# Arbitrary Sampling Fourier Transform and Its Applications in Magnetic Field Forward Modeling
## 5 Results
### Experiments
#### 5.3.1 Weak Magnetic Continuous Medium  
The continuous medium model was designed a with background magnetic field of 50,000 nT, a magnetic inclination of 45\({}^{\circ}\), and a magnetic declination of 5.9\({}^{\circ}\). The calculation area was 20 km \(\times\) 20 km \(\times\) 10 km, with a range of \(-\)10-10 km in the x direction, \(-\)10-10 km \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Methods**} & \multicolumn{2}{c}{\(\begin{array}{c}\textbf{N}\textbf{x}\times\textbf{N}\textbf{y}\textbf{ or}\\ \textbf{N}\textbf{k}\textbf{x}\times\textbf{N}\textbf{k}\textbf{y}\end{array}\)} & \multirow{2}{*}{\(\textbf{N}\textbf{ or}\textbf{K}_{e}\)} & \multicolumn{4}{c}{**Rrms (\%)**} & \multirow{2}{*}{**Time (s)**} \\ \cline{5-6}  & & & & \(B_{ax}\) & & \(B_{ay}\) & & \\ \hline \multirow{3}{*}{Gauss-FFT} & \multirow{3}{*}{201 \(\times\) 201} & \(N=2\) & 5.89 & 5.89 & 3.95 & 6.42 \\  & & \(N=3\) & 0.58 & 0.58 & 1.32 & 15.05 \\  & & \(N=4\) & 0.08 & 0.08 & 0.24 & 30.35 \\ \hline \multirow{3}{*}{Standard-FFT} & \multirow{3}{*}{301 \(\times\) 301} & \(K_{e}=2\) & 1.76 & 1.76 & 4.97 & 2.86 \\  & & \(K_{e}=5\) & 0.23 & 0.23 & 0.67 & 11.12 \\  & & \(K_{e}=8\) & 0.07 & 0.07 & 0.23 & 39.51 \\ \hline \multirow{3}{*}{AS-FT} & \(51\times 51\) & / & 1.21 & 1.21 & 1.08 & 0.14 \\  & & \(101\times 101\) & / & 0.09 & 0.09 & 0.23 & 0.60 \\  & & \(201\times 201\) & / & 0.02 & 0.02 & 0.11 & 2.53 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Efficiency and accuracy statistics of three Fourier transform algorithms (Nz = 101). in the y direction, and 0-10 km in the z direction. The horizontal magnetic susceptibility changed continuously, and the relationship between magnetic susceptibility and space is shown in Figure 6. The relationship between magnetic susceptibility of the abnormal body and space coordinates is:  
\[\chi=0.02\times e^{-5\times 10^{-8}\times(x^{2}+y^{2})}\text{SI}. \tag{27}\]  
The \(x\), \(y\), and \(z\) directions are sampled uniformly. There were 101 nodes in each direction. The spectrum distribution of the anomaly field at different depths of the model is shown in Figure 7. By analyzing the spectral distribution of the anomaly field to varying depths of the model and selecting the wav enumer, the efficiency and accuracy of the numerical simulation can be guaranteed. In Figure 7, the horizontal ordinate adopts non-logarithmic and logarithmic coordinates, and the maximum value is the cut-off frequency, which is obtained according to the sampling theorem. It can be seen that the spectral energy decays to 0 at the cut-off frequency. The spectral energy is different at different depths, but the basic trend is the same. To analyze the 3D distribution of the spectrum, the distribution of the magnetic anomaly spectrum at the position of z = 3 km was taken, as shown in Figure 8. It can be seen through the spectrum distribution that the continuous model spectrum is concentrated, the changing trend is gentle, and there is almost no vibration. To reduce the number of wavenumbers, it is not necessary to sample to the cut-off frequency, which can also ensure the algorithm's accuracy. Combined with Figures 7 and 8, the wavenumber selection range can be set as \(-1.55\times 10^{-3}\)-\(1.55\times 10^{-3}\), with uniform sampling. The numerical simulation results based on Gauss-FFT (4 Gaussian points) with \(201\times 201\times 101\) sampling points were used as a reference to measure the calculation accuracy of the AS-FT algorithm when different mesh divisions and wavenumbers were selected. The sampling points in the spatial domain and wavenumber domain were \(51\times 51\times 101\) and \(101\times 101\times 101\), respectively, and the samples are uniformly sampled. The accuracy is shown in Table 3. It can be seen that \(51\times 51\times 101\) nodes achieved high accuracy, and \(101\times 101\times 101\) nodes achieved higher accuracy. Figure 6: Magnetic susceptibility varying with horizontal coordinates.