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IGFS 2014 pp 93-100 | Cite as

An Ellipsoidal Analogue to Hotine’s Kernel: Accuracy and Applicability

  • Otakar NesvadbaEmail author
  • Petr Holota
Conference paper
Part of the International Association of Geodesy Symposia book series (IAG SYMPOSIA, volume 144)

Abstract

In this paper a mathematical apparatus is discussed that involves effects of the flattening of the Earth in the determination of the gravity potential. It rests on the use of Green’s function of the second kind (Neumann’s function) constructed for Neumann’s boundary value problem in the exterior of an oblate ellipsoid of revolution. The apparatus has a natural tie to the reproducing kernel of Hilbert’s space of functions harmonic in the solution domain considered. For at least one of the points (arguments of the kernel) inside the solution domain an expression of the reproducing kernel is developed. However, for both the points (arguments) on the ellipsoidal boundary a practical use of the kernel represented by means of series of ellipsoidal harmonic is not possible. Therefore the application of an approximate closed formula as the integral kernel is discussed and tested. A quality enhancement, if compared with the use of the spherical apparatus, is demonstrated by means of closed loop simulations. The paper contributes to methods related to the geoid (quasigeoid) computations.

Keywords

Gravimetric boundary value problem Neumann’s function Oblate spheroid Reproducing kernel 

Notes

Acknowledgements

The work on this paper was partly supported by the European Regional Development Fund (ERDF), project “NTIS – New Technologies for Information Society”, European Centre of Excellence, CZ.1.05/1.1.00/02.0090 and also by the Czech Science Foundation through Project No. 14-34595S. All this support is gratefully acknowledged.

References

  1. Abramowitz M, Stegun IA (1964) Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover Publications, New YorkGoogle Scholar
  2. Bjerhammar A, Svensson L (1983) On the geodetic boundary value problem for a fixed boundary surface – a sattelite approach. Bull Géod 57(1–4):382–393CrossRefGoogle Scholar
  3. Carlson BC (1979) Computing elliptic integrals by duplication. Numer Math 33(1):1–16CrossRefGoogle Scholar
  4. Haagmans R, de Min E, van Gelderen M (1993) Fast evaluation of convolution integrals on the sphere using 1D FFT and a comparison with existing methods for Stoke’s integral. Manuscr Geodaet 18(5):227–241Google Scholar
  5. Hofmann-Wellenhof B, Moritz H (2005) Physical geodesy. Springer, New YorkGoogle Scholar
  6. Holota P (1997) Coerciveness of the linear gravimetric boundary-value problem and a geometrical interpretation. J Geodesy 71(10):640–651CrossRefGoogle Scholar
  7. Holota P (2004) Some topics related to the solution of boundary-value problems in geodesy. In: Sansò F (ed) Proceedings from V-th Hotine-Marussi symposium on mathematical geodesy, Matera, Italy, 17–21 June 2002, vol 127. Springer, Berlin/Heidelberg, pp 189–200Google Scholar
  8. Holota P (2011) Reproducing kernel and Galerkin’s matrix for the exterior of an ellipsoid: application in gravity field studies. Stud Geophys Geod 55(3):397–413CrossRefGoogle Scholar
  9. Holota P, Nesvadba O (2014a) Analytical continuation in physical geodesy constructed by means of tools and formulas related to an ellipsoid of revolution. In: EGU general assembly, April 2014Google Scholar
  10. Holota P, Nesvadba O (2014b) Reproducing kernel and Neumann’s function for the exterior of an oblate ellipsoid of revolution: application in gravity field studies. Stud Geophys Geod 58(4):505–535CrossRefGoogle Scholar
  11. Kellogg OD (1953) Foundations of potential theory. Dover Publications, New YorkGoogle Scholar
  12. Koch KR, Pope AJ (1972) Uniqueness and existence for the geodetic boundary-value problem using the known surface of the Earth. Bull Géod 106:467–476CrossRefGoogle Scholar
  13. Meschkowski H (1962) Hilbertsche Räume mit Kernfunktion. Springer, BerlinCrossRefGoogle Scholar
  14. Moritz H (1980a) Advanced physical geodesy. Wichmann, KarlsruheGoogle Scholar
  15. Moritz H (1980b) Geodetic reference system 1980. Bull Géod 54(3):395–405CrossRefGoogle Scholar
  16. Nesvadba O (2010) Reproducing kernels in harmonic spaces and their numerical implementation. In: EGU general assembly, April 2010Google Scholar
  17. Nesvadba O, Holota P (2015) An OpenCL implementation of ellipsoidal harmonics. In: Rizos C, Willis P (eds) Proceedings of the VIII. Hotine-Marussi symposium, vol 142. Springer, Berlin/Heidelberg (in print). http://link.springer.com/chapter/10.1007/1345_2015_59
  18. Pavlis NK, Holmes SA, Kenyon SC, Factor JK (2012) The development and evaluation of the Earth Gravitational Model 2008 (EGM2008). J Geophys Res Solid Earth 117(B04406): 1–38. http://onlinelibrary.wiley.com/doi/10.1029/2011JB008916/abstract
  19. Pavlis NK, Holmes SA, Kenyon SC, Factor JK (2013) Correction to “The development and evaluation of the Earth Gravitational Model 2008 (EGM2008)”. J Geophys Res Solid Earth 118(2633)Google Scholar
  20. Richardson LF, Gaunt JA (1927) The deferred approach to the limit. Part I. Single lattice. Part II. Interpenetrating lattices. Philos Trans R Soc Lond Ser A, Containing Papers of a Mathematical or Physical Character 226(636–646):299–361CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Land Survey OfficePragueCzech Republic
  2. 2.Research Institute of GeodesyTopography and CartographyPrague-EastCzech Republic

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