## 2 Forward Modeling Principle
### Large-Scale SAI-Krylov Algorithm
In this study, a relative error of 5% is used to determine the tolerance \(\epsilon\) for the entire interval. As shown

Figure 1: The influence of initial value and tolerance on the computational results: (**a**) relative error corresponding to different initial values; (**b**) relative error corresponding to different tolerances. in Figure 1b, when the tolerance \(\epsilon\) is less than \(10^{-20}\), the simulated TEM response does not meet the precision criteria. The relative error decreases and stabilizes as the tolerance decreases. When the tolerance is \(10^{-20}\), the relative error curve nearly satisfies the criteria, although some relative errors still exceed 5% at certain times. To properly simulate the TEM response, the tolerance is set to \(10^{-21}\), resulting in a minimum relative error of 3.1%, a maximum relative error of 4.8%, and an average relative error of 4.3%. In summary, this paper sets the tolerance to \(\epsilon=10^{-21}\) and the initial iteration value to 0. Furthermore, this study investigates the optimal tolerance and subspace order for different time periods in the following sections. ### The Effect of Air Conductivity on the Algorithm

The conductivity of air cannot be considered infinitesimal when using numerical methods to simulate the TEM response, and a smaller value must be used to approximate the conductivity of air. Therefore, this section examines the effect of air conductivity on the computational process. Figure 2a shows the relative error curves for different values of \(\sigma_{air}\). As observed in Figure 2a, when \(\sigma_{air}=10^{-3}\,\mathrm{S/m}\), the relative error is relatively high, with the minimum relative error exceeding 20%. When \(\sigma_{air}=10^{-4}\,\mathrm{S/m}\), the relative error is significantly reduced, with the maximum relative error being less than 5%. Although the relative error decreases as air conductivity decreases, the precision of the TEM data does not improve substantially. Figure 2b, on the other hand, shows the calculation times for different values of \(\sigma_{air}\). As the air conductivity decreases, the computation time grows exponentially. The calculation times are 182 s, 359 s, 927 s, 2371 s, and 4873 s. Considering both numerical accuracy and computational efficiency, the air conductivity in this study is set to \(10^{-4}\,\mathrm{S/m}\). Subsequently, this paper briefly analyzes why \(\sigma_{air}\) has a significant effect on the calculation time. It is observed that the coefficient matrix is primarily influenced by \(\mu_{0}\), \(\sigma\), \(\gamma\), and the grid size factor. As discussed earlier, \(\mu_{0}\) and \(\gamma\) are constants with values of \(4\pi\times 10^{-7}\) and \(2.6\times 10^{-7}\), respectively, indicating that the impact of \(\sigma_{air}\) on the coefficient matrix is crucial. Figure 3 shows the matrix condition number for different values of \(\sigma_{air}\). It can be observed that when \(\sigma_{air}\) is reduced by an order of magnitude, the condition number of the coefficient matrix decreases correspondingly. Comparing Figures 2b and 3, it is evident that air conductivity affects the condition number of the coefficient matrix, which in turn adversely impacts the computation time. As shown in Figure 3, when \(\sigma_{air}\) is reduced by an order of magnitude, the condition number of the coefficient matrix increases by an order of magnitude as well. Therefore, comparing Figures 2b and 3 highlights that \(\sigma_{air}\)

Figure 2: The influence of air conductivity on the computational results: (**a**) relative error curves of different \(\sigma_{air}\); (**b**) calculation time of different \(\sigma_{air}\). significantly influences the condition number of the coefficient matrix, which greatly affects the computation time. ### The Strategy of Time Division

Based on the above analysis, we set \(\sigma_{air}=10^{-4}\) S/m, \(\epsilon=10^{-21}\).