The generally shaped polyhedron is a widely used model in gravity field modelling and interpretation. Its induced gravity signal – gravitational potential and its derivatives up to second order – has been studied extensively in the geophysical literature. The class of solutions with special interest is the one which leads to closed analytical expressions, as these offer an exact representation of the gravity signal of finite three-dimensional distributions while being at the same time linked to a flexible means of geometric modelling. Of the several mathematical algorithms available, the line integral approach involves no approximations and is, however, connected with certain singularities, occurring for specific relative positions of the computation point with respect to the polyhedral source. The present contribution analyses the algorithmic details of the polyhedral line integral approach focusing on its geometric and computational aspects. Following the definitions of the individual coordinate systems and the two-step application of the Gauss divergence theorem for each face and polygonal boundary of the source, the geometric insight of the algorithm is presented, which permits a deeper understanding of the significance of the involved numerical singular terms. An overview of the line integral analytical approach for the polyhedral gravity signal is presented with emphasis on its geometric and computational aspects. Matlab code and results for the two considered case studies, a prismatic source and asteroid Eros, are provided as electronic supplement.

中文翻译：

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多面体重力信号线积分解析公式的计算综述

一般形状的多面体是重力场建模和解释中广泛使用的模型。它的诱导重力信号——重力势及其高达二阶的导数——已在地球物理文献中得到广泛研究。特别感兴趣的解决方案类别是导致封闭解析表达式的解决方案，因为它们提供有限三维分布的重力信号的精确表示，同时与几何建模的灵活方法相关联。在可用的几种数学算法中，线积分方法不涉及近似值，但是与某些奇点有关，发生在计算点相对于多面体源的特定相对位置。本贡献分析了多面体线积分方法的算法细节，重点是其几何和计算方面。根据各个坐标系的定义和对源的每个面和多边形边界的高斯散度定理的两步应用，提出了算法的几何洞察力，从而可以更深入地理解所涉及数值的重要性单数条款。概述了多面体重力信号的线积分分析方法，重点介绍了其几何和计算方面。两个考虑过的案例研究的 Matlab 代码和结果，棱镜源和小行星爱神，作为电子补充提供。