# Magnetic Moment Estimation Algorithm Based on Convolutional Neural Network
## 2 Materials and Methods  
### Magnetic Anomaly Detection  
#### 2.1.1 Principles and Methods  
Magnetic sensors can be classified into scalar magnetic sensors and vector magnetic sensors based on the type of field quantity they measure. A vector magnetic field sensor measures the three components of the total magnetic field, providing directional information [23]. However, vector magnetic field sensors are susceptible to steering errors, making them less suitable for use on moving vehicles. In contrast, scalar magnetic field sensors, which measure the modulus of the total magnetic field, are widely used on flying platforms due to their superior stability. As shown in Figure 1, let \(\mathbf{H}_{T}\), \(\mathbf{H}_{E}\), and \(\mathbf{H}_{r}\) represent the total magnetic field, the geomagnetic field, and the magnetic anomaly signal, and the two angles \(\theta\) and \(\varphi\) represent the angles between different vectors, respectively [24]. Using the law of cosines, we can derive the following equation:  
\[\mathbf{H}_{T}=\mathbf{H}_{E}+\mathbf{H}_{r} \tag{1}\]\[|\mathbf{H}_{T}| =\sqrt{|\mathbf{H}_{E}|^{2}+|\mathbf{H}_{r}|^{2}-2|\mathbf{H}_{E}|| \mathbf{H}_{r}|\cos(\pi-\varphi)}\] \[=|\mathbf{H}_{E}|\sqrt{1+\frac{|\mathbf{H}_{r}|^{2}+2|\mathbf{H}_{ E}||\mathbf{H}_{r}|\cos(\varphi)}{|\mathbf{H}_{E}|^{2}}}. \tag{2}\]  
In aeromagnetic surveys, since the magnitude of the geomagnetic field is significantly higher than that of the magnetic anomaly signal, the last term in Equation (2) can be represented as follows:  
\[\frac{|\mathbf{H}_{r}|^{2}+2|\mathbf{H}_{E}||\mathbf{H}_{r}|\cos(\varphi)}{| \mathbf{H}_{E}|^{2}}\to 0 \tag{3}\]  
In Equation (3), the total magnetic field signal can be represented as follows:  
\[\lim_{x\to 0}(1+x)^{\alpha}\sim 1+\alpha\cdot x \tag{4}\]  
from which  
\[|\mathbf{H}_{T}|\approx|\mathbf{H}_{E}|+|\mathbf{H}_{r}|\cos\varphi+\frac{| \mathbf{H}_{r}|^{2}}{2|\mathbf{H}_{E}|}. \tag{5}\]  
As mentioned earlier, the last term in Equation (5) can be omitted. Therefore, the measured magnetic field signal can be expressed as follows:  
\[|\mathbf{H}_{T}|\approx|\mathbf{H}_{E}|+|\mathbf{H}_{r}|\cos\varphi. \tag{6}\]  
According to Equation (6), the total magnetic field signal can be expressed as the magnitude of the geomagnetic field along with the projection of the magnetic anomaly signal in the direction of the geomagnetic field. The process of generating the magnetic anomaly signal and the principle of aeromagnetic measurement are illustrated in Figure 2.  
Figure 1: Vector diagram of signal relationship. Figure 2: Magnetic survey map: (**a**) schematic of magnetic anomaly detection; (**b**) magnetic anomaly signal. 2.1.2. Geomagnetic Field, Magnetic Moment, and Magnetic Anomaly Signals  
As discussed earlier, the magnetic anomaly field is determined by both the geomagnetic field vector and the target magnetic field. To offer a clearer explanation of the magnetic moment angular characteristics, we will now describe the associated angular parameters. According to Figure 3, the angle formed between the magnetic moment and its projection onto the XY plane is defined as the inclination angle, denoted as \(I_{M}\). Meanwhile, the angle between the projection of the magnetic moment onto the XY plane and the X-axis is defined as the declination angle, denoted as \(D_{M}\). Assume that the direction of the geomagnetic field is parallel to the Y-axis, the target is located at \((0,0,0)\), and the observation range spans from \(-30\) to \(30\) along both axes, with a sampling point interval of \(0.1\). The observation plane is set at \(z=10\), and the directions of the total magnetic moment vector are \((1,0,0)\), \((0,1,0)\), \((0,0,1)\), and \((1,1,1)\). The magnetic field intensity is provided in nanoteslas (nT). We simulated the magnetic anomaly field under varying magnetic moment orientations within the coordinate system, allowing us to examine the relationship between the magnetic moment direction, the geomagnetic field direction, and the resulting magnetic anomaly field.