# Three-Dimensional Inversion of the Time-Lapse Resistivity Method on the MPI Parallel Algorithm
## 3 Synthetic Example Tests
### Analysis of the Inversion Results
After time-lapse inversion, the variation ranges of the resistivities of the T\({}_{2}\)-T\({}_{1}\) and T\({}_{3}\)-T\({}_{2}\) models are on the right side of the real variation area, and the variation value is smaller, whereas the variation range of the resistivities of the T\({}_{3}\)-T\({}_{2}\) models is more regular. At the Y = 0 m section, the resistivity of T\({}_{2}\)-T\({}_{1}\) and T\({}_{3}\)-T\({}_{2}\) decreases around B\({}_{2}\) after separate inversion, and the variation range is larger than the real  
Figure 8: The sections of resistivity difference with adjacent time at \(Z\) = 8 m in the results of separate inversion and time-lapse inversion of synthetic data. The top panels (**a**) are schematic diagrams of T\({}_{2}\)-T\({}_{1}\), and the bottom panels (**b**) are schematic diagrams of T\({}_{3}\)-T\({}_{2}\). Figure 9: The sections of resistivity difference with adjacent time at Y = 0 m in the results of separate inversion and time-lapse inversion of synthetic data. The top panels (**a**) are schematic diagrams of T\({}_{2}\)-T\({}_{1}\), and the bottom panels (**b**) are schematic diagrams of T\({}_{3}\)-T\({}_{2}\). variation region. After time-lapse inversion, the variation range of the resistivity is smaller, and it is concentrated in the upper right of the real variation region. To quantitatively evaluate the effect of time-lapse inversion, the model fitting difference \(\delta\) is introduced to calculate the difference between the real model and the inversion result. The calculation of model fitting difference \(\delta\) follows the calculation method of [PERSON] [19]. \[\delta=\frac{1}{M_{p}}\sqrt{\sum_{i=1}^{M_{p}}\left(\frac{m_{i}^{inv}-m_{i}^{ true}}{m_{i}^{true}}\right)^{2}}, \tag{16}\]  
where \(m_{i}^{true}\) represents the true model, \(m_{i}^{inv}\) represents the result of the separate method or joint inversion, and \(M_{p}\) represents the total number of grid cells underground. Table 4 shows the \(\delta\) value statistics of the results of single inversion and time-lapse inversion. We can use (16) to calculate \(\delta\) at three times. Compared with those of the single inversion, the \(\delta\) value at T\({}_{1}\) decreases, the \(\delta\) value at T\({}_{2}\) increases, and the \(\delta\) value at T\({}_{3}\) decreases after time-lapse inversion. This is because the resistivity values at the three times in the time-lapse model have not recovered to the true value; with increasing B\({}_{2}\) scale, the resistivity value decreases, and the time-lapse inversion results in closer values at those three times. At T\({}_{1}\), the resistivity of B\({}_{2}\) decreases and is closer to the true value, resulting in a decrease in the \(\delta\) value at T\({}_{1}\). At T\({}_{2}\), the \(\delta\) value increases because of the influence of high-noise data at T\({}_{3}\). The influence of noise at T\({}_{3}\) is suppressed by T\({}_{2}\), and the \(\delta\) value also decreases. In summary, the inversion results of time-lapse inversion at adjacent times are more similar in value form and can still reflect the change in underground resistivity under high noise interference.