# Three-Dimensional Inversion of the Time-Lapse Resistivity Method on the MPI Parallel Algorithm
## 3 Synthetic Example Tests
### Analysis of the Inversion Results  
We performed both separate inversion and time-lapse inversion on the synthesized data. The final RMS values of single inversion and time-lapse inversion are 1.77 and 1.76, respectively. Compared with that of single inversion, the RMS of time-lapse inversion is slightly smaller because the RMS reflects only the degree of data fitting. In this study, we added high noise to the data, so the RMS value cannot accurately invert the true fitting degree of the true resistivity model. Figures 6 and 7 show the sections at Z = 8 m and Y = 0 m for the separate inversion and time-lapse inversion results, respectively. The black solid line in the figures represents the true location of the anomalous bodies. The analysis of the results shown in Figures 6 and 7 clearly reveals that the time-lapse inversion results are more similar in shape and resistivity values for anomalous bodies B\({}_{1}\) and B\({}_{2}\) at all three times than the separate inversion results are. The results of the separate inversion and time-lapse inversion at the three times points were compared. At T\({}_{1}\), the time-lapse inversion better approximates the shape and resistivity of anomalous body B\({}_{2}\) than the separate inversion does. At T\({}_{2}\), there is no significant change in the shape or resistivity of anomalous bodies B\({}_{1}\) and B\({}_{2}\) between the time-lapse and separate inversions. At T\({}_{3}\), the separate inversion results are heavily influenced by noise, distorting the shapes of B\({}_{1}\) and B\({}_{2}\) and introducing false anomalies around the anomalous bodies. In contrast, the time-lapse inversion significantly reduces these distortions and eliminates most of the false anomalies. To further assess the constraint effect of the time-lapse function, we computed the difference between adjacent time points for both the separate inversion results and the time-lapse inversion results, as shown in Figures 8 and 9. The black solid line represents the model from the previous time step, and the area between the black solid line and the dotted line highlights the model changes relative to the previous time. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Type of Program**} & **Number of** & **Running Time** & \multirow{2}{*}{**Acceleration Ratio**} \\  & **Processes** & **/Minutes** & & \\ \hline Serial program & 1 & 487.26 & 1 \\ \hline  & 5 & 124.3 & 3.92 \\  & 10 & 64.62 & 7.54 \\ MPI parallel & 20 & 36.77 & 12.98 \\ program & 30 & 32.46 & 15.04 \\  & 40 & 31.32 & 15.56 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of the running time of the serial program and the MPI parallel program. Figure 6: The sections at \(Z=8\) m from separate inversion and time-lapse inversion of synthetic data. The separate inversion results are shown in the first column, and the time-lapse inversion results are shown in the second column; the top to bottom panels of (**a-c**) are schematic diagrams at T\({}_{1}\), T\({}_{2}\), and T\({}_{3}\), respectively. Figure 7: The sections at Y = 0 m from separate inversion and time-lapse inversion of synthetic data. The separate inversion results are shown in the first column, and the time-lapse inversion results are shown in the second column; the top to bottom panels of (**a-c**) are schematic diagrams at T\({}_{1}\), T\({}_{2}\), and T\({}_{3}\), respectively. As shown in Figures 8 and 9, the resistivity difference between adjacent times decreases after time-lapse inversion. At the \(Z\) = 8 m section, the resistivity of the T\({}_{2}\)-T\({}_{1}\) model significantly decreases in the real change area after separate inversion, and the reduction range is larger than that in the real change area. The resistivity of the T\({}_{3}\)-T\({}_{2}\) model changes in a disordered manner, which is inconsistent with the real change area and is caused mainly by the influence of high noise at T\({}_{3}\).