# Arbitrary Sampling Fourier Transform and Its Applications in Magnetic Field Forward Modeling
## 5 Results
### Experiments
#### 5.3.3 Strong Magnetic Catastrophe Medium  
We continued to use the sphere model in Figure 10 and set the magnetic susceptibility of the sphere to 1 SI. The other model parameters were consistent with those in Section 5.3.2. It can be seen that the model is under strong magnetic conditions. The spectrum distribution of the anomaly field at different depths of the model is shown in Figure 14. Similarly, we analyzed the spectrum distribution of z = 0 m, 100 m, 200 m, and 250 m planes. Compared with the weak magnetic spectrum distribution in Figure 11, the energy of the strong magnetic spectrum is enhanced by two orders of magnitude, but the spectrum's shape is the same; it is also oscillatory, and it is attenuated to ero at the cut-off frequency. To understand the 3D distribution of the spectrum, the distribution of the magnetic anomaly spectrum at the position of z = 250 m is shown in Figure 15. The strong magnetic catastrophe model spectrum is concentrated according to the spectrum's distribution. Still, the oscillation is intense, the energy is strong, and the attenuation is slow. There is still strong energy at large wavenumbers, which cannot be ignored. Therefore, to ensure accuracy, it is necessary to expand the wavenumber selection range to \(-\)0.25-0.25. After determining the wavenumber selection interval, we changed the spatial domain and wavenumber domain sample methods. The accuracy of numerical simulation under different division methods is shown in Table 5. The number of sample nodes was \(Nx=Ny=Nz=101\), and \(Nkx=Nky=101\), respectively. Figure 14: Spectrum distribution of strong 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. It can be seen from Table 5 that in the case of uniform sampling in both the space and the wavenumber domain, the Rrms errors of \(B_{ax},B_{ay},B_{az}\) are 1.91%, 1.82%, and 1.83%, respectively. In the case of uniform sampling in the spatial domain and logarithmic sampling in the wavenumber domain the Rrms errors of \(B_{ax},B_{ay},B_{az}\) are 0.95%, 1.08%, and 0.95%, respectively. Next, to better fit the abnormal body, non-uniform sampling was selected in the spatial domain. The spatial domain was divided non-uniformly in the horizontal direction, and the sample was densified at the abnormal sphere. The sample interval away from the abnormal sphere was gradually increased, with the minimum grid spacing of 1 m and the maximum grid spacing of 32 m. Its grid is shown in Figure 16a. Piecewise uniform and log-domain sampling were adopted for the wavenumber range of \(-\)0.25-0.25. The strategy of piecewise uniform sampling is to divide into three segments according to the spectral energy distribution, with uniform sampling in each part. Taking the positive wavenumber as an example, 31 nodes were taken from 0 to 0.04, 11 nodes were taken from 0.04 to 0.1, and 11 nodes were taken from 0.1 to 0.25. The negative wavenumber was symmetrical with the positive wavenumber, with a total of 101 nodes, as shown in Figure 16b. When piecewise uniform sampling was adopted, after six iterations, the Rrms errors of the numerical solution of the ground field value \(B_{ax},B_{ay},B_{az}\) were 0.19%, 0.24%, and 0.25% compared with the analytical solution of the strong magnetic sphere, respectively. When piecewise uniform sampling was adopted, after six iterations, the Rrms errors of the numerical solution of the ground field value \(B_{ax},B_{ay},B_{az}\) were 0.17%,  
Figure 15: Spectral distribution of strong magnetic field in wavenumber domain (z = 250 m).