## 4 Synthetic Examples Tests
### Example III: Double Prisms Model in High Noise Background  
The true model in synthetic example III is composed of two prism bodies with a length, width and height of 6 m, 5 m and 5 m, respectively. The burial depth of the top surface of the two abnormal bodies is 5 m, and the interval is 6 m. The resistivity of the abnormal body is \(\rho\)1 = 10 \(\Omega\cdot\)m, and the polarizability is \(\eta\)1 = 0.2. The abnormal body is buried in the uniform half space with resistivity \(\rho\)2 = 100 \(\Omega\cdot\)m and polarizability \(\eta\)2 = 0.01. Figure 5a,b show slices of the model at X = 0 m and \(Z\) = 8 m, respectively. The observation system used was the same as in example I (Figure 5c). Figure 4: Imaging results from separate inversion and joint inversion of synthetic resistivity method and TDIP data for example II. The separate inversion results are shown in (**a,c**), and the joint inversion results are shown in (**b,d**). The top panels (**a,b**) are the vertical slice diagrams of the resistivity model and polarizability model with X = 0 m, and the bottom panels (**c,d**) are plan sections at Z = 8 m in depth. Cross-gradient maps shown in the third column. Figure 5: Sections of the true model in synthetic example III. (**a**) The vertical slice diagram of the model with Y = 0 m; (**b**) the horizontal slice diagram of the model with Z = 8 m; (**c**) schematic diagram of the observation system: the red circle denotes transmitters, the blue triangles denote seismic receivers, and the blue rectangle denotes the projection of the prism on the surface. A 20% Gaussian random error was added to the observed data derived from the model forward. A uniform half-space with a resistivity of 100 \(\Omega\)-m and a polarizability of 0.01 was designed as the initial model for inversion, and the inversion results are shown in Figure 6.  
As shown in Figure 6a,c, the separate inversion results of resistivity are distorted in the background of high noise, and a large number of false anomalies appear around the abnormal bodies, making it difficult to recover the abnormal body to the true model, and difficult to distinguish the two adjacent abnormal bodies. The separate inversion results of polarizability have a small deviation to the left and right. Compared with separate inversion, joint inversion can suppress the influence of high noise, recover the shape and value of the anomalies better, identify the location of the anomalies effectively, and accurately distinguish the two adjacent anomalous bodies (Figure 6a,b). The _RMS_ values of the resistivity method and TDIP obtained in separate inversion were 1.765 and 1.807, respectively, and that in joint inversion were 1.764 and 1.790, respectively. The variation in _RMS_ values of the resistivity method and TDIP is not obvious, which is the same as example II. The values obtained for separate inversion were \(\zeta_{p}\) = 15.03 and \(\zeta_{\eta}\) = 0.45, while those for joint inversion were \(\zeta_{p}\) = 12.3 and \(\zeta_{\eta}\) = 0.4. The value of \(\overline{t}\) in separate inversion was  
Figure 6: Imaging results from separate inversion and joint inversion of synthetic resistivity method and TDIP data for example III. The separate inversion results are shown in (**a,c**), and the joint inversion results are shown in (**b,d**). The top panels (**a,b**) are the vertical slice diagrams of the resistivity model and polarizability model with X = 0 m, and the bottom panels (**c,d**) are plan sections at Z = 8 m in depth. Cross-gradient maps shown in the third column. \(1.09\times 10^{-4}\), while it was \(0.28\times 10^{-4}\) in joint inversion. The values of \(\zeta\) and \(\overline{t}\) decreased to different degrees in the joint inversion.