# Enhancing Precision in Magnetic Map Interpolation for Regions with Sparse Data
## 3 Simulation
### Simulation of Magnetic Field Using Multiple Magnetic Dipoles  
The second test: six spherical magnetic models were used to simulate the local magnetic anomaly region. The parameters are shown in Table 4. The Z-axis was specified as the positive direction, the normal field of the magnetic background was set to 48,000 nT, and the simulated magnetic map with 50 \(\times\) 50 m regional resolution was generated as shown in Figure 5a.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Variogram Models** & **ME (nT)** & **MAE (nT)** & **RMSE (nT)** \\ \hline Gaussian Variogram & 40.4315 & 2.9309 & 4.6730 \\ Exponential & 54.5584 & 3.8303 & 7.1077 \\ Variogram & 9.1834 & 1.7590 & 2.7815 \\ Improved Variogram & 4.5483 & 0.6291 & 0.7985 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparison of Kriging interpolation errors using different variogram models for the simulation of two magnetic dipoles.  
Figure 4: Comparison of the interpolation results and error distribution using different variograms: (**a**) Gaussian variogram; (**b**) exponential variogram; (**c**) spherical variogram; (**d**) improved variogram.  
In order to obtain the magnetic samples on the route during the simulation of the aeromagnetic survey, the magnetic data of the simulated flight along the Y direction were taken as the measured value on the constructed magnetic map, the heading resolution was set to 50 m, and the flight line spacing was set to 500 m. The schematic diagram of the measurement points is shown in Figure 5b.  
The improved variogram model was used to fit the simulated data and then compared with the Gaussian, exponential, and spherical variogram. The comparison of the variogram graph is shown in Figure 6, and the calculation results are shown in Table 5. It can be seen from the fitting effect diagram that the improved variogram has the best continuity at the origin, which not only ensures the fitting accuracy when the lag distance is small but also indicates that the fitting effect is better than those of other models. This lays a foundation for accuracy improvement in the following process.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**No.** & \multicolumn{3}{c}{**Position**} & \multicolumn{1}{c}{**Radius of the Sphere/m**} \\  & **X/m** & **Y/m** & **Z/m** & **Radius of the Sphere/m** \\ \hline
1 & 1000 & 7000 & \(-1000\) & 80 \\
2 & 5000 & 4500 & \(-200\) & 63 \\
3 & 4500 & 6000 & \(-200\) & 63 \\
4 & 4000 & 4500 & \(-250\) & 63 \\
5 & 4000 & 5500 & \(-200\) & 63 \\
6 & 5000 & 5000 & \(-200\) & 14 \\ \hline \end{tabular}
\end{table}
Table 4: Parameters of six spherical combined models.  
Figure 5: Diagrams of the simulated magnetic field with six magnetic dipoles and simulated aeromagnetic measurement points: (**a**) The simulated magnetic field with six magnetic dipoles; (**b**) The simulated aeromagnetic measurement points.  
Figure 6: Variogram graph constructed by different variograms.  
The magnetic map was constructed by different variogram models, as shown in Figure 7. According to the interpolation results, the improved variogram method has a better effect than the others and retains the magnetic field distribution characteristics greatly. From the error distribution, the accuracy of the improved variogram is raised by about 20 nT over the other models, and the gross errors are mainly concentrated near the magnetic anomalies.  
From the comparison (as shown in Table 6), it can be seen that the ME, MAE, and RMSE of the improved variogram have decreased, indicating that the algorithm proposed in this paper is optimal compared with the other three algorithms.