# A Stable and High-Precision Downward Continuation Method of Magnetic Data
## 3 Synthetic Model Test
Finally, we followed Equation (19) to forward calculate the magnetic anomaly at the depth of 40 m under the 0 m observation surface. The equivalent source method and wave number domain iteration method were used for downward continuation of the same distance for comparison. The results for those methods are shown as Figure 7. When the continuation was conducted with large distance, according to Figure 7a, the wave number domain iterative method was disturbed by noise seriously, which led to a huge misfit. The result of the equivalent source method (Figure 7b) had a similar amplitude to the theoretical value, yet the amount of information missed and the abnormal trend were different. The result of the constrained equivalent source method (Figure 7c) was closer to the theoretical value of the anomaly shape and amplitude than others, and also had good noise suppression. The deviation diagrams (Figure 7a',c',b') were obtained by applying theoretical data (Figure 4b) minus the results of different methods, and from that they could be illustrated more intuitively. The deviation of the constrained equivalent source method was minimal, and the high value deviations in other two methods were counterbalanced. Figure 5: The radially averaged logarithm power spectrum of data (blue line) and its fitting by piece-wise lines. The quantitative comparison of each method resulted in differences with the theoretical magnetic data, as Table 2 shows, which demonstrated that the constraint equivalent source method improves the accuracy. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & **Maximum (nT)** & **Minimum (nT)** & \
\begin{tabular}{c} **Mean** \\ **(nT)** \\ \end{tabular} &
\begin{tabular}{c} **RMSE** \\ **(nT)** \\ \end{tabular} \\ \hline Wave number domain iteration & \(357.220\) & \(-273.520\) & \(-1.018\) & \(46.768\) \\ Equivalent source & \(333.511\) & \(-296.890\) & \(0.151\) & \(52.125\) \\ Constrained equivalent source & \(198.991\) & \(-149.11\) & \(0.083\) & \(29.459\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The difference between the results of the downward continuation methods and the theoretical magnetic data. Figure 6: The regularization parameter \(\lambda\) solved by the UPRE function using the CG method. Figure 7: The downward continuation results obtained by (**a**) the wave number domain iteration method, (**b**) the equivalent source downward continuation method, and (**c**) the proposed method, and their differences with the theoretical magnetic anomaly (Figure 4b) are shown in (**a**\({}^{\prime}\)), (**b**\({}^{\prime}\)) and (**c**\({}^{\prime}\)). According to Figure 8, we extracting the data corresponding to the survey lines position (shown as Figure 4b) from the results of the three downward continuation methods, it could be clearly seen that the result of the wavenumber domain iteration method (the black line) deviated greatly from the theoretical value (the blue line) due to the influence of noise. The constrained equivalent source method (the red line) could fit the theoretical value well, and the influence of noise was relatively minimal.