## 3 Theoretical Model Testing  
To verify the predictive capability of the NGO-LSSVM algorithm for three-dimensional models, we designed experiments involving conductive targets embedded in a homogenous half-space. Model I is shown in Figure 8, the central point coordinate of the conductive target is (3000, 5000, 500) m, and the length of the conductive body is 400 m \(\times\) 400 m \(\times\) 200 m. The resistivity of the earth and the conductive target are set to 100 \(\Omega\cdot\)m and 10 \(\Omega\cdot\)m, respectively. To verify the imaging effect of the magnetic gradient in the \(x\)-direction and \(y\)-direction, respectively, two perpendicular survey lines are set above the conductive target, as the dotted lines shown in Figure 8.  
The apparent resistivity and apparent resistivity variation results are obtained using the traditional iteration-matching method and NGO-LSSVM algorithm respectively, as shown in Figure 9. The white box indicates the outline of the conductive abnormal body. In Figure 9,c, a high resistance pole is observed at the left boundary of the object, while a low resistance pole exists at the right boundary. The changing range of the apparent resistivity shows a significant spread. In Figure 9,d, both the apparent resistivity variation in the \(x\)-direction \(\Delta\rho^{\Delta x}\) and \(y\)-direction \(\Delta\rho^{\Delta y}\) obtain the pole at the center of the abnormal body, which can indicate the location of the abnormal body. Compared to the apparent resistivity, the apparent resistivity variations exhibit a more compact changing area. Figure 8: Schematic diagram of the model I with a conductive object: (**a**) XOY plane, (**b**) YOZ plane. Figure 7: Error of apparent resistivity and apparent resistivity variation. (**a**) Relative error of the calculated apparent resistivity, (**b**) error of predicted apparent resistivity variation. To verify the universality of the proposed method, model II was established with two anomalous objects positioned along the \(x\)-axis, as shown in Figure 10. The central point coordinates of the two conductive targets are (2200, 5000, 500) m and (3000, 5000, 500) m, respectively. An \(x\)-direction survey line is set above the conductive targets, as the dotted line shown in Figure 10. The imaging results for model II are shown in Figure 11. Figure 11a exhibits the apparent resistivity calculated from the magnetic field \(H_{z}\) by the traditional bisection iteration method. For both anomalous bodies, the apparent resistivity on the left side is high and the apparent resistivity on the right side is low. Different from the single anomalous body model in Figure 8, there exists an apparent resistivity transition region between the highest and lowest values of apparent resistivity, precisely located in the middle of the two anomalous bodies. The transition region of apparent resistivity makes a gradual change in the distribution of apparent resistivity from near source to far source, exhibiting a monotonic decrease. The center of both anomalous bodies exhibits an apparent resistivity interface, indicating a significant change in apparent resistivity. Figure 11b exhibits the apparent resistivity variation predicted from magnetic field gradient \(H_{z}^{\Delta x}\) by the NGO-LSSVM model. The center of the two anomalous bodies exhibits a maximum value of apparent resistivity variation, indicating a clear localization of the anomalous bodies. The model III was established with two anomalous objects positioned along the \(y\)-axis, as shown in Figure 12. The central point coordinates of the two conductive targets are (3000, 4200, 500) m and (3000, 5000, 500) m, respectively. A \(y\)-direction survey line is set above the conductive targets, as the dotted line shown in Figure 12. The imaging results for model III are shown in Figure 13. Similar to the imaging pattern in Figure 11, the apparent resistivity in Figure 13 has the change interface positioned at the center of the two anomalous bodies.