## 2 Forward Modeling Principle
### Large-Scale SAI-Krylov Algorithm
Figure 12: Cross-sectional views of the electric field \(E_{x}\): (**a**) 0.01 ms; (**b**) 0.25 ms; (**c**) 1 ms; (**d**) 10 ms.

Figure 13: Cross-sectional views of the electric field \(\partial B_{z}/\partial t\): (**a**) 0.01 ms; (**b**) 0.25 ms; (**c**) 1 ms; (**d**) 10 ms.

### Large-Scale Sulfide Model

To further validate the effectiveness and feasibility of the proposed method for large-scale forward modeling, as shown in Figure 9.0 Figure 14a presents the model schematic diagram, and Figure 14b shows the tetrahedral mesh division. This model is derived from a sulfide deposit in Labrador, Canada, which has a shape resembling a sprout-like block of sulfide ore. Based on related drilling data, the length of the ore body in the \(x\)-direction is approximately 400 m, its width in the \(y\)-direction is about 300 m, and its thickness in the \(z\)-direction is about 115 m. In this study, a grounded wire is used as the transmitter with a length of 500 m and a current of 1 A. Both the transmitter and the receiver are located on the surface (terrain effects are not considered), and the receiving area extends from \(-500\) m to 500 m in both the \(x\)- and \(y\)-directions. The uniform half-space resistivity is set to 100 \(\Omega\)-m, and the resistivity of the sulfide ore body is set to 0.01 \(\Omega\)-m. The minimum grid size is 10 m, with the total number of unknowns reaching 8.7 million and a computation time of 17,875 s. On the other hand, the memory usage of this method is only 7.6 GB. Figure 15 shows the contour maps of the horizontal electric field component in the \(xoy\) plane for different transmitter positions. Figure 15a-d correspond to transmitter coordinates at (\(-250\) m, \(-250\) m, 0 m) and (\(-250\) m, 250 m, 0 m); Figure 15e-h correspond to transmitter coordinates at (\(-250\) m, \(-250\) m, 0 m) and (\(250\) m, \(-250\) m, 0 m); Figure 15i-l correspond to transmitter coordinates at (\(250\) m, \(-250\) m, 0 m) and (\(250\) m, \(250\) m, 0 m); and Figure 15m-p correspond to transmitter coordinates at (\(-250\) m, \(250\) m, 0 m) and (\(250\) m, \(250\) m, 0 m). Each transmitter's anomaly contour includes four time steps: 0.01 ms, 0.25 ms, 1.6 ms, and 10.0 ms.

From Figure 15, it can be seen that at \(t\) = 0.01 ms, the electric field is mainly concentrated in the shallow strata and spreads in an elliptical shape from the transmitter to the surrounding area. At \(t\) = 0.25 ms, due to the influence of the low-resistivity body, the electric field amplitude is relatively higher on both sides of the anomaly. As time progresses, the contour lines of the horizontal electric field component gradually align with the shape of the anomaly, and the electric field strength decreases. The white dotted line in the Figure 15 is the contour of the anomalous body in the \(xoy\) plane. Figure 9.1 Three-dimensional large-scale sulfide model: (**a**) model schematic diagram; (**b**) grid division diagram. Figure 16 shows the contour maps of the induced electromotive force (\(z\)-component) in the _xoy_ plane for different transmitter positions, with the transmitter coordinates and observation times held constant. From Figure 16, it can be observed that for transmitters at different positions, when the current is turned off at \(t=0.01\) ms, the electromagnetic anomaly is approximately symmetrically distributed on both sides of the transmitter. At \(t=0.25\) ms, due to the influence of the low-resistivity body, the contour lines of the induced electromotive force show significant distortion at the center of the anomaly. At \(t=1.6\) ms, the contour lines for different transmitter positions align relatively well with the outer boundary of the anomaly, although the degree of alignment varies. As time progresses further, the contour lines of the induced electromotive force gradually concentrate within the anomaly body, and the amplitude decreases. The white dotted line in the Figure 16 is the contour of the anomalous body in the _xoy_ plane.