# An Improved Aeromagnetic Compensation Method Robust to Geomagnetic Gradient
## 2 Principle of Aeromagnetic Compensation  
### Tolles-Lawson Model  
To compensate for the platform-generated interference field, a three-axis fluxgate magnetometer is used to measure the attitude of the platform according to the classical Tolles-Lawson (T-L) model [12; 13]. The reference coordinate system attached to the aircraft is established, as shown in Figure 2, where the origin \(O\) represents the location of the magnetometer, and the \(T\), \(L\), and \(V\) are the transverse, longitudinal, and vertical axis, respectively. \(H_{e}\) represents the geomagnetic field, and \(X\), \(Y\), and \(Z\) are the angles between the geomagnetic field and the three axes, respectively. Figure 1: Map of the global geomagnetic total field intensity at the World Geodetic System-1984 Coordinate System (WGS84) ellipsoid surface for epoch 2020. White asterisks indicate locations of the magnetic dip poles. The direction cosines of the angles between the geomagnetic field and the platform are denoted as \(u_{1}\), \(u_{2}\), and \(u_{3}\), and can be expressed as follows [21]:  
\[ \begin{array}{l}u_{1}=\cos X(t)=\frac{T(t)}{\sqrt{T(t)^{2}+(t)^{2}+V(t)^{2}}}\\ u_{2}=\cos Y(t)=\frac{L(t)}{\sqrt{T(t)^{2}+L(t)^{2}+V(t)^{2}}}\\ u_{3}=\cos Z(t)=\frac{V(t)}{\sqrt{T(t)^{2}+L(t)^{2}+V(t)^{2}}}\end{array} , \tag{1}\]  
According to the T-L model, the platform-generated interference field can be grouped into three types, namely the permanent, the induced, and the eddy current magnetic fields. Define \(H_{d}(t)\) as the total intensity of the interference field, then \(H_{d}(t)\) can be expressed as follows:  
\[H_{d}(t)=H_{PERM}(t)+H_{IND}(t)+H_{EDDY}(t). \tag{2}\]  
The permanent magnetic field \(H_{PERM}(t)\) caused by the permanent magnetism of the aircraft's ferromagnetic parts can be expressed as follows:  
\[H_{PERM}(t)=c_{1}u_{1}+c_{2}u_{2}+c_{3}u_{3}=\sum\limits_{i=1}^{3}A_{i}(t)c_{ i}, \tag{3}\]  
where \(c_{i}\) (\(i\) = 1, 2, 3) is the compensation coefficient of the permanent magnetic field, and \(A_{i}(t)\) (\(i\) = 1, 2, 3) is the variable constituted by the direction cosine. The induced magnetic field \(H_{IND}(t)\) is created in paramagnetic parts by the geomagnetic field, and it can be expressed as follows:  
\[ \begin{array}{l}H_{IND}(t)=H_{e}(t)(c_{4}{u_{1}}^{2}+c_{5}u_{1}u_{2}+c_{6}u_ {1}u_{3}+c_{7}{u_{2}}^{2}+c_{8}u_{2}u_{3}+c_{9}{u_{3}}^{2})\\ =\sum\limits_{i=4}^{9}A_{i}(t)c_{i}\end{array} , \tag{4}\]  
where \(c_{i}\) (\(i\) = 4,..., 9) is the compensation coefficient of the induced magnetic field, and \(A_{i}(t)\) (\(i\) = 4,..., 9) is the variable constituted by the direction cosine and \(H_{e}(t)\). The eddy current, generated by the maneuvers of the metal components of the aircraft, is proportional to the time change rate of the flux through these components. The eddy current interference field \(H_{EDDY}(t)\) can be expressed as follows:  
\[ \begin{array}{l}H_{EDDY}(t)=H_{e}(t)(c_{10}{u_{1}}{u_{1}}^{\prime}+c_{11}{u_ {1}}{u_{2}}^{\prime}+c_{12}{u_{1}}{u_{3}}^{\prime}+c_{13}{u_{2}}{u_{1}}^{ \prime}\\ +c_{14}{u_{2}}{u_{2}}^{\prime}+c_{15}{u_{2}}{u_{3}}^{\prime}+c_{16}{u_{3}}{u_{ 1}}^{\prime}+c_{17}{u_{3}}{u_{2}}^{\prime}+c_{18}{u_{3}}{u_{3}}^{\prime})= \sum\limits_{i=4}^{9}A_{i}(t)c_{i}\end{array} , \tag{5}\]  
where \({u_{1}}^{\prime}\), \({u_{2}}^{\prime}\), and \({u_{3}}^{\prime}\) are the time derivatives of \(u_{1}\), \(u_{2}\), and \(u_{3}\), respectively, \(c_{i}\) (\(i\) = 10,..., 18) is the compensation coefficient of the eddy current magnetic field, and \(A_{i}(t)\) (\(i\) = 10,..., 18) is the variable constituted by the direction cosine and \(H_{e}(t)\). Figure 2: The definition of the reference coordinate system.