## 2 Related Work
### The Compensation Influence of the Fluxgate Magnetometer Noise  
In aeromagnetic compensation, the T-L compensation model is as follows:  
\[H_{I}=H_{IP}+H_{II}+H_{IE} \tag{8}\]  
\[H_{IP}=c_{1}\frac{T}{H_{e}}+c_{2}\frac{L}{H_{e}}+c_{3}\frac{V}{H_{e}} \tag{9}\]  
\[H_{II}=\left(c_{4}\frac{T^{2}}{H_{e}}+c_{5}\frac{L^{2}}{H_{e}}+c_{6}\frac{V^{ 2}}{H_{e}}+c_{7}\frac{TL}{H_{e}}+c_{8}\frac{TV}{H_{e}}+c_{9}\frac{LV}{H_{e}}\right) \tag{10}\]  
\[ \begin{array}{c}H_{IE}=c_{10}T\left(\frac{T}{H_{e}}\right)^{\prime}+c_{11}L \left(\frac{L}{H_{e}}\right)^{\prime}+c_{12}V\left(\frac{V}{H_{e}}\right)^{ \prime}+c_{13}T\left(\frac{L}{H_{e}}\right)^{\prime}+c_{14}T\left(\frac{V}{H_ {e}}\right)^{\prime}+\\ c_{15}L\left(\frac{V}{H_{e}}\right)^{\prime}+c_{16}L\left(\frac{T}{H_{e}} \right)^{\prime}+c_{17}V\left(\frac{T}{H_{e}}\right)^{\prime}+c_{18}V\left( \frac{L}{H_{e}}\right)^{\prime}\end{array}  \tag{11}\]  
\[H_{e}=\sqrt{T^{2}+L^{2}+V^{2}} \tag{12}\]  
where \(H_{IP}\) is permanent magnetic field, \(H_{II}\) is induction field, and \(H_{IE}\) is the eddy current field. \(H_{e}\) is the total magnetic field measured by the fluxgate magnetometer, and \(c_{i}\) is the coefficient of the T-L model. \(T\), \(L\), and \(V\) represent the true value of the magnetic field on the three axes X, Y, and Z of the fluxgate magnetometer. The magnetic interference of the aircraft platform is a quantity \(H_{I}\left(T,L,V\right)\) related to the magnetic field value on the three-axis of the fluxgate. When the magnetic field value on the three-axis deviates from the real value, the deviation of the aircraft interference magnetic field can be expressed as follows:  
\[\Delta H_{I}=H_{I}(T+\Delta T,L+\Delta L,V+\Delta V)-H_{I}(T,L,V). \tag{13}\]  
where \(T\), \(L\), and \(V\) are the true values of the magnetic field on the three axes of the fluxgate magnetometer while \(\Delta T\), \(\Delta L\), and \(\Delta V\) represent the deviations between the measured values and the true values on the three axes of the fluxgate magnetometer, respectively. Taylor expansion of \(H_{I}\) (\(T+\Delta T,L+\Delta L,V+\Delta V\)) yields the following:  
\[H_{I}(T+\Delta T,L+\Delta L,V+\Delta V)=H_{I}(T,L,V)+\frac{\partial H_{I}}{ \partial T}\Delta T+\frac{\partial H_{I}}{\partial L}\Delta L+\frac{\partial H _{I}}{\partial V}\Delta V+o(T,L,V) \tag{14}\]  
\(o(T,L,V)\) is a higher-order term of the Taylor expansion, so the first-order linear Taylor expansion of the magnetic interference is as follows:  
\[\Delta H_{I}=H_{I}(T+\Delta T,L+\Delta L,V+\Delta V)-H_{I}(T,L,V)=\frac{ \partial H_{I}}{\partial T}\Delta T+\frac{\partial H_{I}}{\partial L}\Delta L +\frac{\partial H_{I}}{\partial V}\Delta V. \tag{15}\]  
It can be found that the error of the magnetic interference model is linearly related to the fluxgate measurement errors \(\Delta T\), \(\Delta L\), and \(\Delta V\). The larger the measurement error is, the larger the error of the magnetic interference model of the aircraft platform is. So, the fluxgate magnetometer noise can affect the compensation coefficient solved and reduce the compensation effect. In order to verify the influence of the fluxgate magnetometer noise on the compensation effect, we compare the influence of fluxgate noise with different amplitude on compensation effect through the Hainan experimental flight data in 2019. The sampling rate is 10 Hz. Add gaussian white noise with an amplitude of 1%, 5%, and 10% of stability amplitude of the fluxgate magnetic field to the fluxgate magnetometer data and the OPM data do not add noise. Then, use the fluxgate magnetometer data and OPM data to solve the compensation coefficients so as to compare the influence of the fluxgate magnetometer noise on compensation effect. The compensation coefficients solved by the fluxgate magnetometer data that added different amplitudes of noise are shown in Table 1.