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On the Two Different Formulas for the 3D Rectangular Prism Effect in Gravimetry

  • R. Karcol
  • R. Pašteka
Article
  • 54 Downloads

Abstract

The direct problem of simple geometrical bodies plays an important role in the gravimetrical processing and modelling tools. We focused on the 3D rectangular prism, which is widely used in such processes. Even though the solution for this body is well known, there are still some issues about it, which are not answered or not answered completely in the available literature, e.g. the presence of the singularities in the source-free points or a continuity of the solutions. We present the singularity-free solution valid for the each position of the calculation point. Next, the analysis of the two basic types of the formulae for the 3D rectangular prism’s gravitational effect is held on. We discuss the ways of their derivation, the validity and the problems connected with them. Later, special attention is paid at the problems with the citation of these two formulae types within the gravimetrical literature.

Keywords

Gravimetry direct problem 3D rectangular prism 

Notes

Acknowledgements

This work was supported by the project of the Scientific and Grant Agency of Slovak Republic (VEGA-1/0462/16).

References

  1. Banerjee, B., & Das Gupta, S. P. (1977). Gravitational attraction of a rectangular parallelepiped. Geophysics, 42(5), 1053–1055.CrossRefGoogle Scholar
  2. Çavşak, H. (2012). Effective calculation of gravity effects of uniform triangle polyhedral. Studia Geophysica et Geodaetica, 56, 185–195.CrossRefGoogle Scholar
  3. D’Urso, M. G. (2014). Analytical computation of gravity effects for polyhedral bodies. Journal of Geodesy, 88, 13–29.CrossRefGoogle Scholar
  4. Everest, G. (1830). An account of the measurement of an arc of the meridian between the parallels of 18°3′ and 24°7′. London.Google Scholar
  5. Gradshteyn, I. S., & Ryzhik, I. M. (1962). Table of integrals, series and products. Moscow: State Publishing House of Physical and Mathematical Literature (in Russian).Google Scholar
  6. Haáz, I. B. (1953). Relation between the potential of the attraction of the mass contained in a finite rectangular prism and its first and second derivatives. Geofizikai Kőzlemenyek 2(7) (in Hungarian).Google Scholar
  7. Hamayun, Prutkin, I., & Tenzer, R. (2009). The optimum expression for the gravitational potential of polyhedral bodies having a linearly varying density distribution. Journal of Geodesy, 83, 1163–1170.CrossRefGoogle Scholar
  8. Kellogg, D. O. (1929). Foundations of potential theory. Berlin: Springer.CrossRefGoogle Scholar
  9. Li, X., & Chouteau, M. (1998). Three-dimensional modeling in all space. Surveys in Geophysics, 19, 339–368.CrossRefGoogle Scholar
  10. MacMillan, W. D. (1930). The theory of the potential. New York: McGraw-Hill.Google Scholar
  11. Mader, K. (1951). Das Newtonsche Raumpotential prismatischer Körper und seine Ableitungen bis zur dritten Ordnung. Sonderheft 11 der Österreichischen Zeitschrift für Vermessungswesen. Wien: Österreichischer Verein für Vermessungswesen.Google Scholar
  12. Nagy, D. (1966). The gravitational attraction of a rectangular prism. Geophysics, 31(2), 362–371.CrossRefGoogle Scholar
  13. Nagy, D. (1973). A chart for the computation of the gravitational attraction of a right rectangular prism. Pure and Applied Geophysics, 102(1), 5–14.CrossRefGoogle Scholar
  14. Nagy, D., Papp, G., & Benedek, J. (2000). The gravitational potential and its derivatives for the prism. Journal of Geodesy, 74, 552–560.CrossRefGoogle Scholar
  15. Rektorys, K. (1968). The compendium of the applied mathematics (in Czech). Prague: Publishing House of Technical Literature.Google Scholar
  16. Sorokin, L. V. (1951). Gravimetry and gravimetrical prospecting. Moscow: State Technology Publishing (in Russian).Google Scholar
  17. Werner, R. A. (2017). The solid angle hidden in polyhedron gravitation formulations. Journal of Geodesy, 91, 307–328.CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of Applied and Environmental GeophysicsComenius University BratislavaBratislavaSlovakia

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