# Arbitrary Sampling Fourier Transform and Its Applications in Magnetic Field Forward Modeling
## 5 Results
### Experiments
#### 5.3.2 Weak Magnetic Catastrophe Medium  
A weak magnetic catastrophe medium model was designed, with a background magnetic field of 50,000 nT, a magnetic inclination of 45\({}^{\circ}\), and a magnetic declination of 5.9\({}^{\circ}\). The model is shown in Figure 10. The \(x,y,\) and \(z\) directions are sampled uniformly in the spatial domain, and the number of nodes in each of the three directions is 101. Figure 10: Sphere catastrophe model. The calculation area is 500 m \(\times\) 500 m \(\times\) 500 m, with a range of \(-\)250 m–250 m in the x direction, \(-\)250 m–250 m in the y direction, and 0–500 m in the z direction. The center of the abnormal sphere model is located at (0 m, 0 m, 250 m). The radius of the sphere is 100 m. The magnetic susceptibility of the sphere is 0.01 SI. \begin{table}
\begin{tabular}{c c c c c} \hline \multicolumn{5}{c}{**Number of Sample**} & \multicolumn{3}{c}{**Rrms Error (\%)**} \\ \hline
**Spatial Domain** & **Wavenumber Domain** & **\(\mathbf{B_{ax}}\)** & **\(\mathbf{B_{ay}}\)** & **\(\mathbf{B_{az}}\)** \\ \(\mathbf{N}\mathbf{x}\times\mathbf{N}\mathbf{y}\) & \(\mathbf{N}\mathbf{x}\times\mathbf{N}\mathbf{y}\) & **\(\mathbf{B_{ax}}\)** & **\(\mathbf{B_{ay}}\)** & **\(\mathbf{B_{az}}\)** \\ \hline
51 \(\times\) 51 & 51 \(\times\) 51 & 0.23 & 0.73 & 0.99 \\
101 \(\times\) 101 & 101 \(\times\) 101 & 0.03 & 0.06 & 0.08 \\ \hline \end{tabular}
\end{table} 
Table 3: Accuracy statistics of different sampling points between \(-\)1.55 \(\times\) 10\({}^{-3}\)–1.55 \(\times\) 10\({}^{-3}\) in the wavenumber domain (z = 0 m, Nz = 101, uniform sampling in both spatial and wavenumber domain). Figure 9: Comparison of continuum model between AS-FT and Gauss-FFT results (z = 0 m). The spectrum distribution of the anomaly field at different depths of the model is shown in Figure 11. Since the abnormal body is in the center, the spectral distribution of the abnormal field obtained by numerical simulation is symmetrical about z = 250 m, so the spectral distribution of z = 0 m, 100 m, 200 m, and 250 m planes was taken for analysis. Compared with the spectral distribution in Figure 7 of the continuous model, the concentration of the spectrum becomes weaker, and the spectrum oscillates at large wavenumbers but attenuates to 0 at the cut-off frequency. To understand the 3D spectrum, the distribution of the magnetic anomaly spectrum at the position of z = 250 m is shown in Figure 12. According to the distribution of the spectrum, the distribution of the weak magnetic catastrophe model spectrum is relatively concentrated. Still, with the increase in the wavenumber, the distribution of the spectrum fluctuates slightly. Considering the efficiency and accuracy, the wavenumber selection range is set as \(-\)0.1-0.1. The following tests analyzed the influence of uniform and log-domain uniform sampling on the accuracy. It can be seen from Table 4 that the samples are uniformly sampled in the range of \(-\)0.1-0.1, and the Rrms errors of the ground z = 0 m plane \(B_{ax},B_{ay},B_{az}\) are 0.11%, 0.17%, and 0.22%, respectively. The samples are uniformly sampled in the log domain of \(-\)0.1-0.1, and the Rrms errors of the ground z = 0 m plane \(B_{ax},B_{ay},B_{az}\) are 0.05%, 0.05%, and 0.06%, respectively. Figure 11: Spectrum distribution of weak magnetic field at different depths. The horizontal ordinate of figures (**a**,**c**,**e**) is non-logarithmic, and the horizontal ordinate of figures (**b**,**d**,**f**) is logarithmic. The comparison between the numerical simulation results of log-domain sampling and the analytical solution of the sphere is shown in Figure 13. The results show that the AS-FT algorithm has high calculation accuracy for the catastrophe model.