## 2 Forward Modeling Principle
### Large-Scale SAI-Krylov Algorithm
It can be observed that when the relaxation factor is \(\omega=1\), the computation efficiency is the highest. Therefore, based on the previous analysis, the optimal relaxation factor is \(\omega=1\) for all three periods. \begin{table}
\begin{tabular}{c c c c} \hline
**Time Interval** & \([\mathbf{10^{-5}}\)**,\(\mathbf{10^{-4}}\)**]**s** & \([\mathbf{10^{-4}}\)**,\(\mathbf{10^{-3}}\)**]**s** & \([\mathbf{10^{-3}}\)**,\(\mathbf{10^{-2}}\)**]**s** \\ \hline Subspace order \(m\) & \(m_{1}=60\) & \(m_{2}=130\) & \(m_{3}=320\) \\ \hline \end{tabular}
\end{table} 
Table 1: The SAI-Krylov subspace order \(m\) in different time intervals. ### Comparison of Computational Performance

The TEM forward modeling parameters are as follows: air layer conductivity \(\sigma_{air}=10^{-4}\) S/m, the tolerances are \(\epsilon=10^{-21},10^{-18},10^{-16}\) for the three time intervals, and the SAI-Krylov subspace order is \(m=320\). This paper uses the time-segmented method,the SAI-Krylov subspace method [27], and the finite element method with an iterative method to simulate the TEM response, respectively. For spatial discretization, all three methods use the same grid. The backward Euler strategy is employed for the iterative (FEM) method. To simulate the TEM response after turning off the current, 13 different iteration time steps, \(\Delta t^{k}=2^{k-1}\times 10^{-8}\)s, where \(k=1,2,\ldots,13\), are used for the iterative method (FEM). During the computation process, 1300 linear equations need to be solved. For the FEM direct method, when the number of unknowns is small, this method offers remarkably high computational efficiency. However, as the number of grids increases, memory usage grows exponentially. When the number of unknowns reaches 5,275,768, the method can no longer be used on the workstation due to memory limitations. For the iterative method (FEM), the computation time has increased dramatically as the number of unknowns has grown. It is important to note that the time step used in this paper is not optimal. Using a suitable time step could further enhance the technique, but determining the optimal time step has always been a challenging task. The memory usage for the three iterative approaches is nearly the same, but the time-segmented SAI-Krylov subspace method offers the best computational efficiency. The computation time and memory usage are shown in Table 3. These three techniques can yield TEM responses with the same precision (less than 5%), but there are significant differences in calculation time and memory usage. ## 3 Numerical Experiments

The above content introduces the large-scale forward modeling method and its parameter optimization strategy. Next, this paper first validates the accuracy of the proposed method using a layered model and a three-dimensional vertical contact band model. Then, two complex geological models are used to study the propagation characteristics of the electromagnetic field in the _xoz_ and _xoy_ planes, respectively. ### Layer Model

The layered model is shown in Figure 7. It consists of a conductive layer with a burial depth of 100 m, a thickness of 20 m, and a conductivity of 10 \(\Omega\cdot\)m, situated in a half-space with a conductivity of 200 \(\Omega\cdot\)m. A grounded wire is used as the transmitter, with a length of 100 m. The coordinates of the two endpoints are (\(-\)50 m, \(-\)50 m, 0 m) and (\(-\)50 m, \(-\)50 m, 0 m), and the receiver is located at (0 m, 0 m, 0 m). The time range for the forward