# Accurate computation of gravitational field of a tesseroid

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## Abstract

We developed an accurate method to compute the gravitational field of a tesseroid. The method numerically integrates a surface integral representation of the gravitational potential of the tesseroid by conditionally splitting its line integration intervals and by using the double exponential quadrature rule. Then, it evaluates the gravitational acceleration vector and the gravity gradient tensor by numerically differentiating the numerically integrated potential. The numerical differentiation is conducted by appropriately switching the central and the single-sided second-order difference formulas with a suitable choice of the test argument displacement. If necessary, the new method is extended to the case of a general tesseroid with the variable density profile, the variable surface height functions, and/or the variable intervals in longitude or in latitude. The new method is capable of computing the gravitational field of the tesseroid independently on the location of the evaluation point, namely whether outside, near the surface of, on the surface of, or inside the tesseroid. The achievable precision is 14–15 digits for the potential, 9–11 digits for the acceleration vector, and 6–8 digits for the gradient tensor in the double precision environment. The correct digits are roughly doubled if employing the quadruple precision computation. The new method provides a reliable procedure to compute the topographic gravitational field, especially that near, on, and below the surface. Also, it could potentially serve as a sure reference to complement and elaborate the existing approaches using the Gauss–Legendre quadrature or other standard methods of numerical integration.

### Keywords

Gravitational field Numerical differentiation Numerical integration Split quadrature Tesseroid## Notes

### Acknowledgements

The author appreciates valuable suggestions and fruitful comments by Prof. Uieda and two anonymous referees to improve the quality of the article.

## Supplementary material

### References

- Anderson EG (1976) The effect of topography on solutions of Stokes’ problem. Unisurv S-14 Report, School of Surveying. University of New South Wales, KensingtonGoogle Scholar
- Bailey DH, Jeyabalan K, Li XS (2005) A comparison of three high-precision quadrature schemes. Exp Math 14:317–329CrossRefGoogle Scholar
- Conway JT (2015) Analytical solution from vector potentials for the gravitational field of a general polyhedron. Celest Mech Dyn Astron 121:17–38CrossRefGoogle Scholar
- D’Urso MG (2014) Analytical computation of gravity effects for polyhedral bodies. J Geod 88:13–29CrossRefGoogle Scholar
- Dziewonski AM, Anderson DL (1981) Preliminary reference Earth model. Phys Earth Planet Inter 25:297–356CrossRefGoogle Scholar
- Fukushima T (2014) Computation of a general integral of Fermi–Dirac distribution by McDougall–Stoner method. Appl Math Comput 238:485–510Google Scholar
- Fukushima T (2016a) Numerical computation of gravitational field of infinitely thin axisymmetric disc with arbitrary surface mass density profile and its application to preliminary study of rotation curve of M33. Mon Not R Astron Soc 456:3702–3714CrossRefGoogle Scholar
- Fukushima T (2016b) Mosaic tile model to compute gravitational field for infinitely thin non axisymmetric objects and its application to preliminary analysis of gravitational field of M74. Mon Not R Astron Soc 459:3825–3860CrossRefGoogle Scholar
- Fukushima T (2016c) Numerical computation of gravitational field for general axisymmetric objects. Mon Not R Astron Soc 462:2138–2176CrossRefGoogle Scholar
- Fukushima T (2016d) Numerical integration of gravitational field for general three-dimensional objects and its application to gravitational study of grand design spiral arm structure. Mon Not R Astron Soc 463:1500–1517CrossRefGoogle Scholar
- Fukushima T (2017a) Numerical computation of electromagnetic field for general static and axisymmetric current distribution. Comput Phys Commun 221:109–117CrossRefGoogle Scholar
- Fukushima T (2017b) Precise and fast computation of gravitational field of general finite body and its application to gravitational study of asteroid Eros. Astron J 154:145CrossRefGoogle Scholar
- Garcia-Abdeslem J (2005) Gravitational attraction of a rectangular prism with density varying with depth following a cubic polynomial. Geophysics 70:J39–J42CrossRefGoogle Scholar
- Grombein T, Heck B, Seitz K (2013) Optimized formulae for the gravitational field of a tesseroid. J Geod 87:645–660CrossRefGoogle Scholar
- Heck B, Seitz K (2007) A comparison of the tesseroid, prism and point-mass approaches for mass reductions in gravity field modelling. J Geod 81:121–136CrossRefGoogle Scholar
- Heiskanen WA, Moritz H (1967) Physical geodesy. Freeman, San FranciscoGoogle Scholar
- Hirt C, Kuhn M (2014) Band-limited topographic mass distribution generates full-spectrum gravity field: gravity forward modeling in the spectral and spatial domains revisited. J Geophys Res Solid Earth 119:3646–3661CrossRefGoogle Scholar
- Hirt C, Rexer M (2015) Earth 2014: 1 arc-min shape, topography, bedrock and ice-sheet models-available as gridded data and degree-10,800 spherical harmonics. Int J Appl Earth Obs Geoinf 39:103–112CrossRefGoogle Scholar
- Jekeli C (2007) Potential theory and static gravity field of the earth. In: Schubert G (ed) Treatise on geophysics, vol 3, 2nd edn. Elsevier, AmsterdamGoogle Scholar
- Karcol R (2011) Gravitational attraction and potential of spherical shell with radially dependent density. Stud Geophys Geod 55:21–34CrossRefGoogle Scholar
- Kellogg OD (1929) Foundations of potential theory. Springer, BerlinCrossRefGoogle Scholar
- Kennett BLN (1998) On the density distribution within the Earth. Geophys J Int 132:374–382CrossRefGoogle Scholar
- Klees R, Lehmann R (1998) Calculation of strongly singular and hypersingular surface integrals. J Geod 72:530–546CrossRefGoogle Scholar
- Kuhn M, Hirt C (2016) Topographic gravitational potential up to second-order derivatives: an examination of approximation errors caused by rock-equivalent topography (RET). J Geod 90:883–902CrossRefGoogle Scholar
- Laske G, Masters G, Ma Z, Pasyanos M (2013) Update on CRUST1.0—a 1-degree Global Model of Earth’s Crust. In: Geophysical research abstracts, vol 15, Abstract EGU2013-2658Google Scholar
- MacMillan WD (1930) The theory of the potential. McGraw-Hill, New YorkGoogle Scholar
- Martinec Z (1988) Boundary value problems for gravimetric determination of a precise geoid. Springer, BerlinGoogle Scholar
- Mori H (1985) Quadrature formulae obtained by variable transformation and DE rule. J Comput Appl Math 12&13:119–130CrossRefGoogle Scholar
- Nagy D, Papp G, Benedek J (2000) The gravitational potential and its derivatives for the prism. J Geod 74:552–560CrossRefGoogle Scholar
- Novak P, Grafarend EW (2005) Ellipsoidal representation of the topographical potential and its vertical gradient. J Geod 78:691–706CrossRefGoogle Scholar
- Press WH, Teukolsky SA, Vetterling WT, Flannery BP (2007) Numerical recipes: the art of scientific computing, 3rd edn. Cambridge University Press, CambridgeGoogle Scholar
- Roussel C, Verdun J, Cali J, Masson F (2015) Complete gravity field of an ellipsoidal prism by Gauss–Legendre quadrature. Geophys J Int 203:2220–2236CrossRefGoogle Scholar
- Smith DA, Robertson DS, Milbert DG (2001) Gravitational attraction of local crustal masses in spherical coordinates. J Geod 74:783–795CrossRefGoogle Scholar
- Stacey FD, Davis PM (2008) Physics of the Earth, 4th edn. Cambridge University Press, CambridgeCrossRefGoogle Scholar
- Tachikawa T, Hato M, Kaku M, Iwasaki A (2011) Characteristics of ASTER GDEM version 2. In: Proceedings on IEEE international geoscience and remote sensing symposium, pp 3657–3660Google Scholar
- Takahashi H, Mori H (1973) Quadrature formulae obtained by variable transformation. Numer Math 21:206–219CrossRefGoogle Scholar
- Takahashi H, Mori H (1974) Double exponential formulae for numerical integration. Publ RIMS Kyoto Univ 9:721–741CrossRefGoogle Scholar
- Tscherning CC (1976) Computation of the second-order derivatives of the normal potential based on the representation by a Legendre series. Manuscr Geod 1:71–92Google Scholar
- Uieda L, Barbosa VCF, Braitenberg C (2016) Tesseroids: forward-modeling gravitational fields in spherical coordinates. Geophysics 81:F41–F48CrossRefGoogle Scholar
- Vanicek P, Krakiwsky EJ (1982) Geodesy: the concepts. North-Holland Publishing Company, AmsterdamGoogle Scholar
- Waldvogel J (1979) The Newtonian potential of homogeneous polyhedra. J Appl Math Phys (ZAMP) 30:388–398CrossRefGoogle Scholar
- Wild-Pfeiffer F (2008) A comparison of different mass elements for use in gravity gradiometry. J Geod 82:637–653CrossRefGoogle Scholar
- Wu L (2016) Efficient modelling of gravity effects due to topographic masses using the Gauss—FFT method. Geophys J Int 205:160–178CrossRefGoogle Scholar
- Yu Y, Baoyin H (2015) Modeling of migrating grains on asteroids surface. Astrophys Space Sci 355:43–56CrossRefGoogle Scholar