Accurate computation of gravitational field of a tesseroid
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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.
KeywordsGravitational field Numerical differentiation Numerical integration Split quadrature Tesseroid
The author appreciates valuable suggestions and fruitful comments by Prof. Uieda and two anonymous referees to improve the quality of the article.
- 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
- Fukushima T (2014) Computation of a general integral of Fermi–Dirac distribution by McDougall–Stoner method. Appl Math Comput 238:485–510Google Scholar
- Heiskanen WA, Moritz H (1967) Physical geodesy. Freeman, San FranciscoGoogle 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
- 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
- Press WH, Teukolsky SA, Vetterling WT, Flannery BP (2007) Numerical recipes: the art of scientific computing, 3rd edn. Cambridge University Press, CambridgeGoogle 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
- 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
- Vanicek P, Krakiwsky EJ (1982) Geodesy: the concepts. North-Holland Publishing Company, AmsterdamGoogle Scholar