## 5 Application  
The three-dimensional joint inversion algorithm was applied to groundwater detection. The Miyun District of Beijing is the area with the most abundant groundwater resources, which plays an important role in the urban water supply system of Beijing. The Quaternary pore water in the region mainly exists of sand and gravel formed by the impact of the Chaobai River, which is a single aquifer. There are many monitoring wells for regularly monitoring groundwater levels and electrical conductivity (EC). The area between monitoring well MY2 (40\({}^{\circ}\)21\({}^{\prime}\)51\({}^{\prime\prime}\), 116\({}^{\circ}\)48\({}^{\prime}\)9\({}^{\prime\prime}\)) and the river was chosen as the study area. Table 2 shows the statistical table of the groundwater level and EC value of MY2 in the past year. As shown in Table 2, the variation in the groundwater table is not regular, and the EC value is nonlinearly correlated with the groundwater table value. The project has two purposes: (1) to determine the groundwater level between the MY2 monitoring well and Chaobai River; (2) to determine whether there are abnormal bodies that cause changes in the EC value of groundwater. To achieve the purposes above, five measuring lines (L1-L5) were arranged in the research area. To display the research results more intuitively, the projection of the midpoint of the measuring line at the riverbank was taken as the origin O. The direction of the river is positive on the X-axis, and the direction on the right bank of the river is positive on the Y-axis. A rectangular coordinate system was established to obtain the distribution of the measuring lines (Figure 7). L1 to L5 correspond to Y = 30 m, 70 m, 100 m, 130 m and 170 m, respectively. The receiving electrodes were spaced at 3 m equal intervals along the black measuring line. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Parameter**} & \multirow{2}{*}{**Example**} & \multicolumn{2}{c}{**Separate Inv**} & \multicolumn{2}{c}{**Joint Inv**} \\ \cline{3-6}  & & \(\rho\) & \(\eta\) & \(\rho\) & \(\eta\) \\ \hline \multirow{3}{*}{\(\zeta\)} & I & 7.59 & 0.64 & 4.02 & 0.47 \\  & II & 17.89 & 0.65 & 3.59 & 0.44 \\  & III & 15.03 & 0.45 & 12.3 & 0.4 \\ \hline \multirow{3}{*}{\(\overline{t}\)} & I & \(5.74\times 10^{-4}\) & \multicolumn{2}{c}{\(2.34\times 10^{-4}\)} \\  & II & \(4.67\times 10^{-4}\) & \multicolumn{2}{c}{\(3.73\times 10^{-4}\)} \\  & III & \(1.09\times 10^{-4}\) & \multicolumn{2}{c}{\(0.28\times 10^{-4}\)} \\ \hline \hline \end{tabular}
\end{table} 
Table 1: \(\zeta\) and \(\overline{t}\) values of separate and joint inversion. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Data** & **October 2021** & **November 2021** & **December 2021** & **January 2022** & **April 2022** & **June 2022** & **August 2022** \\ \hline water level (m) & 17.68 & 16.7 & 19.37 & 16.54 & 16.88 & 17.35 & 17.4 \\ EC (\(\mu\)s/cm) & 866 & 842 & 832 & 657 & 827 & 868 & 765 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Groundwater level and EC values of borehole MY2. The GDD full waveform high power IP equipment was used for data acquisition, which can measure the apparent resistivity and apparent polarizability. The technical specifications of the measuring equipment are as follows: steady current accuracy 0.001; 10 A transmission current; 150-2400 V voltage; resolution 1 \(\upmu\)V; accuracy 0.8%; operating temperature range: \(-45\)\({}^{\circ}\)C to \(-60\)\({}^{\circ}\)C. In the process of data acquisition, we used a three-pole device for measurement. When each transmitting electrode transmits, all receiving electrodes located on the same line receive simultaneously; the power supply cycle is 8 s; each data point is superimposed at least 10 times; 20 windows are used in decay curve reconstruction. After preprocessing the collected data, a total of 24,540 data points were chosen as the measured data, and then separate inversion and joint inversion were carried out.