## 3 Synthetic Test
### Inversion Results for Time-Lapse  
#### 3.3.1 Inversion Parameters  
Theoretically, when RMS = 0, the resistivity model derived from inversion aligns with the underground resistivity distribution. However, it is not possible to achieve an inversion result with RMS = 0 due to the presence of noise. Figure 6 illustrates a smooth decrease in RMS with an increasing number of iterations, along with a smooth descent curve. This indicates the stable convergence of the inversion process and reflects the stability of the algorithm. Analyzing the RMS reduction curves of Test 1 and Test 2 in Figure 6 reveals that the incorporation of the time-lapse regularization factor \(\beta\) decelerates the rate of RMS reduction in data fitting. This phenomenon can be attributed to the diminished influence of the data term in the inversion process, a consequence of integrating the time-lapse term. In the synthetic data test, the deviation value \(\alpha\) remains constant. However, in Test 3, certain measurement points are impacted by high noise, leading to a higher initial RMS value compared to Test 1, as depicted in Equation (11). Test 3 underscores the challenge of fitting high-noise data, resulting in a more gradual reduction in RMS. Based on the selected plotting results in Figure 6, profiles at z = 200 m and x = 0 are separately depicted in Figure 7. In Figure 7, the red box corresponds to the range of high-resistivity anomaly blocks at each time step, while the blue box corresponds to the range of low-resistivity anomaly blocks. The results indicate that the 3D time-lapse electric field inversion algorithm for CSAMT has, to varying degrees, recovered the positions and resistivity values of the designated high and low resistivity anomaly blocks. Figure 7a displays the inversion results of Test 1 with \(\beta=0\), where the time-lapse term does not impact the inversion results. Figure 7b illustrates the inversion results of Test 2 with a time lapse factor of 0.1. When comparing the horizontal slice results in the two figures, it is observed that the high resistivity volume range within the red box is closer to the theoretical position, and the false anomaly range around the blue box is significantly reduced. \(\beta\) is a regularization factor that theoretically spans from 0 to \(\infty\). However, if the value is excessively large, it prevents the inversion from converging, whereas if it is too small, it fails to provide a constraining effect. Upon examining Figure 6, it is evident from the RMS descent curves of Test 1 and Test 2 that the inclusion of a time-lapse term hampers the convergence rate of the inversion process. Despite Test 1 having a lower RMS value compared to Test 2, the results of Test 2 align more closely with the theoretical model, suggesting that the time-lapse term facilitates the inversion process. Figure 6: RMS reduction curve with iteration count. We evaluated the algorithm's effectiveness in mitigating inverse crime [29] by varying the grid size and number and performing the time-lapse electric field inversion in Test 2. The inversion results are depicted in Figure 8a. The algorithm demonstrates stability and does not exhibit any prominent issues of inverse crime. Figure 6 depicts the RMS decline curve of the inversion crime test, represented by the green curve. Figure 7: Synthetic data testing results for the time-lapse electric field inversion. (**a**) The red and blue boxes represent the theoretical locations; \(\text{Nx}=60\), \(\text{Ny}=60\), \(\text{Nz}=42\). (**b**) The red and blue boxes represent the theoretical locations; \(\text{Nx}=60\), \(\text{Ny}=60\), \(\text{Nz}=42\). In Figure 8b, the data yielded in Test 3 were highly noisy and polluted, and the time-lapse inversion accurately recovered the locations of high and low resistivity anomaly bodies. Moreover, in Test 3, the time-lapse data at three different noise levels, 1t, 2t, and 3t, were iteratively inverted simultaneously, mitigating the challenge of comparing independently inverted data at each time step due to varying fitting degrees.