## 2 Materials and Methods
### SSIP Equivalent Inversion Method  
Forward modeling serves as the foundation for inversion in geophysics, where finite element method (FEM) simulations are conducted prior to inversion. Originally developed in elasticity mechanics, FEM is particularly suitable for geophysical problems involving complex property distributions. Its standardized solution process and other advantages have led to widespread application in geophysical studies. In the limiting cases of low frequency \(f_{L}\to 0\) and high frequency \(f_{H}\rightarrow\infty\), it can be concluded that the limiting chargeability equals the limiting polarizability. Since chargeability reflects the intensity of induced polarization effects and is equivalent to polarizability and metal factor in anomaly detection, this study treats spread-spectrum IP data as DC IP data for inversion. Based on [PERSON]'s induced polarization theory, the geoelectric model is jointly described using resistivity and polarizability parameters, with polarizability defined within the range [0,1) and assumed to vary significantly less than resistivity. For the inversion of resistivity and polarizability data, resistivity inversion is typically completed first, followed by polarizability inversion with the resistivity model fixed. First, the objective function for resistivity inversion is constructed under the least squares criterion as follows:  
\[\Phi_{\rho}(\mathbf{m}_{\rho})=\big{\|}\mathbf{W}_{\rho}\big{[}\mathbf{d}_{\rho a}-\mathbf{d}_{ \rho c}(\mathbf{m}_{\rho})\big{]}\big{\|}_{2}^{2}+\lambda_{\rho a}\big{\|}\mathbf{S}\bm {m}_{\rho}\big{\|}_{2}^{2}+\lambda_{\rho b}\big{\|}\mathbf{B}\big{[}\mathbf{m}_{\rho}- \mathbf{m}_{\rho b}\big{]}\big{\|}_{2}^{2} \tag{3}\]  
where the first term on the right side is the data misfit term and \(\mathbf{W}_{\rho}\) is the data weighting matrix used to suppress interference noise in the data. \(\mathbf{d}_{\rho a}=ln(\rho_{a})\), where \(\rho_{a}\) is the measured apparent resistivity, and \(\mathbf{d}_{\rho c}(m_{\rho})=ln(\rho_{c})\), where \(\rho_{c}\) is the modeled apparent resistivity. The model parameters satisfy \(\mathbf{m}_{\rho}=ln(\rho)\), and \(\rho\) represents the subsurface resistivity distribution. The second term on the right side represents the model smoothness constraint term, which suppresses abrupt variations between model parameters. Here, \(\lambda_{\rho s}\) denotes the smoothness constraint factor for model parameters, playing a crucial role in inversion resolution and stability, while \(\mathbf{S}\) represents the smoothness constraint matrix. The third term on the right-hand side constitutes the model background constraint term, which serves to drive the inversion model toward a homogeneous model. Here, \(\lambda_{\rho b}\) represents the background or known property constraint factor for model parameters, while \(\mathbf{B}\) denotes the background or known property constraint matrix (a diagonal matrix). Differentiating both sides of Equation (3) with respect to the model parameters and setting the derivatives equal to zero yields the following generalized linear inversion equation for resistivity:  
\[(\mathbf{J}_{\rho}^{T}\mathbf{W}_{\rho}{}^{T}\mathbf{W}_{\rho}\mathbf{I}_{\rho}+\lambda_{\rho s }\mathbf{S}^{T}\mathbf{S}+\lambda_{\rho b}\mathbf{B}^{T}\mathbf{B})\Delta\mathbf{m}_{\rho}=\mathbf{ J}^{T}\mathbf{W}_{\rho}{}^{T}\Delta\mathbf{d}_{\rho}-\lambda_{\rho s}\mathbf{S}^{T}\mathbf{S}\mathbf{m}_{ \rho}+\lambda_{\rho b}\mathbf{B}^{T}\mathbf{B}(\mathbf{m}_{\rho b}-\mathbf{m}_{\rho}) \tag{4}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Measurement Array Length & 1200 m \\ Current Electrode Span & 5200 m \\ Minimum Electrode Spacing (Current) & 80 m \\ Minimum Electrode Spacing (Potential) & 40 m \\ Array Shift Distance & 400 m \\ Ground Resistance & \textless{}2000 \(\Omega\) \\ Power Supply Current & \textless{}10 A \\ Transmission Frequency & 0.0625 Hz to 1 Hz \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Key configuration parameters of the SSIP survey array.