## 2 Forward Modeling Principle
### Large-Scale SAI-Krylov Algorithm
In this study, the off-time is segmented into three periods: \([10^{-5},10^{-4}]\)s, \([10^{-4},10^{-3}]\)s, and \([10^{-3},10^{-2}]\)s. To more accurately simulate the later electromagnetic signals, the subspace order must be increased according to the principles of the Krylov subspace algorithm and the propagation behavior of the TEM diffusion field. In other words, the Krylov subspace method requires only a minimal subspace order to simulate the early electromagnetic signals. Figure 4 shows the relative error for different Krylov subspace orders \(m\). Figure 4a presents the relative error curve for the Krylov subspace order in the time interval \([10^{-5},10^{-4}]\)s. The accuracy of the calculation results does not meet the criteria when \(m=50\), as shown in Figure 4a, with the maximum relative error exceeding 10%. However, as the Krylov subspace order is increased further, the relative error drops below 5%, and the relative error shows little change with further increases in \(m\). Figure 3: The matrix condition number of different \(\sigma_{air}\). Figure 4b presents the relative error curve for the Krylov subspace order in the time interval \([10^{-4},10^{-3}]\)s. When \(m=120\), as shown in Figure 4b, the precision of the computation results does not meet the criteria, with the maximum relative error exceeding 15%. Figure 4c shows the relative error curve for the Krylov subspace order in the time interval \([10^{-3},10^{-2}]\)s. The features of Figure 4c are similar to those of Figure 4a,b and therefore will not be discussed in detail. The order of the subspace in different time intervals can be determined in this paper using the aforementioned analysis, as shown in Table 1.

### Optimal Tolerance \(\epsilon\) and Relaxation Factor \(\omega\) in Different Time Intervals

According to TEM propagation theory, the early diffusion field contains more high-frequency signals, while the later diffusion field is dominated by low-frequency components. As a result, the early time step is smaller, and the later time step is larger when the backward Euler method is used to solve the electromagnetic response. In constructing the orthogonal basis of the Krylov subspace, a time-step selection strategy is also employed in this study. The orthogonal basis of the Krylov subspace, as discussed earlier, is a dense matrix with \(m\) columns, and \(\mathbf{V}_{m1}\), \(\mathbf{V}_{m2}\), and \(\mathbf{V}_{m3}\) significantly influence the early, middle, and late forward responses, respectively. Consequently, different tolerances \(\epsilon\) can be applied when solving for \(\mathbf{V}_{m1}\), \(\mathbf{V}_{m2}\), and \(\mathbf{V}_{m3}\) to optimize computational efficiency. Figure 5 shows the relative error for different tolerance values \(\epsilon\). Figure 5a illustrates that when the time interval is \([10^{-5},10^{-4}]\)s and \(\epsilon=10^{-21}\), the relative error is relatively large, with the minimum relative error exceeding 5%. Figure 5b,c exhibit similar characteristics to Figure 5a. In the time interval \([10^{-4},10^{-3}]\)s, \(\epsilon=10^{-18}\) satisfies the numerical accuracy requirements. Similarly, \(\epsilon=10^{-16}\) meets the numerical accuracy criteria in the time interval \([10^{-3},10^{-2}]\)s.

To summarize, the tolerances \(\epsilon\) discussed in this paper are \(\epsilon=10^{-21}\), \(\epsilon=10^{-18}\), and \(\epsilon=10^{-16}\) for the three separate time intervals. Following this, the paper will examine the effect of varying the relaxation factor across different time intervals on both the accuracy and computational time. Table 2 shows the TEM results with different relaxation factors \(\omega\). The relaxation factor \(\omega\) has little impact on the accuracy of the TEM response, but has a noticeable effect on the convergence speed. When the relaxation factor \(\omega=1\), the convergence speed is the fastest across all three time periods. Figure 6 shows the relationship between the forward modeling computation time and the relaxation factor for different time intervals.