# Arbitrary Sampling Fourier Transform and Its Applications in Magnetic Field Forward Modeling
## 5 Results
### Theory of Magnetic Field Modeling in Space-Wavenumber Domain  
The modeling theory of magnetic fields in the space-wavenumber domain reduces the 3D partial differential equation satisfied by the magnetic potential to a 1D ordinary differential equation independent of different wavenumbers by performing a 2D Fourier transform in the horizontal direction that reduces the amount of calculation and storage requirements. The vertical direction is reserved as the spatial domain, and the density of the elements can be adjusted according to the actual situation, which can consider the calculation accuracy and efficiency [37].  
The magnetic potential \(U_{a}\) and magnetization \(\mathbf{M}\) of the magnetic anomaly field meet the 3D partial differential equation in the spatial domain [38]:  
\[\
abla^{2}U_{a}(x,y,z)=\
abla\cdot\mathbf{M}. \tag{23}\]  
where (\(x\), \(y\), \(z\)) are the coordinates of observation points; \(\
abla^{2}\) is the Laplace operator and \(\
abla^{2}=\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y ^{2}}+\frac{\partial^{2}}{\partial z^{2}}\); \(\
abla\cdot\mathbf{M}\) refers to the divergence of magnetization \(\mathbf{M}\) and \(\
abla\cdot=\partial/\partial x+\partial/\partial y+\partial/\partial z\).  
Figure 4: Time required for one transformation of 1D, 2D, and 3D AS-FT. (**a**) Calculation time of 1D AS-FT under different node numbers. (**b**) Calculation time of 2D AS-FT under different node numbers. (**c**) Calculation time of 3D AS-FT under different node numbers.  
The magnetization \(\mathbf{M}\) is expressed as  
\[\mathbf{M}=\chi(\mathbf{H}_{0}+\mathbf{H}_{a})+\mathbf{J}_{r}. \tag{24}\]  
where \(\mathbf{H}_{0}(\mathrm{A}/\mathrm{m})\) is the background magnetic field, and \(\mathbf{H}_{a}(\mathrm{A}/\mathrm{m})\) is the magnetic field of the induction magnetization of the magnetic body, that is, the intensity of the abnormal field. The total field is \(\mathbf{H}\), obtained by \(\mathbf{H}=\mathbf{H}_{0}+\mathbf{H}_{a}\). \(\mathbf{J}_{r}(\mathrm{A}/\mathrm{m})\) is the residual magnetization.  
The horizontal 2D Fourier transform was performed on Equation (23), and a 1D ordinary differential equation was obtained, satisfied by the abnormal magnetic potential in the space-wavenumber domain  
\[\frac{\partial^{2}\widetilde{U}_{a}}{\partial z^{2}}-k^{2}\widetilde{U}_{a}= ik_{x}\widetilde{M}_{x}+ik_{y}\widetilde{M}_{y}+\frac{\partial\widetilde{M}_{z}}{ \partial z}. \tag{25}\]  
The 1-D FEM was used to solve the above ordinary differential equations. The magnetic field solution can be obtained using the derivative relationship between the magnetic potential and the magnetic field.  
It should be pointed out that when the magnetic susceptibility of the abnormal body is less than 0.01 SI, it is a weak magnetic situation, and the abnormal field \(\mathbf{H}_{a}\) can be ignored. The literature [37] analyzes the numerical simulation algorithm of the weak magnetic field in the space-wavenumber domain based on Gauss-FFT and the standard expanded FFT method, which will not be repeated here. The weak magnetic field can be obtained by directly solving Equation (25).  
When the magnetic susceptibility of the abnormal body is larger than 0.01 SI, it is a strong magnetic situation, \(\mathbf{H}_{a}\) cannot be ignored, and Equation (25) can be solved by the iterative method [26].