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

authored by: Miao Lin, Heiner Denker
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 (V) and vector (V_x, V_y, V_z) 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 10-4m2s-2 for V, 10-5mGal for V_x, 10-7mGal for V_y, and 10-4mGal 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.
Organisation(s): Institute of Geodesy
Type: Article
Journal: Journal of geodesy
Volume: 93
Pages: 723-747
No. of pages: 25
ISSN: 0949-7714
Publication date: 01.05.2019
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Geophysics, Geochemistry and Petrology, Computers in Earth Sciences
Electronic version(s): https://doi.org/10.1007/s00190-018-1193-4 (Access: Closed)
: Details in the research portal "Research@Leibniz University"

BibTeX

@article{e095172e1e95450e963fa0ef4c266198,
title = "On the computation of gravitational effects for tesseroids with constant and linearly varying density",
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{\textquoteright}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{\textquoteright}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 (V) and vector (Vx, Vy, Vz) 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 10-4m2s-2 for V, 10-5mGal for Vx, 10-7mGal for Vy, and 10-4mGal for Vz. 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{\textquoteright}s surface using the US Standard Atmosphere 1976.",
keywords = "Adaptive subdivision, Atmospheric effect, Gravity forward modeling, Linearly varying density, Tesseroid, Topographic reduction",
author = "Miao Lin and Heiner Denker",
note = "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).",
year = "2019",
month = may,
day = "1",
doi = "10.1007/s00190-018-1193-4",
language = "English",
volume = "93",
pages = "723--747",
journal = "Journal of geodesy",
issn = "0949-7714",
publisher = "Springer Verlag",
number = "5",
}