## 1 Introduction  
Wire rope is a slender and vulnerable ferromagnetic component that bears the most load and is related to the safety of life and property. It is widely used in mining, oil, elevators, cableway, transportation, ports, and several other vital fields due to its fatigue resistance and high tensile strength [1]. However, wire rope is susceptible to defects such as broken wire, corrosion, wear, and fatigue during the long-term use process. What is worse, these defects could occur in the internal of the wire rope, which cannot be found using the visual inspection of traditional methods [2, 3]. There are growing appeals for effective wire rope detection. With the continuous improvement of safety awareness, wire rope inspection has been used in many scenes. The practical goal in lots of applications is to realize the safe use of wire rope and avoid wastage caused by prematurely being replaced [4, 5]. At present, most scenes have to take the measure of regular replacement of wire rope, which not only contributes to much pressure on resources but also cannot eliminate the risk of sudden fracture. To realize the real-time performance of wire rope inspection without affecting normal production activities, wire rope inspection must develop from off-line handheld to in-service inspection. In-service detection equipment runs under high load for 24 h [6]. The existing methods of detecting wire rope include ultrasonic [7], visual [8], X-ray, electromagnetic method [9], magnetostriction, and so on. The electromagnetic method is the most commonly used in practical applications, and it contains magnetic flux leakage (MFL) methods [10], main flux methods [11], eddy current [12; 13], alternating current field measurement, and pulsed eddy current [14]. Among these kinds of methods, the MFL method has a significant effect in addition to simple theory and easy implementation in practical applications. The damage can be determined by measuring the MFL signal [15]. Until now, using the MFL method, the existing detection lift-off distance is very small (2 mm-5 mm) [16]. [PERSON] and [PERSON] proposed an equation for determining the crack volume when the lift-off is known using the 1/4 RMS algorithm to detect a far-side crack. Besides, the lift-off distance in their study can reach 11 mm [17]. [PERSON] et al. observed the MFL field of a defect varies drastically with lift-off by simulations, and its lift-off distance reaches 10 mm [18]. [PERSON] et al. introduced the defect quantization errors caused by mechanical vibration and electromagnetic noises to analyze the influence of lift-off values and electromagnetic noises [19]. A lift-off-tolerant MFL sensor is developed and tested by [PERSON] et al. based on the magnetic field focusing effect of ferrite cores. It showed high sensitivity at a lift-off distance of 5.0 mm [20]. [PERSON] and [PERSON] acquired the induced current of the coil sensor and optimum values of parameters in magnetizing and sensing units numerically. Finally, the 4.8 mm lift-off distance of the coil sensor was verified [21]. Combined with the thickness of magnetic sensitive elements, the inner surface of the sensor head is usually directly attached to the detected steel wire rope. As a result, the magnetic flux leakage sensor will scrap off the oil stain on the steel wire rope's surface. Therefore, it is crucial to realize that the magnetic flux leakage sensor can detect wire rope defects at a large lift-off distance. However, with the increasing distance, the field strength of the leakage magnetic field decreases exponentially. Even if the magnetization is enhanced by increasing the number of magnetizers, there is still a problem of weak damage signal overwhelmed by the noise interference at a large lift-off distance, such as strand signal, swing signal, and background noise signal [22]. The frequency of wire rope damage signal is affected by many factors, such as damage type, size, and running speed of wire rope, making signal processing and recognition difficult [23].