Fast 3-D Magnetic Anomaly Forward Modeling Based on Integral Equation

Efficient and accurate 3-D numerical modeling of the total field magnetic anomaly is a key problem that restricts the 3-D fast inversion imaging and geological interpretation of large-scale magnetic data. To address this problem, we develop a fast numerical modeling method for 3-D magnetic anomaly based on integral equation. First, we derive a new analytical expression for prismatic total field magnetic anomaly, which has the advantages of compact structure and low computational cost. Then, we take advantage of the Toeplitz property of the kernel matrix when the grid is uniformly discretized in the horizontal directions to reduce the redundant computation and memory requirement without losing accuracy. In addition, the grid discretization in the vertical direction is flexible and can be finer at a shallower depth and coarser at a deeper depth, taking into account accuracy and efficiency. The 2-D FFT is used to achieve fast discrete convolution of the kernel matrix and magnetic susceptibility, which greatly improves the efficiency of the discrete convolution. In the numerical examples, a cube model is designed to verify the correctness of this method. The results demonstrate that the computation time of our algorithm is reduced by 1–2 orders of magnitude, and the memory requirement is decreased by approximately one order of magnitude compared with the conventional fast magnetic anomaly numerical modeling method based on block-Toeplitz–Toeplitz-block (BTTB) matrix, while achieving the same accuracy. Finally, the applicability of our algorithm is tested by calculating the terrain effect on an airborne magnetic data built in a real complex topography model.