collision sound is large, but because of the obstruction of the soil, the specific position of the weld cannot be determined directly on the ground, so the PIG cannot be accurately tracked either [21]. There is also another above-ground tracking method based on an extremely-low frequency (ELF) magnetic field, which has better positioning accuracy and stronger general applicability. PIG carries the transmitter, which can radiate the ELF magnetic field, while the tracking receiver above the ground locates PIG by detecting the ELF magnetic field signal [23]. The AGM based on the ELF magnetic signal has been widely used in practical engineering [24]. It is also suitable for tracking the PIG with unconventional shapes [25]. The most common way to analyze the ELF-AGM is the magnetic dipole model (MDM) method [26]. [PERSON] established a MDM for the PIG transmitter, and proposed the pipeline global position system to realize long-distance tracking of the pipeline based on the principle of multi-satellite measurement [27].