# The Generalized Skin Depth for Polarized Porous Media Based on the Cole-Cole Model
## 5 Skin Depth Imaging of Polarized Porous Media
### Depth Interpretation of a Three-Dimensional Chargeable Body  
We took basalt as an example, and a three-dimensional chargeable body model was set up, as shown in Figure 18. A graphite ore of dimensions \(200\times 200\times 200\) m was buried inside the basalt. The burial depth was h = 200 m below the ground. The polarization parameters of basalt were given by Reference [23] as follows: \(\sigma_{\infty}=0.01\) S/m, c = 0.82, \(\eta=0.28\), and \(\tau=0.00005\) s. The polarization parameters of graphite ore were given by Reference [22] as follows: \(\sigma_{\infty}=0.01\) S/m, c = 0.82, \(\eta=0.28\), and \(\tau=0.00005\) s. To show that the skin depth for polarized porous media is universal, we added 50 Hz power frequency noise to the response and extracted the polarization parameters in the case of power frequency noise, as shown in Table 3. The traditional skin depth formula and generalized skin depth formula were used to image the graphite ore, as shown in Figure 19.  
As shown in Figure 19, the black dotted line represents the location of the graphite ore in the theoretical model (Figure 18). Due to the noise introduced in the response, the depths of the upper edge and the lower edge of the graphite ore body had large errors. The generalized skin depth formula was used to image the depth of the graphite ore. The depth of the upper edge of the graphite ore body was 175 m, and the depth of the lower edge was approximately 373 m. Compared with the theoretical model, the relative error of the depth of the upper edge was 12.5%, and that of the lower edge was 6.75%. The traditional skin depth formula was also used to image the depth of the graphite ore. The \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**IP Parameters**} & \multicolumn{2}{c}{**Theoretical Model**} & \multicolumn{2}{c}{**Extraction Parameter**} \\ \cline{2-5}  & **Basalt** & **Graphite** & **Basalt** & **Graphite** \\ \hline \(\sigma_{\infty}\) (S/m) & 0.01 & 0.0479 & 0.0135 & 0.0491 \\ c & 0.82 & 0.6 & 0.8155 & 0.57 \\ \(\eta\) & 0.28 & 0.308 & 0.2498 & 0.334 \\ \(\tau\) (s) & 0.00005 & 2 & 0.00005 & 1.93 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Extraction results of the polarization parameters of the 3D polarization model with noise. Figure 18: 3D polarization model. depth of the upper edge of the graphite ore body was 159 m, and the depth of the lower edge was 360 m. Compared with the theoretical model, the relative error of the upper edge was 20.5%, and the relative error of the lower edge was 10%. It can be seen that, with added noise, compared with the traditional skin depth formula, the generalized polarization media skin depth formula had a higher accuracy in the interpretation of the graphite ore depth. In the actual measurement, due to the interference of noise, it is often necessary to filter the measurement data. We used the Butterworth filter to remove noise, and image interpretation of the filtered results is shown in Figure 20.  
As shown in Figure 20, the depth of the graphite ore was imaged by the generalized skin depth formula. The depth of the upper edge of the graphite ore body was 185 m, and the depth of the lower edge was approximately 385 m. Compared with the theoretical model, the relative error of the upper edge depth was 7.5%, and that of the lower edge depth was 3.75%. The traditional skin depth formula was used to image the depth of the graphite ore. The depth of the upper edge of the graphite ore body was 175 m, and the depth of the lower edge was 376 m. Compared with the theoretical model, the relative error of the upper edge was 12.5%, and the relative error of the lower edge was 6%. It can be seen that the electromagnetic wave had a deeper propagation depth in the polarized medium, and the traditional skin depth could not explain the location of graphite ore. Under IP effects, using the generalized skin depth formula to determine the depth of graphite ore was more accurate.