Fast and high accuracy 3D magnetic anomaly forward modeling based on BTTB matrix

Given the low efficiency of the traditional three dimensional (3D) magnetic forward modeling method in the spatial domain, we propose a fast and high-accuracy 3D forward algorithm to calculate the magnetic anomaly Delta T and its gradient on a horizontal observation surface generated by the underground source. The vertical cuboid is selected as a fundamental mesh element to discretize the source region. The observation points align with the center points of the discretized cuboids on the horizontal plane. The improved algorithm has three advantages: (1) The analytical solution formula without singularity is used to calculate the magnetic anomaly generated by the field source on the horizontal observation plane to ensure accuracy. (2) The BTTB (Block-Toeplitz Toeplitz-Block) matrix is constructed for compressing the kernel matrix according to its structural properties, and intermediate variables are pre-calculated and stored to improve efficiency. (3) The product of the susceptibility vector and the kernel matrix with the structure of the BTTB matrix is transformed into a two-dimensional discrete convolution by using the FFT (Fast Fourier Transform) to improve the computational efficiency further. The model tests show that the improved algorithm is about five orders of magnitude faster than the traditional analytic method and about two orders of magnitude faster than the existing Gauss-FFT algorithm with 8 Gaussian nodes. In the model test, the maximum absolute error is about 10(-6) nT, and the memory usage of the kernel matrix at inversion is reduced by about five orders of magnitude. These indicate that the improved algorithm has the advantages of high precision, high efficiency, and low storage. Finally, the forward algorithm is applied to an inversion of magnetic anomaly Delta T. The result shows that the 3D magnetic susceptibility obtained by the inversion is consistent with the input model, and the computational time and memory usage are greatly reduced, which verifies the practicability of the improved algorithm.