# Enhancing Precision in Magnetic Map Interpolation for Regions with Sparse Data
## 3 Simulation
### Simulation of Magnetic Field Using Two Magnetic Dipoles  
The first test: two spherical magnetic models were used to simulate the local magnetic anomaly region. The parameters are shown in Table 1. The normal field of the magnetic background was set to 18,000 nT, and the simulated magnetic map with 50 \(\times\) 50 m regional resolution was generated as shown in Figure 3a.  
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 X direction were taken as the measured value on the constructed magnetic map, and the noise of 0.1 nT was added. The heading resolution was set to 250 m, and the flight line spacing was set to 1000 m. The schematic diagram of the measurement points is shown in Figure 3b.  
The relevant parameters obtained by the method proposed in this paper are shown in Table 2. At the same time, the magnetic map was constructed with different variogram models, as shown in Figure 4. From the results, the improved variogram method proposed in this paper has a better effect than others and greatly retains the magnetic field distribution characteristics. From the error distribution, the error of the improved variogram is distributed in the edge region due to the little information near there.  
At the same time, in order to verify the effectiveness of the algorithm, we use the maximum absolute error (ME), the mean absolute error (MAE), and the root-mean-square error (RMSE) to evaluate the accuracy. ME responses are the magnitude of the error, and MAE and RMSE responses are the distribution of the error. Their formulae are as follows:  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**No.** & \multicolumn{2}{c}{**Position**} & \multicolumn{2}{c}{**Radius of the Sphere/m**} \\  & **X/m** & **Y/m** & **Z/m** & \\ \hline
1 & 100 & 3000 & \(-\)4000 & 500 \\
2 & 7500 & 7500 & 4000 & 500 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of two spherical magnetic models.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters of Model** & \(C\) & \(a\) & \(\lambda\) \\ \hline Improved model & 4883.74 & 3305.24 & 0.5425 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameters of the improved variogram model.  
Figure 3: Diagrams of the simulated magnetic field with two magnetic dipoles and simulated aeromagnetic measurement points. (**a**) The simulated magnetic field with two magnetic dipoles; (**b**) The simulated aeromagnetic measurement points.  
\[\begin{split}\text{ME}=\max_{i\in(1,\text{m})}|z_{i}-\hat{z}_{i}|\\ \text{MAE}=\frac{1}{m}\sum\limits_{i=1}^{m}|z_{i}-\hat{z}_{i}|\\ \text{RMSE}=\sqrt{\frac{1}{m}\sum\limits_{i=1}^{m}\left(z_{i}- \hat{z}_{i}\right)^{2}}\end{split} \tag{15}\]  
where the parameter \(m\) is the number of points in the Kriging interpolation, the parameter \(z_{i}\) is the true value of the magnetic field, and the parameter \(\hat{z}_{i}\) is the magnetic value after interpolation.  
From the comparison (as shown in Table 3), 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 better than the others.