## 2 Forward Modeling Principle
### Large-Scale SAI-Krylov Algorithm
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Unknowns**} & \multicolumn{2}{c}{**Time-Segmented Method**} & \multicolumn{2}{c}{**Direct Method (FEM)**} & \multicolumn{2}{c}{**Iterative Method (FEM)**} \\ \cline{2-7}  & **Time** & **Memory** & **Times** & **Memory** & **Time** & **Memory** \\ \hline
211,725 & 266 s & 0.4 G & 55 s & 2.4 G & 1808 s & 0.4 G \\ \hline
689,789 & 997 s & 0.8 G & 241 s & 8.1 G & 2996 s & 0.8 G \\ \hline
1,574,640 & 2730 s & 1.5 G & 851 s & 21.6 G & 17,384 s & 1.5 G \\ \hline
3,160,300 & 6065 s & 2.8 G & 2255 s & 51.5 G & 37,166 s & 2.8 G \\ \hline
5,275,768 & 10,815 s & 4.9 G & \(\times\) & \(\times\) & 62,987 s & 4.9 G \\ \hline
8,550,025 & 16,286 s & 7.5 G & \(\times\) & \(\times\) & 104,414 s & 7.5 G \\ \hline
12,642,086 & 27,783 s & 11.2 G & \(\times\) & \(\times\) & 151,442 s & 11.2 G \\ \hline
17,690,940 & 39,679 s & 16.2 G & \(\times\) & \(\times\) & 217,680 s & 16.2 G \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The comparison between the time-segmented SAI-Krylov solver and the other solvers, and “\(\times\)” means beyond the calculation power of the computer. simulation is from 10\({}^{-5}\) s to 10\({}^{-2}\) s, with a total of 31 time gates. The minimum grid size is 10 m, resulting in 238,964 unknowns and a computation time of 352 s.

Figure 8 presents the computational results for this model, where Figure 8a and Figure 8b represent the horizontal component of the electric field and the vertical component of the induced electromotive force, respectively. Figure 8c shows the relative error compared to the one-dimensional results. The relative errors for both the horizontal component of the electric field and the vertical component of the induced electromotive force are less than 5%, validating the effectiveness and accuracy of the numerical simulation method proposed in this paper. ### Three-Dimensional Vertical Contact Model

The three-dimensional vertical contact band model is used to further validate the forward modeling method proposed in this paper, as shown in the schematic diagram in Figure 9. The resistivity of air is set to \(\rho_{air}=10^{4}\)\(\Omega\cdot\)m, and a low-resistivity cover layer of 50 m thickness is placed on the surface, with a resistivity of 10 \(\Omega\cdot\)m. Below the cover layer,

Figure 8: The computational results of the layered model: (**a**) \(E_{x}\) data; (**b**) \(\partial B_{z}/\partial t\) data; (**c**) relative error. Figure 7: Layer model. the resistivity of the strata on the left side is 100 \(\Omega\cdot\)m, and the resistivity of the strata on the right side is 300 \(\Omega\cdot\)m. A step-shaped geological body with a resistivity of 1 \(\Omega\cdot\)m is located in the middle of the strata. This geological body has a length of 400 m along the \(x\)-axis, a width of 100 m along the \(y\)-axis, and a thickness of 500 m along the \(z\)-axis. The accuracy of the method is tested using both a loop source and a grounded wire source. The loop source is placed on the surface, with a side length of 100 m, and its center is located at (0 m, 50 m, 0 m). The four receiver points are positioned along the \(y\)-axis at 50 m, 150 m, 450 m, and 1050 m. The grounded wire source is placed along the \(x\)-axis, with its endpoints at (\(-\)50 m, 0 m, 0 m) and (50 m, 0 m, 0 m), and three receiver points are positioned along the \(y\)-axis at 200 m, 400 m, and 1000 m. The minimum grid size is 5 m, resulting in 1,357,187 unknowns and a computation time of 2171 s.

Figure 10a and Figure 10b show the induced electromotive force (z-component) curves at different receiver points for the loop source and grounded wire source, respectively. In Figure 10a, the electromagnetic signal at the receiver point (0 m, 50 m, 0 m) decays normally. However, for the receiver points (0 m, 150 m, 0 m), (0 m, 450 m, 0 m), and (0 m, 1050 m, 0 m), which are located outside the loop source, the corresponding decay curves exhibit a turning point, with the sign of the electromagnetic field reversing before and after the turning point.