Fast 3D Large‐Scale Forward Modelling of Vector and Tensor Magnetic Fields Using the FFT‐Based Spectral Method

ABSTRACT Traditional partial differential equation (PDE)‐based numerical methods for modelling three‐dimensional (3D) magnetic potential require solving a large, sparse linear system using either matrix inversion or iterative solvers. The corresponding computational cost can present considerable challenges for large‐scale 3D magnetic inversions. Additionally, the accuracy of the computed magnetic anomalies, such as the magnetic field vector and magnetic gradient tensor, derived through finite‐element or finite‐difference approximations, may degrade due to the inherent limitation of the numerical differentiations of the magnetic potential. To address these challenges, we present a fast Fourier transform (FFT)‐based spectral scheme aimed at effectively simulating the boundary value problem associated with the 3D magnetic potential. By applying the 3D FFT technique along three Cartesian axes to Poisson's equation, the corresponding PDE in the spatial domain is converted to an algebraic equation in the wavenumber domain, allowing for the straightforward computation of the magnetic potential. Additionally, the vector and tensor magnetic fields can be computed with high precision by utilizing the differential operators of the Fourier transform for spatial derivatives. Through applications to two synthetic and SEG/EAGE salt models, we validate the numerical accuracy and computational efficiency of our proposed method. Compared with other approaches in similar classes, such as the hybrid wavenumber‐domain finite‐element method, the proposed FFT‐based spectral method has a higher accuracy and requires lower computational cost and time. Thus, the new method is superior in the forward and inverse modelling of large‐scale magnetic problems.