Method of Successive Approximations in Solving Geodetic Boundary Value Problems: Analysis and Numerical Experiments

  • P. Holota
  • O. Nesvadba
Conference paper
Part of the International Association of Geodesy Symposia book series (IAG SYMPOSIA, volume 137)


After an introductory note reviewing the role and the treatment of boundary problems in physical geodesy, the explanation rests on the concept of the weak solution. The focus is on the linear gravimetric boundary value problem. In this case, however, an oblique derivative in the boundary condition and the need for a numerical integration over the whole and complicated surface of the Earth make the numerical implementation of the concept rather demanding. The intention is to reduce the complexity by means of successive approximations and step by step to take into account effects caused by the obliqueness of the derivative and by the departure of the boundary from a more regular surface. The possibility to use a sphere or an ellipsoid of revolution as an approximation surface is discussed with the aim to simplify the bilinear form that defines the problem under consideration and to justify the use of an approximation of Galerkin’s matrix. The discussion is added of extensive numerical simulations and tests using the ETOPO5 boundary for the surface of the Earth and gravity data derived from the EGM96 model of the Earth’s gravity field.


Earth’s gravity field Geodetic boundary-value problems Variational methods Weak solution Numerical methods Galerkin’s approximations Iteration and convergence problems 



The presentation of the paper at the VII Hotine-Marussi Symposium, Rome, Italy, July 6–10, 2009, was sponsored by the Ministry of Education, Youth and Sports of the Czech Republic through Projects No. LC506. The computations were performed in CINECA (Bologna) within the Project HPC-EUROPA\(++\) (RII3-CT-2003–506079), supported by the European Community – Research Infrastructure Action under the FP6 “Structuring the European Research Area” Programme. All this support as well as the possibility to discuss the topic with Prof. F. Sansò, Milano, is gratefully acknowledged. Thanks go also to anonymous reviewers for valuable comments.


  1. Bers L, John F, Schechter M (1964) Partial differential equations. Wiley, New YorkGoogle Scholar
  2. Holota P (1991) On the iteration solution of the geodetic boundary-value problem and some model refinements. Travaux de l’A.I.G., Tome 29, Paris, 1992, pp 260–289Google Scholar
  3. Holota P (1997) Coerciveness of the linear gravimetric boundary-value problem and a geometrical interpretation. J Geodes 71:640–651CrossRefGoogle Scholar
  4. Holota P (1999) Variational methods in geoid determination and function bases. Phys Chem Earth Solid Earth Geodes 24(1):3–14CrossRefGoogle Scholar
  5. Holota P (2000) Direct method in physical geodesy. In: Schwarz KP (ed) Geodesy Beyond 2000, IAG Symposia, vol 121, Springer, New York, pp 163–170Google Scholar
  6. Holota P (2001a) Variational methods in the recovery of the gravity field – Galerkin’s matrix for an ellipsoidal domain. In: Sideris MG (ed) Gravity, Geoid and Geodynamics 2000, IAG Symposia, vol 123, Springer, New York, pp 277–283Google Scholar
  7. Holota P (2001b) Variational methods in the representation of the gravitational potential. Cahiers du Centre Européen de Géodynamique et de Séismologie, vol 20, Luxembourg, 2003, pp 3–11Google Scholar
  8. Holota P (2004) Some topics related to the solution of boundary-value problems in geodesy. In: Sansò F (ed) V Hotine-Marussi symposium on mathematical geodesy, IAG Symposia, vol 127, Springer, New York, pp 189–200Google Scholar
  9. Holota P (2005) Successive approximations in the solution of weakly formulated geodetic boundary-value problem. In: Sansò F (ed) A window on the future of geodesy, IAG Symposia, vol 128, Springer, New York, pp 452–458CrossRefGoogle Scholar
  10. Holota P, Nesvadba O (2008). Model refinements and numerical solutions of weakly formulated boundary-value problems in physical geodesy. In: Xu P, Liu J, Dermanis A (eds) VI Hotine-Marussi symposium on theoretical and computational geodesy, IAG Symposia, vol 132, Springer, pp 320–326Google Scholar
  11. Krarup T (1969) A contribution to the mathematical foundation of physical geodesy. Danish Geodetic Institute, Publ. No. 44, CopenhagenGoogle Scholar
  12. Lemoine FG, Kenyon SC, Factor JK, Trimmer RG, Pavlis NK, Chinn DS, Cox CM, Klosko SM, Luthcke SB, Torrence MH, Wang YM, Williamson RG, Pavlis EC, Rapp RH, Olson TR (1998) The development of the joint NASA GSFC and National Imagery and Mapping Agency (NIMA) geopotential model EGM96. NASA/TP-1998–206861, NASA, GSFC, Greenbelt, MarylandGoogle Scholar
  13. Moritz H (2000) Geodetic Reference System 1980. The Geodesist’s Handbook. J Geodes 74:128–133Google Scholar
  14. Nečas J (1967) Les méthodes directes en théorie des équations elliptiques. Academia, PragueGoogle Scholar
  15. Nesvadba O, Holota P, Klees R (2007) A direct method and its numerical interpretation in the determination of the Earth’s gravity field from terrestrial data. In: Tregoning P, Rizos C (eds) Dynamic Planet, IAG Symposia, vol 130, Chap. 54, Springer, New York, pp 370–376Google Scholar
  16. Neyman YuM (1979) A variational method of physical geodesy. Nedra, Moscow (in Russian)Google Scholar
  17. Rektorys K (1974) Variaicný́ metody v inizenýrských problémech a v problémech matematické fyziky STNL, Praha. (In Czech)Google Scholar
  18. Rektorys K (1977) Variational Methods. Reidel, Dordrecht-BostonCrossRefGoogle Scholar
  19. Sansò F (1986) Statistical methods in physical geodesy. In: Sünkel H (ed) Mathematical and numerical techniques in physical geodesy. Lecture notes in earth sciences, vol 7, Springer, New York, pp 49–155Google Scholar
  20. Sansò F, Sacerdote F (2011) Least Squares, Galerkin and BVPs applied to the determination of global gravity filed models. In: Mertikas SP (ed) Gravity, Geoid and Earth Observations, IAG Symposia, vol 135, Springer, New York, pp 511–517Google Scholar
  21. Tscherning CC (1975) Application of collocation. Determination of a local approximation to the anomalous potential of the Earth using “Exact” astro-gravimetric collocation. In: Brosowski B, Martensen E (eds) Methoden und Verfahren der Mathematischen Physik, Band 14, Bibliographisches Institut AG, Mannheim-Wien-Zürich, pp 83–110Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Research Institute of Geodesy, Topography and CartographyPraha-východCzech Republic
  2. 2.Land Survey OfficePraha 8Czech Republic

Personalised recommendations