## 4 Experimental Results
Moreover, as shown in Figure 16b,e,h, the Y-component, \(\Delta\)V\({}_{1,\mathrm{Y}}\), does not differ significantly along the center line of the crack, indicating that the VDM model is valid. Figure 15: Distribution of 3-axial TMR signals. (**a**) X-, (**b**) Y-, (**c**) Z-directional component on the 0\({}^{\circ}\) specimen; (**d**) X-, (**e**) Y-, (**f**) Z-directional component on the 30\({}^{\circ}\) specimen; and (**g**) X-, (**h**) Y-, (**i**) Z-directional component on the 60\({}^{\circ}\) specimen. To investigate the quantitative trend of MFD based on the angle direction to the depth of cracks, Figures 17-19 show the MFD distribution along a line passing through the center of the specimen (y = 5.1 mm) in Figures 15 and 16. In each graph, \(\overset{\rightarrow}{\Delta V}_{2}(x,y)\) is the value expressed by Equation (49). \(\Delta\overset{\rightarrow}{V}_{1,\text{ avr}}(x,y)\) is the average value expressed by Equation (50). This variable was introduced to observe the correlation between the position of x and the crack inclination angle when \(\overset{\rightarrow}{\Delta V}_{2}(x,y)=0\). In addition, M, N, and r are the number of data points in the x-axis direction (M = 5000), number of data points in the y-axis direction (N = 2500), and spatial resolution (r = 0.004 mm), respectively. The thick solid line near the center of the longitudinal axis (x = 9.3-9.5 mm) is the central position of the crack observed under the microscope. The solid vertical lines at x = 1.4 mm and x = 15.9 mm indicate the positions of both corners of the specimen. \[\Delta\overset{\rightarrow}{V}_{2}(x,y)=\Delta\overset{\rightarrow}{V}_{1}(x,y)-\Delta\overset{\rightarrow}{V}_{1,\text{ avr}}(x,y) \tag{49}\]  
\[\Delta\overset{\rightarrow}{V}_{1,\text{avr}}(x,y)=\frac{1}{\text{MN}}\Bigg{(} \sum_{i=1}^{M}\sum_{j=1}^{N}V_{1,X}(ir,jr)\overset{\rightarrow}{a}_{x}+\sum_ {i=1}^{M}\sum_{j=1}^{N}V_{1,Y}(ir,jr)\overset{\rightarrow}{a}_{y}+\sum_{i=1}^ {M}\sum_{j=1}^{N}V_{1,Z}(ir,jr)\overset{\rightarrow}{a}_{Z}\Bigg{)} \tag{50}\]  
Comparing \(V_{\text{raw,x}}(x,5.1)\) in Figure 17a and \(\Delta V_{1,x}(x,5.1)\) in Figure 17b, the background signal of about \(-0.9\) V (approximately \(-0.18\) G) is applied uniformly. As the overall depth angle increased, the peak value also increased. This means that \(\overset{\rightarrow}{B}_{\text{CDM,c}}(x,z)\) in Figure 9 exhibits a more accurate trend than \(\overset{\rightarrow}{B}_{\text{CDM,d}}(x,z)\) in Figure 10, i.e., for the same depth, the area of magnetization wides according to the inclination angle, which means that a larger MFD distribution is obtained as a result. Figure 16: Distribution of 3-axial TMR signals subtracted background data. (**a**) X-, (**b**) Y-, (**c**) Z-directional component on the 0\({}^{\circ}\) specimen; (**d**) X-, (**e**) Y-, (**f**) Z-directional component on the 30\({}^{\circ}\) specimen; and (**g**) X-, (**h**) Y-, (**i**) Z-directional component on the 60\({}^{\circ}\) specimen. In contrast, the position of the peak was deflected to the right (direction with an acute angle) according to the depth angle. Therefore, if the central position of the crack can be specified by penetrant inspection, the crack direction to depth based on the correlation with the peak position can be estimated. However, it is difficult to estimate the crack direction if only the MFD distribution is provided without knowing the exact location of the crack. Furthermore, according to \(\Delta\mathrm{V}_{2,\mathrm{x}}(\mathrm{x},5.1)\) in Figure 17c, the left-right symmetry collapses with respect to the peak according to the depth direction. To understand the left-right symmetry of \(\Delta\mathrm{V}_{2,\mathrm{x}}\), an angle estimation function defined by Equation (51), \(\Delta\mathrm{x}_{2,\mathrm{x}}\), was proposed.