## 4 Synthetic Examples Tests
### Example I: A Single Prism Model  
The true model in example I is a prismatic body with a length, width and height of 10 m, 10 m and 7 m, respectively, and the buried depth of the top surface is 5 m. The resistivity of the abnormal body is \(\rho\)1 = 10 \(\Omega\cdot\)m, and the polarizability is \(\eta\)1 = 0.2. The  
Figure 1: Flow chart of three-dimensional joint inversion of the resistivity method and TDIP. abnormal body is buried in the uniform half space with resistivity \(\rho\)2 = 100 \(\Omega\cdot\)m and polarizability \(\eta\)2 = 0.01. Figure 2a,b show slices of the model at X = 0 m and Z = 8 m. The observation system is shown in Figure 2c: the electrodes are arranged in a rectangular square matrix. When each transmitting electrode transmits, all adjacent receiving electrodes in the X direction receive at the same time to ensure that enough information can be obtained in three-dimensional space. The model response was obtained through forward calculation, and a total of 22,819 data points were chosen as the observed data. A Gaussian random error of 5% was added to the observed data, which was used to test the algorithms of separate inversion and joint inversion. A uniform half-space with a resistivity of 100 \(\Omega\cdot\)m and a polarizability of 0.01 was designed as the initial model for inversion. In the inversion, the regularization factor was set to a constant. To obtain a better result, we used multiple sets of different values of \(\lambda_{\sigma}\), \(\lambda_{\eta}\), \(\beta_{\sigma}\) and \(\beta_{\eta}\) for testing. Ultimately, it was found that the inversion results are best when the values of \(\Phi_{m}\) and \(\Phi_{cg}\) are about one-tenth of \(\Phi_{d}\). In this case, the value range of \(\lambda\) is 0.1-10, and the value range of \(\beta\) is 10-1000. Larger \(\lambda\) and \(\beta\) will result in the _RMS_ not converging, and smaller \(\lambda\) and \(\beta\) will result in \(\Phi_{m}\) and \(\Phi_{cg}\) being unable to play the role of constrained inversion. The final inversion results are shown in Figure 3.  
The inversion results were analyzed with horizontal and vertical slicing diagrams of the resistivity and polarizability models. As shown in Figure 3a,c, the separate inversion results differed from the true model in terms of convergence degree and boundary recovery. Compared with the single inversion, the results of joint inversion are closer to the true model and converge better at the boundary (Figure 3b,d). Joint inversion overcomes some shortcomings of separate inversion, and improves the accuracy of inversion to a certain extent. The _RMS_ values of the resistivity method and TDIP obtained in separate inversion were 0.724 and 0.849, respectively, and that in joint inversion were 0.722 and 0.802, respectively. The variation in the _RMS_ value of the resistivity method is not obvious, while that of TDIP had a small decrease, which indicates that the TDIP results of joint inversion are better than separate inversion. Figure 2: Sections of the true model in synthetic example I. (**a**) The vertical slice diagram of the model with X = 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. Since there was little difference in \(RMS\) values of the resistivity method, to quantitatively evaluate the inversion results, we evaluated the model misfit between the true model and the joint inversion model. The calculation of model perturbation \(RMS\) refers to the method proposed by [PERSON] [13], which is defined as follows:  
\[\zeta=\frac{1}{M_{p}}\sqrt{\sum_{i=1}^{M_{p}}\left(\frac{m_{i}^{inv}-m_{i}^{true }}{m_{i}^{true}}\right)^{2}}, \tag{18}\]  
where \(m_{i}^{true}\) is the true model, \(m_{i}^{inv}\) is the result of the separate method or joint inversion, and \(M_{p}\) is the total number of grid cells where the prism is located.