## 3 Numerical Simulations
### Complex Model Testing  
In this section, the convergence of the PSO-GD algorithm is further examined based on a relatively complex model (see Figure 7), which has a background electrical resistance of 50 \(\Omega\cdot\)m and contains three anomalies: one shallow high-resistance (250 \(\Omega\cdot\)m) anomaly, one deep high-resistance (250 \(\Omega\cdot\)m) anomaly, and one deep low-resistance (10 \(\Omega\cdot\)m) anomaly. The shallow anomaly has a dimension of 600 m \(\times\) 600 m \(\times\) 250 m, with its top surface located at a depth of 20 m. Each of the two deep anomalies has a dimension of 1200 m \(\times\) 800 m \(\times\) 500 m, with its top surface located at a depth of 500 m. Gaussian random noise is added at a 20% level to the forward simulation data to generate an inversion-fitting objective. On this basis, the GD, PSO, and PSO-GD algorithms are respectively used to invert the data. A uniform half space with a resistance of 60 \(\Omega\cdot\)m is used as the initial model for the GD algorithm. Parameter \(\alpha\) in the PSO and PSO-GD algorithms is set to 0.2-0.9. The HPSO-TVAC algorithm is used to update the PSO model.  
Figure 5: Horizontal sectional views of the inversion results at different noise levels (z = 500 m).  
Figure 6: Convergence of the residual errors of the fits at different noise levels.  
The inversion result yielded by the GD algorithm in Figure 8 can be used to satisfactorily reconstruct the shallow high-resistance structure, but fails to reflect the deep high- and low-resistance anomalies. The inversion result produced by the PSO algorithm in Figure 9 accurately shows the location of each deep anomaly. However, the boundary information for the anomalies requires further correction, and the model is not adequately smooth. In comparison, the inversion result given by the PSO-GD algorithm is smoother and closer to the original model (Figure 10). In addition, note that due to the addition of noise and the relativity of global convergence, some false anomalous points appear in the deep area or at the edges where the constraint on the data decreases in the three inversion results. Figure 11 shows the fitting error curves and uncertainty analysis corresponding to the three algorithms. Compared to the conventional PSO algorithm, the PSO-GD algorithm has a markedly faster convergence rate at the late stage and ultimately yields a smaller fitting error. In an uncertainty analysis, the new algorithm has a higher capacity for model recovery and a more stable effect. This numerical test demonstrates the reliability of the new algorithm in the inversion of SATEM data and lays a foundation for its real-world exploration applications.  
Figure 8: Inversion result yielded by the GD algorithm. (**a**) Horizontal sectional view at z = 150 m. (**b**) Horizontal sectional view at z = 750 m. (**c**) Vertical sectional view at x = 0 m.  
Figure 7: Schematic diagrams of the transceiver and the locations of the anomalies.  
Figure 10: Inversion result yielded by the PSO-GD algorithm. (**a**) Horizontal sectional view at z = 150 m. (**b**) Horizontal sectional view at z = 750 m. (**c**) Vertical sectional view at x = 0 m.  
Figure 9: Inversion result yielded by the PSO algorithm. (**a**) Horizontal sectional view at z = 150 m. (**b**) Horizontal sectional view at z = 750 m. (**c**) Vertical sectional view at x = 0 m.