## 4 Lithospheric Magnetic Field Model of the Tibetan Plateau
### Error Analysis  
Considering that the inversion model adopts data sets of different heights, the intensity of the lithospheric magnetic field varies significantly at different height layers. Therefore, errors are fitted at CHAMP and Swarm satellite altitudes separately to analyze the consistency between the R-SCHA model fitting value and the original observation data. The model inversion coefficients in Regions 1 and 2 are used to calculate the R-SCHA model inversion values in two regions and to seamlessly splice the overlapped region. The difference between the model inversion value and the original observation over the Tibetan Plateau and the root mean square error (RMSE) are calculated. Table 2 lists the statistical results of the errors of X, Y, and Z geomagnetic elements.  
Table 2 shows that model fitting error decreases as altitude increases. The model fitting error in the CHAMP satellite altitude is slightly larger than that of Swarm satellite. However, the _RMSE_ of all geomagnetic components are better than 0.3 nT.  
Figure 6 shows the difference values (\(\Delta Z\)) between the R-SCHA and CHAMP satellite at orbital altitude of 330 \(\pm\) 5 km (a) and Swarm satellite at 470 \(\pm\) 5 km (b). Figure 6 shows that the maximum errors are less than 0.8 and 0.6 nT at the height layers, respectively. This comparative analysis indicates that the inversion model is consistent with the original observation. The R-SCHA algorithm can accurately describe the distribution characteristics of the lithospheric magnetic field in the Tibetan Plateau. Figure 6 shows a slight shock in the boundary regions of two independent spherical caps. The shock may be related to the large extension degree of the Mehler basis function. On the edge of the spherical cap, the value of the Mehler basis function increases constantly with extension degree. This trend affects the fitting accuracy of the edge data. The lack of boundary data and uneven distribution also strongly influence the fitting effect.  
\begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Satellite Altitude** & **Item** & \(\Delta X/nT\) & \(\Delta Y/nT\) & \(\Delta Z/nT\) \\ \hline CHAMP & Min & \(-\)1.070 & \(-\)1.419 & \(-\)0.829 \\ (250–340 km) & Max & 0.918 & 1.177 & 0.803 \\  & RMS & **0.235** & **0.285** & **0.196** \\ Swarm & Min & \(-\)1.051 & \(-\)0.882 & \(-\)0.406 \\ (450–510 km) & Max & 0.627 & 0.961 & 0.583 \\  & RMS & **0.156** & **0.185** & **0.120** \\ \hline \hline \end{tabular}
\end{table}
Table 2: Fitting errors of X, Y, and Z components on the height layer of CHAMP and Swarm.  
Figure 5: Flowchart showing the processing procedure for the overlapping area of the adjacent spherical caps.  
To further verify the stability of the model in the radial direction based on the model coefficient, the error between inversion model and EMM2015 (16\({}^{\circ}\)\(\sim\)720\({}^{\circ}\)) at an altitude range of 250-500 km every 25 km is computed based on a 0.5\({}^{\circ}\)\(\times\) 0.5\({}^{\circ}\) grid resolution. Given that the absolute error of the model decreases as height increases, RMSE (in %) is used to show the statistical result in Figure 7b. Figure 7 does not indicate an obvious increase of errors (in %) from 350 km to 450 km, although the data lack constraints at these intermediate altitudes. The R-SCHA model can provide a stable spatial continuation in the radial direction with a reasonable setting of the truncation degree.