# Arbitrary Sampling Fourier Transform and Its Applications in Magnetic Field Forward Modeling
## 1 Introduction  
Fourier transform is an essential tool in the geophysical field, and is of great significance for high-precision numerical simulation and near-surface high-resolution imaging [1, 2, 3]. It converts time signals to the frequency domain or spatial signals to the wavenumber domain. It can make fast convolution, target recognition, and other algorithms easy to implement. It is also widely used in waveform analysis, sequence, linear-equation-solving disciplines, and engineering fields [4, 5, 6]. In 1965, the release of the Cooley-Tukey fast Fourier transform (FFT) algorithm reduced the complexity of the Fourier transform from \(O(N^{2})\) to \(O(N\log_{2}(N))\) (whererepresents the scale of the problem), opening up a new field for digital signal processing [7]. [PERSON] proposed that when \(N\) is a prime number, DFT calculation can be transformed into cyclic convolution by rearranging the data, which significantly improves the efficiency of the discrete Fourier transform algorithm [8]. [PERSON] used convolution to calculate DFT, showed a deep correlation between convolution and Fourier transform, and clarified the minimum number of multiple convolutions required for cyclic convolution. The above Fourier transform requires uniform sampling [9]. To realize the Fourier transform algorithm with non-uniform sampling, [PERSON] and [PERSON] developed a fast Fourier transform algorithm on non-uniform nodes based on the interpolation of the quick multipole method [10]. [PERSON] improved the NUFFT algorithm proposed by [PERSON] and [PERSON] [11]. Subsequently, some scholars conducted further relevant research to improve the accuracy and efficiency of the non-uniform fast Fourier transform (NUFFT) algorithm [12; 13]. With the significant improvement of NUFFT algorithm's accuracy and efficiency, a series of NUFFT open-source libraries were proposed accordingly [14; 15; 16; 17; 18; 19; 20]. [PERSON] et al. further optimized NUFFT based on the \"semicircular kernel index\" and developed a parallel version (FINUFFT), which is significantly more efficient than NUFFT and has the advantage of small memory occupation [21]. The optimized NUFFT algorithm provides a suitable means for numerical problems requiring non-uniform sampling. Fourier transform with uniform or non-uniform sampling provides a good tool for solving many practical problems. Fourier transform is widely used in geophysical numerical simulation. FFT provides an efficient way to solve geophysical problems, but its edge effects are apparent, affecting data processing [22]. To overcome the edge effects of FFT, [PERSON] and [PERSON] proposed Gauss-FFT technology based on shift sampling [23], which significantly improved the calculation accuracy of FFT and weakened the edge effects, realizing efficient and high-precision calculation of three-dimensional gravity and magnetic field forward modeling [24; 25]. [PERSON] and [PERSON] applied the Gauss-FFT method to the numerical simulation of a 3D strong magnetic field integral equation. They obtained a high-precision solution using the iterative method, avoiding the influence of zero wavenumbers and improving the forward modeling efficiency [26]. [PERSON] et al. applied non-uniform Fourier transform to electromagnetic scattering problems, which has advantages in solving the issues of non-uniform scatterers and sharp discontinuities [27]. [PERSON] et al. used Gauss-FFT and NUFFT to solve the gravity potential based on the prism model and found that NUFFT solved the singularity problem at zero wavenumbers and had high accuracy [28]. [PERSON] et al. proposed an FFT algorithm based on cubic spline interpolation that can sample flexibly and apply its results to the forward modeling of gravitational and magnetic potential fields. However, this method's discrete Fourier transform coefficients are related to the values at discrete points. The efficiency is not dominant for the algorithm that needs iteration [29]. [PERSON] et al.