# An Improved Aeromagnetic Compensation Method Robust to Geomagnetic Gradient
## 3 Proposed Aeromagnetic Compensation Model  
The total field intensity \(H_{T}\) measured by the optically pumped magnetometer is the superposition of the geomagnetic field \(H_{e}\) and the platform interference field \(H_{d}\), hence the platform interference field \(H_{d}\) can be expressed as follows:  
\[H_{d}=H_{T}-H_{e}. \tag{10}\]  
It is difficult to obtain an accurate platform interference field in practice. A widely used method is to apply the bandpass filter to Equation (10), which can be expressed as follows:  
\[bpf(H_{d})=bpf(H_{T})-bpf(H_{e}). \tag{11}\]  
where \(bpf()\) is a bandpass finite impulse response (FIR) filter operator. The passband of the filter is determined by the frequency characteristics of the aircraft's maneuvers, which are measured by the three-axis fluxgate. The power spectrum of the direction cosines \(u_{1}\), \(u_{2}\), and \(u_{3}\) are estimated, as shown in Figure 4. It can be seen that the maneuvering frequency of the aircraft during the calibration flight is concentrated between 0.15 Hz and 0.5 Hz, which provides a reference for the design of the bandpass filter in data preprocessing. On the other hand, the frequency band of magnetic anomaly detection (MAD) is about 0.04 Hz-0.4 Hz. Therefore, the cut-off frequency of the bandpass filter is set between 0.04 Hz and 0.5 Hz. For the traditional compensation method, the geomagnetic gradient is usually considered to be negligent, and the following equation can be obtained:  
\[bpf(H_{d})=bpf(H_{T})-bpf(H_{e})\approx bpf(H_{T}). \tag{12}\]  
However, for a high-precision aeromagnetic survey, it is critical to consider the influence of the geomagnetic gradient when estimating the compensation coefficients. As suggested by the IGRF model, the geomagnetic field can be expressed by the gradient of a scalar potential, and the potential function can be described as a finite series expansion of spherical harmonic coefficients, as follows:  
\[V(r,\theta,\phi,t)=a\sum_{n=1}^{N}\sum_{m=0}^{n}\left(\frac{a}{r}\right)^{n+1} \left[g_{n}^{m}(t)\cos m\phi+h_{n}^{m}(t)\sin m\phi\right]P_{n}^{m}(\cos\theta), \tag{13}\]  
where \(r\) is the radial distance from the center of the Earth, \(\theta\) and \(\phi\) are the geocentric co-latitude and longitude, respectively, \(t\) represents time, \(\alpha\) = 6371.2 km is the approximate mean radius of the Earth. \({g_{n}}^{m}(t)\) and \({h_{n}}^{m}(t)\) are the Gauss coefficients, \({P_{n}}^{m}(\cos\theta)\) is Schmidt  
Figure 4: The power spectral density of the direction cosines. semi-normalized associated Legendre functions of degree \(n\) and order \(m\). \(N\) is the number of expansion terms of spherical harmonic function. The spherical coordinate system is defined in Figure 5. By deriving the scalar potential of the internal magnetic field, the zonal component \(X\), meridional component \(Y\), and radial component \(Z\) of the geomagnetic field in the spherical coordinate system can be obtained as follows:  
\[X=\frac{1}{r}\frac{\partial V}{\partial\theta},Y=-\frac{1}{r\sin\theta}\frac{ \partial V}{\partial\phi},Z=\frac{\partial V}{\partial r}. \tag{14}\]  
The geomagnetic field \(H_{e}\) can be expressed as follows:  
\[H_{e}=\sqrt{X^{2}+Y^{2}+Z^{2}}. \tag{15}\]  
According to the aforementioned analysis, Equation (15) can be rewritten as a function related to the year, longitude, latitude, and height, as follows:  
\[H_{e}=\psi(year,lat,lon,height), \tag{16}\]  
where \(\psi\) is a model that can calculate the geomagnetic field by using the IGRF coefficients. Based on the magnetic field data from the magnetic survey satellites, observatories, and other magnetic survey programmers around the world, the IGRF coefficients are computed every five years by the IAGA. The 13 th Generation IGRF, the latest version released in December 2019, represents the geomagnetic field from 2020 to 2025. The longitude, latitude, and height can be converted to the parameters \(r\), \(\phi\), and \(\theta\) in Equation (13) by coordinate transformation.