## 3 Simulating MFL Signals Using FEA
### FEA Cases
The first includes \(d_{g}/t\) equal to 20% and \(d_{p}/t\) equal to 30, 40, 50, 60, 70, or 80%. The second group includes \(d_{g}/t\) equal to 30% and \(d_{p}/t\) equal to 40, 50, 55, 60, 70, or 80%. The last group includes \(d_{g}/t\) equal to 40% and \(d_{p}/t\) equal to 50, 60, 70, or 80%. It should be noted that \(d_{p}/t\) is assumed to be at least 10% larger than \(d_{g}/t\) in all three groups. This results in 64 PIC defects in total with varying sizes. In terms of the corrosion location, five values of \(h\) (\(-\)50, \(-\)25, 0, 25, and 50 mm) and five values of \(\phi\) (\(-\)10, \(-\)5, 0, 5, and 10 degrees) are considered. The permutation of 64 defects and 25 locations leads to 1600 cases for each of PIC\({}_{\text{in}}\) and PIC\({}_{\text{ex}}\). It follows that a dataset consisting of 9600 cases (1600 \(\times\) 6 = 9600) is generated in this study. This large dataset is intended to facilitate the training and validation of the CNN model, enabling comprehensive analysis and robust model development. The developed FEA model meshes using 4-node tetrahedral elements, where the minimum size is 4 mm, and the maximum mesh size is 55 mm after a convergence study. Specifically, grids inside the 160 mm \(\times\) 160 mm area are finely resolved at 1 mm intervals. At each grid point, the value of \(B\) is calculated via interpolation between two mesh nodes. The system of nonlinear equations is solved using the Newton-Raphson method. As an illustration, Figure 6 depicts the 3D magnetic flux density information (i.e., \(B_{x}\), \(B_{y}\), and \(B_{z}\)) within the 160 mm \(\times\) 160 mm area of a selected FE model. The proposed FEA model was validated by comparing the simulation results with experimental data from a CMFL tool reported by [PERSON] and [PERSON] [34] and an AMFL tool reported by [PERSON] and [PERSON] [35]. Readers are referred to [PERSON] and [PERSON] [19] for details of the validation. MFL signals in practical applications inevitably include noise stemming from factors such as non-uniform pipe wall thickness and differences in sensor lift-off values [36]. These noises can impact the accuracy of the inverse model, necessitating the inclusion of noises in the FEA-simulated MFL signals. The signal-to-noise (SNR) ratio serves as an important parameter for quantifying the noise in the three-dimensional MFL signal. Equation (3) defines SNR in this study [37]:  
\[SNR=10\text{log}_{10}\left(\frac{\sum_{n=1}^{160}\sum_{m=1}^{160}\left[B_{x}( n,m)^{2}+B_{y}(n,m)^{2}+B_{z}(n,m)^{2}\right]}{\sum_{n=1}^{160}\sum_{m=1}^{160} \left[w_{x}(n,m)^{2}+w_{y}(n,m)^{2}+w_{z}(n,m)^{2}\right]}\right) \tag{3}\]  
Figure 6: FEA-obtained \(B_{x}\), \(B_{y}\), and \(B_{z}\) corresponding to a PIC defect on the internal pipe surface (\(w_{g}=50\) mm, \(l_{g}=50\) mm, \(d_{g}/t=40\%\), \(r_{p}=4\) mm, \(d_{p}/t=80\%\), \(h=0\) mm and \(\phi=0\) degree). where \(w_{x}\), \(w_{y}\) and \(w_{z}\) denote the noises along three directions of the pipe, namely circumferential, radial and longitudinal directions, respectively, and (\(n\), \(m\)) (where \(n\), \(m\) = 1, 2, \(\ldots\), 160) represents the coordinate of each grid (with 1 mm resolution) in the 160 mm \(\times\) 160 mm area, with a total of 25,600 grids. By considering a representative value of SNR equal to 20 [38], three 160 \(\times\) 160 matrices of Gaussian-distributed white noises, which are assumed to be independent of each other, are simulated and then applied to \(B_{x}\), \(B_{y}\), and \(B_{z}\), respectively. Each matrix assumes a mean value of zero, with standard deviations adjusted through trial and error until achieving the predefined signal-to-noise ratio. Let \(\sigma_{n}\) denote the standard deviation of \(w_{x}\); the standard deviations of \(w_{y}\) and \(w_{z}\) are assumed to be 0.75\(\sigma_{n}\) and 0.25\(\sigma_{n}\), respectively.