Surveys in Geophysics

, Volume 39, Issue 3, pp 365–399 | Cite as

Evaluation of Optimal Formulas for Gravitational Tensors up to Gravitational Curvatures of a Tesseroid

  • Xiao-Le Deng
  • Wen-Bin Shen


The forward modeling of the topographic effects of the gravitational parameters in the gravity field is a fundamental topic in geodesy and geophysics. Since the gravitational effects, including for instance the gravitational potential (GP), the gravity vector (GV) and the gravity gradient tensor (GGT), of the topographic (or isostatic) mass reduction have been expanded by adding the gravitational curvatures (GC) in geoscience, it is crucial to find efficient numerical approaches to evaluate these effects. In this paper, the GC formulas of a tesseroid in Cartesian integral kernels are derived in 3D/2D forms. Three generally used numerical approaches for computing the topographic effects (e.g., GP, GV, GGT, GC) of a tesseroid are studied, including the Taylor Series Expansion (TSE), Gauss–Legendre Quadrature (GLQ) and Newton–Cotes Quadrature (NCQ) approaches. Numerical investigations show that the GC formulas in Cartesian integral kernels are more efficient if compared to the previously given GC formulas in spherical integral kernels: by exploiting the 3D TSE second-order formulas, the computational burden associated with the former is 46%, as an average, of that associated with the latter. The GLQ behaves better than the 3D/2D TSE and NCQ in terms of accuracy and computational time. In addition, the effects of a spherical shell’s thickness and large-scale geocentric distance on the GP, GV, GGT and GC functionals have been studied with the 3D TSE second-order formulas as well. The relative approximation errors of the GC functionals are larger with the thicker spherical shell, which are the same as those of the GP, GV and GGT. Finally, the very-near-area problem and polar singularity problem have been considered by the numerical methods of the 3D TSE, GLQ and NCQ. The relative approximation errors of the GC components are larger than those of the GP, GV and GGT, especially at the very near area. Compared to the GC formulas in spherical integral kernels, these new GC formulas can avoid the polar singularity problem.


Gravity forward modeling Tesseroid Topographic effects Gravitational curvatures 



We are very grateful to Prof. Rycroft and two anonymous reviewers for their valuable comments and suggestions, which greatly improved the manuscript. This study is supported by National 973 Project China (Grant No. 2013CB733305), NSFCs (Grant Nos. 41631072, 41721003, 41429401, 41574007, 41210006, 41174011, 41128003, 41021061), DAAD Thematic Network Project (Grant No. 57173947), NASG Special Project Public Interest (Grant No. 201512001) and Key Laboratory of GEGME fund (Grant No. 16-02-02).


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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Geodesy and GeomaticsWuhan UniversityWuhanChina
  2. 2.State Key Laboratory of Information Engineering in Surveying, Mapping and Remote SensingWuhan UniversityWuhanChina

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