## 3 Verification of Theoretical Analysis and Numerical Simulation
### Comparison of Homogeneous and Inhomogeneous Whole-Space Medium  
Based on the above analytical and numerical results, the response curve presents different attenuation laws in different media. Taking the induced voltage of the receiver coil as an example, the numerical simulation results of homogeneous and inhomogeneous whole-space media under the three-dimensional condition can be obtained, as shown in Figure 11. As for the homogeneous whole-space medium, the transient electromagnetic response results can be calculated by modifying the resistivity of the water body to the resistivity of surrounding rock. By comparing the response curve of the inhomogeneous and homogeneous whole-space medium, there are obvious differences between the two response curves, and it can be quantitatively divided into three stages according to the response time. In Stage I, the induced voltage amplitude in the inhomogeneous whole-space medium is slightly smaller than that in the homogeneous whole-space medium, the difference between the values is not very large, and both of the response curves almost maintain the same attenuation rate. In Stage II, the induced voltage amplitude in the inhomogeneous whole-space medium is larger than that in the homogeneous whole-space medium, and the difference between the values is dynamic. At the same time, the difference between the inhomogeneous and homogeneous whole-space medium increases first and then decreases, which means that the attenuation rate of the response curve in the inhomogeneous whole-space medium illustrates the same trend. The difference level, defined as the logarithm of the value in the inhomogeneous whole-space medium divided by the value in the homogeneous whole-space medium, is greater than zero, and its maximum value can reach up to 2.11. In Stage III, the induced voltage amplitude in the inhomogeneous whole-space medium is almost equal to that in the homogeneous whole-space medium, and both of the response curves keep the same attenuation rate again. Judging from the phenomenon depicted in Figure 11, three key points of the response curve in the inhomogeneous and homogeneous whole-space medium can be obtained. The first point is the uplift point, and its time is approximately 0.4 us, which is at the end of Stage I and the beginning of Stage II. The uplift point exhibits that the amplitude of the response curve in the inhomogeneous whole-space medium is about to exceed the amplitude of the response curve in the homogeneous whole-space medium, and it also indicates that the diffusion of the induced eddy begins to transform from the surrounding rock to the water body with low resistivity. The second point is the peak point, and its time is approximately 16 us, which is somewhere in the middle of Stage II. The peak point demonstrates that the amplitude of the response curve in the inhomogeneous whole-space medium reaches the maximum relative to the amplitude of the response curve in the homogeneous whole-space medium at the same time, which means that the induced eddy  
Figure 11: Comparison of homogeneous and inhomogeneous whole-space medium. is mainly concentrated on the water body with low resistivity and its amplitude reaches the relative maximum. The third point is the regression point, and its time is approximately 80 us, which is at the end of Stage II and the beginning of Stage III. The regression point shows that the amplitude of the response curve in the inhomogeneous whole-space medium is about to tend the amplitude of the response curve in the homogeneous whole-space medium, which illustrates that the diffusion of the induced eddy begins to transform from the water body with low resistivity to the surrounding rock. To summarize, the diffusion law of the secondary magnetic field in the homogeneous half-space medium is investigated through two different methods, including theoretical analysis and numerical simulation. Furthermore, the rationality and accuracy of the numerical simulation are verified.