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Journal of Geodesy

, Volume 93, Issue 5, pp 723–747 | Cite as

On the computation of gravitational effects for tesseroids with constant and linearly varying density

  • Miao LinEmail author
  • Heiner Denker
Original Article

Abstract

The accurate computation of gravitational effects from topographic and atmospheric masses is one of the core issues in gravity field modeling. Using gravity forward modeling based on Newton’s integral, mass distributions are generally decomposed into regular mass bodies, which can be represented by rectangular prisms or polyhedral bodies in a rectangular coordinate system, or tesseroids in a spherical coordinate system. In this study, we prefer the latter representation because it can directly take the Earth’s curvature into account, which is particularly beneficial for regional and global applications. Since the volume integral cannot be solved analytically in the case of tesseroids, approximation solutions are applied. However, one well-recognized issue of these solutions is that the accuracy decreases as the computation point approaches the tesseroid. To overcome this problem, we develop a method that can precisely compute the gravitational potential \(\left( V\right) \) and vector \(\left( V_x, V_y, V_z\right) \) on the tesseroid surface. In addition to considering a constant density for the tesseroid, we further derive formulas for a linearly varying density. In the near zone (up to a spherical distance of 15 times the horizontal tesseroid dimension from the computation point), the gravitational effects of the tesseroids are computed by Gauss–Legendre quadrature using a two-dimensional adaptive subdivision technique to ensure high accuracy. The tesseroids outside this region are evaluated by means of expanding the integral kernel in a Taylor series up to the second order. The method is validated by synthetic tests of spherical shells with constant and linearly varying density, and the resulting approximation error is less than \(10^{-4}\,\hbox {m}^2\,\hbox {s}^{-2}\) for V, \(10^{-5}\,\hbox {mGal}\) for \(V_x\), \(10^{-7}\,\hbox {mGal}\) for \(V_y\), and \(10^{-4}\,\hbox {mGal}\) for \(V_z\). Its practical applicability is then demonstrated through the computation of topographic reductions in the White Sands test area and of global atmospheric effects on the Earth’s surface using the US Standard Atmosphere 1976.

Keywords

Tesseroid Gravity forward modeling Adaptive subdivision Topographic reduction Atmospheric effect Linearly varying density 

Notes

Acknowledgements

We thank the anonymous reviewers for their constructive comments that helped to significantly improve the manuscript. This work was financially supported by the German Research Foundation (DFG) within CRC 1128 “Relativistic Geodesy and Gravimetry with Quantum Sensors (geo-Q)”, project C04. Most of the figures were plotted by the Generic Mapping Tools (GMT; Wessel and Smith 1998).

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut für Erdmessung (IfE)Leibniz Universität HannoverHannoverGermany

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