## Data processing Flow Chart  
Figure 4: Magnetic data processing flow chart. The left branch shows the iterative process for computing correction coefficients using the data of rotation tracks, after the synchronization of the time delay. The convergence check ensures the reliability of the calibration coefficients. The right branch illustrates the subsequent processing steps after the matrix computation. Since the USV used in this study is constructed with aluminum alloy, a paramagnetic material that exhibits negligible intensities of VM, we approached the problem by eliminating this term from the equation. \[H_{\text{obs}}=A\cdot R\cdot H_{\text{env}}+H_{p} \tag{2}\]  
This equation directly expresses how the hull's magnetic characteristics and attitude changes affect the measurements. The equation proposed in [25] is a transformed version of Equation (2):  
\[B\cdot H_{\text{obs}}+H_{\text{pb}}=R\cdot H_{\text{env}} \tag{3}\]  
where B is A\({}^{-1}\), and H\({}_{\text{pb}}\) represents \(-A^{-1}\cdot H_{p}\). Although these equations are mathematically equivalent, they differ in the process of optimizing the least squares parameters. The formulation proposed by [26] is identical to that of [24] but includes an additional P matrix to correct for the non-orthogonality between the sensor and the platform. Furthermore, it presents a method for correction by calculating only the principal components (xx, yy, zz) of the susceptibility tensor A and the permanent magnetization component \(H_{p}\). \[H_{\text{obs}}=A\cdot P\cdot R\cdot H_{\text{env}}+H_{p} \tag{4}\]  
In this study, we separated the USV's 360-degree rotation intervals (Figure 1b) and analyzed data from each rotation section individually to correct the vessel's magnetization using four different methods. The magnetometer records magnetic field data in a ship-fixed coordinate system, whereas the Earth's magnetic field is expressed in the Earth coordinate system. When the USV rotates 360 degrees, the yaw angle varies from 0 to 360 degrees, while roll and pitch angles fluctuate within a relatively narrow range. Although this rotation pattern predominantly affects the x and y components of the surrounding geomagnetic field, strategic maneuvering of the USV can induce sufficient variations in the z-axis measurements. We can maximize the changes in the z-component measurements from the correction maneuver, enabling comprehensive correction of all three magnetic components through the least squares method. Since the magnetic data is not a square matrix, an inverse matrix does not exist. Therefore, we computed A and H\({}_{\text{p}}\) from Equation (2) [24], and B and H\({}_{\text{pb}}\) from Equation (3) [25] using the pseudo-inverse via Singular Value Decomposition (SVD) for the least squares method. For Equation (4) [26], we first calculated the non-orthogonality matrix, and then computed the principal components of the A matrix and the H\({}_{\text{p}}\) array using the pseudo-inverse via Singular Value Decomposition (SVD) for the least squares method. The correction coefficients derived from each method are summarized in Table 1.  
Each matrix A and B is a 3 \(\times\) 3 matrix representing the sum of the susceptibility tensor of the vessel and Earth's magnetic field vector components, where xx, yy, and zz components of the tensor should exhibit greater than 1 since the vessel has positive susceptibility. However, the two matrices A and B exhibit some discrepancies in each component. To assess the corrected results from the two correction equations, we used the Normalized Root Mean Square Error (NRMSE) to quantitatively compare the corrected magnetic field data with the International Geomagnetic Reference Field (IGRF) model by the following equation. \[NRMSE=\sqrt{\Sigma\bigg{(}\frac{H_{\text{corr}}-H_{\text{env}}}{H_{\text{env }}}\bigg{)}^{2}/N} \tag{5}\]where \(H_{corr}\) represents the corrected magnetic field vector component, \(H_{env}\) denotes the corresponding component of the IGRF model, and N is the total number of data points.