# A Stable and High-Precision Downward Continuation Method of Magnetic Data
## 3 Synthetic Model Test  
A synthetic model was established to investigate the feasibility of the proposed method. The results were compared with the wave number domain iterative method and the unconstrained equivalent source downward continuation method. Table 1 shows the model properties. The inducing magnetic field had an inclination of 90\({}^{\circ}\) and a declination of 0\({}^{\circ}\); the observation surface was 150 m \(\times\) 150 m with data points located on a grid with 151 rows \(\times\) 151 columns. The synthetic model consisted of five cuboids with different space sizes, properties, and depths. The distance between survey points and survey lines was 1 m. The model's spatial distribution is shown in Figure 3.  
Figure 4 shows the magnetic data induced by the above model measured at different heights, of which the magnetic data (in Figure 4a) measured at 40 m was used as the initial magnetic field for downward continuation. An accurate downward continuation of this data to 0 m should correspond well to the theoretical magnetic data, as seen in Figure 4b. Further, the magnetic data at 40 m contained Gaussian random noise with standard deviation of 5% of the maximum data amplitude. This noisy data could be applied to test the stability of the downward continuation method. In this test, two other methods, including the wave number domain iterative method and the unconstrained equivalent source downward continuation method, were also performed for comparison. \begin{table}
\begin{tabular}{c c c} \hline \hline
**No.** & **Spatial Location (m)** & **Magnetization (A/m)** \\ \hline Model 1 & x\({}_{1}=0\), x\({}_{2}=40\), y\({}_{1}=-20\), y\({}_{2}=20\), z\({}_{1}=-100\), z\({}_{2}=-60\) & 0.9 \\ Model 2 & x\({}_{1}=-50\), x\({}_{2}=-30\), y\({}_{1}=-80\), y\({}_{2}=80\), z\({}_{1}=-60\), z\({}_{2}=-20\) & 0.6 \\ Model 3 & x\({}_{1}=0\), x\({}_{2}=60\), y\({}_{1}=50\), y\({}_{2}=90\), z\({}_{1}=-60\), z\({}_{2}=-30\) & 0.9 \\ Model 4 & x\({}_{1}=40\), x\({}_{2}=-20\), y\({}_{1}=-75\), y\({}_{2}=-45\), z\({}_{1}=-40\), z\({}_{2}=-20\) & 0.6 \\ Model 5 & x\({}_{1}=-15\), x\({}_{2}=15\), y\({}_{1}=-70\), y\({}_{2}=-40\), z\({}_{1}=-30\), z\({}_{2}=-10\) & 0.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physical properties of synthetic model. Figure 4: (**a**) Theoretical magnetic anomaly at the height of 40 m with random noise added; (**b**) theoretical magnetic anomaly at the height of 0 m.  
Figure 3: Three\(-\)dimensional perspective view of the synthetic model. The equivalent source depth could be estimated by analyzing the radial logarithmic power spectrum (Figure 5). In this model test, the radially averaged logarithmic power spectrum was divided into four segments for piecewise fitting. Each fitting line represents a depth that can be estimated by Equation (20) [24]:  
\[h=\frac{P(r_{1})-P(r_{2})}{4\pi(r_{2}-r_{1})} \tag{20}\]  
where \(r_{1}\) and \(r_{2}\) are the start and end radial frequency of the fitting line, respectively, and \(p(r_{1})\) and \(p(r_{2})\) are the corresponding radial power. We chose the depth represented by the black line that was the deepest; correspondingly, \(r_{1}=0\), \(p(r_{1})=44.21\), \(r_{2}=0.012\) and \(p(r_{2})=33.91\). Then, following Equation (20) the result was 80.61 m. After that, the equivalent source was set at this depth. The lower space at this depth was divided into 200 \(\times\) 200 cube cells, and each cell size was 1 m \(\times\) 1 m \(\times\) 5 m.  
The theoretical magnetic data (Figure 4a) were utilized to invert the susceptibility parameters of the equivalent source, and the measured data from three survey lines, as Figure 4b shows, were used as constraint data in the inversion process. According to Equation (12), the regularization parameter \(\lambda\) is obtained by the UPRE function, and the relationship of \(\lambda\) and value of Equation (12) is the blue line in Figure 6; furthermore, the red circle is the minimum value of UPRE, and therefore we chose the corresponding value (\(\lambda=8309.94\)) to solve Equation (13).