We present an accurate method for the calculation of gravitational potential (GP), vector (GV), and gradient tensor (GGT) of a tesseroid, considering a density model in the form of a polynomial up to cubic order along the vertical direction. The method solves volume integral equations for the gravitational effects due to a tesseroid by the Gauss–Legendre quadrature rule. A two-dimensional adaptive subdivision technique, which automatically divides the tesseroids near the computation point into smaller elements, is applied to improve the computational accuracy. For those tesseroids having small vertical dimensions, an extension technique is additionally utilized to ensure acceptable accuracy, in particular for the evaluation of GV and GGT. Numerical experiments based on spherical shell models, for which analytical solutions exist, are implemented to test the accuracy of the method. The results demonstrate that the new method is capable of computing the gravitational effects of the tesseroids with various horizontal and vertical dimensions as well as density models, while the evaluation point can be on the surface of, near the surface of, outside the tesseroid, or even inside it (only suited for GP and GV). Thus, the method is attractive for many geodetic and geophysical applications on regional and global scales, including the computation of atmospheric effects for terrestrial and satellite usage. Finally, we apply this method for computing the topographic effects in the Himalaya region based on a given digital terrain model and the global atmospheric effects on the Earth’s surface by using three polynomial density models which are derived from the US Standard Atmosphere 1976.

中文翻译：

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使用垂直方向密度可变的 Tesseroid 进行重力场建模

我们提出了一种计算 tesseroid 的引力势 (GP)、矢量 (GV) 和梯度张量 (GGT) 的准确方法，考虑到沿垂直方向达到三次阶的多项式形式的密度模型。该方法通过 Gauss-Legendre 求积法则求解由 tesseroid 引起的引力效应的体积积分方程。应用二维自适应细分技术，自动将计算点附近的网格划分为更小的元素，以提高计算精度。对于那些具有小垂直尺寸的 tesseroids，额外使用扩展技术来确保可接受的精度，特别是对于 GV 和 GGT 的评估。基于球壳模型的数值实验，存在解析解，被执行以测试该方法的准确性。结果表明，新方法能够计算具有各种水平和垂直维度以及密度模型的tesseroids的引力效应，而评估点可以在tesseroid的表面、表面附近、外部或甚至在里面（仅适用于 GP 和 GV）。因此，该方法对区域和全球范围内的许多大地测量和地球物理应用很有吸引力，包括计算地面和卫星使用的大气效应。最后，