# A Stable and High-Precision Downward Continuation Method of Magnetic Data
## 4 Application to Measured Magnetic Data  
To further investigate the effectiveness of the constrained strategy, an example from the South China Sea Area was presented. Specifically, the data acquisition ship was the \"Ocean No. 6\" multifunctional geological and geophysical survey ship affiliated with the Guangzhou Marine Geological Survey, equipped with a deep-water gravity and magnetic exploration system with underwater automatic depth determination and attitude control functions. This system was used to measure the magnetic anomalies of eight survey lines corresponding to each other on the sea surface and underwater. The measured magnetic data of each survey line are shown in Figure 9, and the corresponding depth of four underwater survey lines are shown in Figure 10.  
Figure 8: Results of different downward continuation methods along with the survey line1 (**a**), line 2 (**b**) and with line 3 (**c**). In each panel, the blue line shows the theoretical value of the downward continuation, the black line is the result of the wave number domain iteration method, the green line represents the result of the equivalent source method, and the red line is for the result of the equivalent source constraint method. The surface and underwater measured data were interpolated into a regular grid, as shown in Figure 11. The radially averaged logarithmic power spectrum of sea surface measured data is shown in Figure 12, divided into four segments for piecewise fitting. Equation (20) was used to estimate the depth of each frequency band, and in this paper, the deepest depth of 2706.8 m (the black line) was set as the equivalent layer depth, where \(p(r_{1})=46.13\), \(r_{2}=0.00021\), and \(p(r_{2})=39.02\). Eventually, the lower space of this depth was divided into 400 \(\times\) 400 cube cells, and each cell size was 1 \(\times\) 1 \(\times\) 5 m.  
According to Equation (14), the regularization parameter \(\lambda\) is obtained by the UPRE function, and the relationship of \(\lambda\) and value of Equation (14) is the blue line in Figure 13; furthermore, the red circle is the minimum value of UPRE, and therefore we chose the corresponding value of \(\lambda\) to solve Equation (15). Then, the equivalent layer forward modeling results were used to fit the measured magnetic anomalies on the sea surface to obtain the equivalent source parameters, and finally the water surface magnetic (Figure 11a) anomaly was determined to be downward continuation by the wave number domain iteration method, the unconstrained equivalent source method, and the constrained equivalent source method, respectively. The constrained data were the four underwater survey lines data, and as Figure 11b shows, the result of downward continuation amplitude was set up to be consistent with the range of the measured value and then was compared with the underwater measured data, as Figure 14 shows. Figure 10: The depth of survey lines. Figure 9: Sea surface and underwater measure magnetic data along with survey lines. Figure 11: Measured magnetic anomaly (**a**) at sea surface and (**b**) at the depth of 2000 m under the sea surface. Figure 12: The radially-averaged logarithm power spectrum of the input data (blue line) and its fitting by piece-wise lines. Figure 13: The regularization parameter \(\lambda\) solved by the UPRE function using the CG method. The magnetic anomaly values corresponding to the positions of the four underwater survey lines in the continuation results of the three methods were extracted and then compared with the actual measured values, as shown in Figure 15. Intuitively, the abnormal trend of the wave number domain iteration method (the black line) was similar to the measured value (the blue line); however, there were amplitude oscillations due to the noise.