# Electromagnetic Field Distribution and Data Characteristics of SUTEM of Multilayer Aquifers
## 3 Forward Modeling Mathematical Method of SUTEM
### Comparative Analysis of One-Dimensional Forward Simulation Accuracy  
To analyze the accuracy of the one-dimensional SUTEM forward program (LMF), we compared it with the surface-borehole TEM program written by Professor [PERSON] [31]. First, the accuracy of the two programs was compared with the analytical solution of a uniform half-space [23]. Figure 5 shows the comparison results of the LMF program and that of [31] for the uniform half-space. The resistivity of the uniform half-space is set to 100 \(\Omega\)-m, the rectangular transmitting loop is set to 500 m \(\times\) 500 m, and the equivalent circular transmitting loop radius is \(a\ =\ \sqrt{250000/\pi}\). This radius is received at the center point of the transmitting coil. Figure 5 shows the \(\partial B_{z}/\partial t\) signal changing with time. Our result and that of [31] basically coincide with the analytical data. Figure 5 shows the relative error calculated against the analytical solution. In the early data stage, the relative errors of the two numerical calculations are relatively large, mainly because the analytical solution considers a circular emission loop, while the numerical solutions solve for rectangular emission loops (and the early secondary field is greatly affected by the shape of the emission loop). In the late stage, the relative error curves tend to zero. Because the algorithm of [31] uses the G-S transform, its accuracy during the late-stage calculation is poor, and the relative error of our program is smaller.  
Next, we establish the H and K geoelectric models, with model parameters as shown in Table 1. The size of the transmitting loop is 500 m \(\times\) 500 m, and the center point of the transmitting coil is the coordinate origin. Because we mainly consider SUTEM detection research, the receiving point is located underground, and two receiving points are established for comparison at the coordinates (0, 0, 0) and (0,0, \(-\)200).  
Figure 6a,b show the attenuation data and relative error data of the receiving point located \(-\)200 m underground. It can be seen that the attenuation curves are basically the same, only the relative error increases in the late stage. From analyzing the comparison results and considering the analytical solution in Figure 5, it can be seen that the program of [31] shows relatively large errors in the late data stage, which indicates that the subsurface calculation results of the LMF program meet the accuracy requirements.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**H**} & \multicolumn{3}{c}{**K**} \\ \hline Resistivity (\(\Omega\)-m) & 100 & 10 & 100 & 10 & 100 & 10 \\ \hline Thickness (m) & 100 & 50 & / & 100 & 50 & / \\ \hline \hline \end{tabular}
\end{table}
Table 1: The parameters of the H and K models.  
Figure 5: Comparison results of 1D simulation programs and analytical solution. Here, (**a**) shows the decay curves and (**b**) shows the relative error curves.  
Through the above comparison, we demonstrate that the accuracy of the LMF program we have compiled meets the simulation requirements and can simulate the electromagnetic field at any position of the transmitting coil in space.