Based on a brief review of forward algorithms for the computation of topographic gravitational and magnetic effects, including spatial, spectral and hybrid-domain algorithms working in either Cartesian or spherical coordinate systems, we introduce a new algorithm, namely the CP-FFT algorithm, for fast computation of terrain-induced gravitational and magnetic effects on arbitrary undulating surfaces. The CP-FFT algorithm, working in the hybrid spatial-spectral domain, is based on a combination of CANDECOMP/PARAFAC (CP) tensor decomposition of gravitational integral kernels and 2D Fast Fourier Transform (FFT) evaluation of discrete convolutions. By replacing the binomial expansion in classical FFT-based terrain correction algorithms using CP decomposition, convergence of the outer-zone computation can be achieved with significantly reduced inner-zone radius. Additionally, a Gaussian quadrature mass line model is introduced to accelerate the computation of the inner zone effect. We validate our algorithm by computing the gravitational potential, the gravitational vector, the gravity gradient tensor, and magnetic fields caused by densely-sampled topographic and bathymetric digital elevation models of selected mountainous areas around the globe. Both constant and variable density/magnetization models, with computation surfaces on, above and below the topography are considered. Comparisons between our new method and space-domain rigorous solutions show that with modeling errors well below existing instrumentation error levels, the calculation speed is accelerated thousands of times in all numerical tests. We release a set of open-source code written in MATLAB language to meet the needs of geodesists and geophysicists in related fields to carry out more efficiently topographic modeling in Cartesian coordinates under planar approximation.

基于对用于计算地形重力和磁力效应的正演算法的简要回顾，包括在笛卡尔坐标系或球坐标系中工作的空间、光谱和混合域算法，我们引入了一种新算法，即 CP-FFT 算法，用于快速计算地形引起的重力和磁力对任意起伏表面的影响。在混合空间-光谱域中工作的 CP-FFT 算法基于引力积分核的 CANDECOMP/PARAFAC (CP) 张量分解和离散卷积的二维快速傅立叶变换 (FFT) 评估的组合。通过使用 CP 分解替换经典的基于 FFT 的地形校正算法中的二项式展开，可以通过显着减小内区半径来实现外区计算的收敛。此外，引入高斯正交质量线模型以加速内区效应的计算。我们通过计算全球选定山区的密集采样地形和测深数字高程模型引起的重力势能、重力矢量、重力梯度张量和磁场来验证我们的算法。考虑了在地形上、上方和下方具有计算表面的恒定和可变密度/磁化模型。我们的新方法与空间域严格解决方案之间的比较表明，建模误差远低于现有仪器误差水平，在所有数值测试中计算速度都加快了数千倍。我们发布了一套用 MATLAB 语言编写的开源代码，以满足相关领域的大地测量学家和地球物理学家在平面近似下更高效地进行笛卡尔坐标地形建模的需求。