## 3 Filtering Method
### Principle of Filtering Method  
According to Figure 4, the maximum of an MFL in the x direction of a defect is directly above the defect, while the maximum and minimum of the MFL in the z direction is on both sides of it. Two sensors are installed in the front and rear: a sensor which is used to detect the LMF in the z direction is in the front, and the other sensor, which is used to detect the LMF in the x direction, is installed in the rear. When the rear sensor is directly above a defect, its output is the maximum. At this time, the output of the front sensor is negative. The absolute value of the difference between the outputs is larger than that of each output. Without any defect, the lift-off interference in x and z directions are in the same direction. The smaller the lift-off, the greater the output in both x and z directions. In a no-defect area, if one sensor measures the LMF in the x direction and the other sensor measures the LMF in the z direction, the absolute value of the difference between their output signals is smaller than that of each output. Two sensors which detect the x and z directions MFL, respectively, are a pair, and several pairs of sensors are arranged in y the direction, as shown in Figure 2. If the distance between a pair of sensors is reasonable, the smaller the absolute value, the more likely it is that no defect is measured. As described in Section 3.1, there are always several sensors in a row that do not detect the LMF of a defect. It can be considered that the pair of sensors with the smallest absolute value of the difference of their outputs does not detect the LMF of a defect, and their outputs are taken as a reference signal that does not contain the defect signal but only the lift-off interference. The absolute value of the difference between the outputs of the two sensors should be as large as possible to find the reference signal accurately. If a sensor detects the maximum in the x direction and the other sensor detects the minimum in the z direction, the absolute value of the difference between the outputs of the two sensors is the maximum. According to Equation (1), if the lift-off is \(z_{0}\), which remains unchanged, \(H_{x}(x,z_{0})\) is the maximum when x = 0. According to Figure 4, the minimum of \(H_{z}(x,z_{0})\) occurs on the positive half-axis of x. Due to the small defect, the magnetic dipole model can be introduced to simplify Equation (2) [19]. \[H_{z}(x,z)=\frac{\sigma_{\mathrm{ms}}}{4\pi}\ln\frac{\left[\left(x-a\right)^{2 }+z^{2}\right]}{\left[\left(x+a\right)^{2}+z^{2}\right]} \tag{4}\]  
As the sensors are close to each other and their lift-offs are similar, \(z=z_{0}\). The derivative of Equation (4) is taken to x and set as equal to 0. \[H_{z}^{\prime}(x,z_{0})=0 \tag{5}\]  
Equation (5) is then solved, that is, \(H_{z}(x,z_{0})\) is the maximum when \(x=-\sqrt{a^{2}+z_{0}^{2}}\), and \(H_{z}(x,z_{0})\) is the minimum when \(x=\sqrt{a^{2}+z_{0}^{2}}\). Obviously, if the distance between two sensors is \(\sqrt{a^{2}+z_{0}^{2}}\), when the rear sensor detects the maximum in the x direction, the front sensor detects the minimum in the z direction. In actual detection, the spacing, which is denoted as 1 between two sensors, is determined as the minimum width and depth of the defect that should be detected. Figure 5 shows the distribution of LMFs of the defects with different sizes. The distance of 10 is calculated according to the minimum defect. For a large defect, its LMF maximum in the x direction is larger than that of a small defect. When a sensor is direct above a larger defect and its output is the maximum in the x direction, the other sensor with a distance of 10 from this sensor does not measure the minimum in the z direction. Nevertheless, its output is smaller than the minimum of the minimal defect. In other words, the absolute value of the difference between the outputs of a pair of sensors is greater than that of a small defect. Therefore, the spacing of the two sensors set to a small defect is also suitable for detecting large defects.