## 2 Related Work
### T-L Model
magnetic bars also have the same effect. So, the induced magnetic field interference can be expressed as  
\[H_{dd}=\frac{H_{e}}{|H_{e}|}\cdot\left(H_{ix}+H_{iy}+H_{iz}\right)= \begin{bmatrix} cos\alpha\\ cos\beta\\ cos\gamma\end{bmatrix} ^{T}\cdot \begin{bmatrix}M_{xx}&M_{xy}&M_{xz}\\ M_{yx}&M_{yy}&M_{yz}\\ M_{zx}&M_{zy}&M_{zz}\end{bmatrix} \cdot|H_{e}| \begin{bmatrix}cos\alpha\\ cos\beta\\ cos\gamma\end{bmatrix}  \tag{1}\]  
\(H_{ix}\), \(H_{iy}\) and \(H_{iz}\), respectively, represent the induced magnetic fields of the three equivalent magnetic bars. \(M_{xx}\), \(M_{xy}\), \(M_{xz}\), respectively, represent the magnetic field action coefficients of the equivalent magnetic bar in the X direction magnetized by the X component of the geomagnetic field. \(M_{yx}\), \(M_{yy}\) and \(M_{yz}\), respectively, represent the magnetic field action coefficients of the equivalent magnetic bar in the Y direction magnetized by the Y component of the geomagnetic field and \(M_{zx}\), \(M_{zy}\) and \(M_{zz}\), respectively, represent the magnetic field action coefficients of the equivalent magnetic bar in the Z direction magnetized by the Z component of the geomagnetic field. (3) Eddy current field: It comes from the eddy current effect generated by the cutting of the geomagnetic field by the conducting materials of the body in the process of movement. According to the properties of the eddy current field, its value is related to the external magnetic field and the changing rate of aircraft attitude. Eddy current field interference can be expressed as  
\[H_{ed}=\frac{\mathbf{H_{e}}}{|\mathbf{H_{e}}|}\cdot\mathbf{H_{ed}}= \begin{bmatrix}cos\alpha \\ cos\beta\\ cos\gamma\end{bmatrix} ^{T}\cdot \begin{bmatrix}b_{11}&b_{12}&b_{13}\\ b_{21}&b_{22}&b_{23}\\ b_{31}&b_{32}&b_{33}\end{bmatrix} \cdot|\mathbf{H_{e}}| \begin{bmatrix}\frac{d\alpha \beta}{dt}\\ \frac{d\alpha\beta}{dt}\\ \frac{d\alpha\beta}{dt}\end{bmatrix}  \tag{2}\]  
where \(b_{ij},i=1,2,3;j=1,2,3\) represent the eddy field action coefficients. So, a compensation model with 18 coefficients is established [22]:  
\[H_{I}=\sum\
olimits_{i=1}^{3}p_{i}c_{i}+H_{f}\sum\
olimits_{i=1}^{3}\sum \
olimits_{j=1}^{3}a_{ij}c_{i}c_{j}+H_{f}\sum\
olimits_{i=1}^{3}\sum\
olimits _{j=1}^{3}b_{ij}\dot{c}_{i}c_{j} \tag{3}\]  
where \(p_{i},a_{ij},b_{ij}\) are the coefficients of the T-L model; \(H_{f}\) is the total magnetic field intensity measured using the fluxgate magnetometer; \(c_{i}\) is the cosine of the angle between the three axes of the aircraft coordinate system, which can be measured from the three axes of the fluxgate magnetometer \(H_{i}\), \(i=1,2,3\); the expression is (4); \(\dot{c}_{i}\) is time derivative of \(c_{i}\):  
\[c_{i}=\frac{H_{i}}{\sqrt{H_{1}^{2}+H_{2}^{2}+H_{3}^{2}}},i=1,2,3 \tag{4}\]  
Then, the measured value of OPM can be expressed as the superposition of the geomagnetic field and aircraft magnetic interference as follows:  
\[H_{M}=H_{I}+H_{E} \tag{5}\]  
where \(H_{E}\) is the geomagnetic field and \(H_{M}\) is the measured value of OPM. Generally, the earth's magnetic field is slowly changing, and its frequency is extremely low. Therefore, the Earth's low-frequency magnetic field and high-frequency interference can be eliminated using a suitable bandpass filter. After bandpass filtering, the magnetic data only contains magnetic anomaly target signals and aircraft magnetic interference. In the detection of aeromagnetic targets, the magnetic anomaly signals generated by the detected targets are usually in the extremely low-frequency band (<1 Hz). In order to match the target recognition, we usually pay attention to the low-frequency magnetic field with the frequency range of 0.04-0.3 Hz. In this paper, the band of the bandpass filter is set to 0.04-0.3 Hz. N data points are obtained using a calibration flight; then, the T-L linear model can be expressed as a matrix as follows:  
\[B=AC+z \tag{6}\]where \(B=[B(1),B(2),\ldots,B(n)]^{T}\) represents n scalar magnetometer field data after preproprocessing. \(\text{A}=