Up and Down Through the Gravity Field

  • F. SansóEmail author
  • M. Capponi
  • D. Sampietro
Living reference work entry
Part of the Springer Reference Naturwissenschaften book series (SRN)


Die Kenntnis des Schwerefeldes hat weitreichende Anwendungen in den Geo- wissenschaften, insbesondere in Geodsie und Geophysik. Unser Anliegen in diesem Beitrag ist die Beschreibung von Eigenschaften zur Fortpflanzung des Potentials oder seiner relevanten Funktionale nach oben und nach unten. Die Fortpflanzung des Potentials nach oben (“upward continuation”) ist stets ein wohlgestelltes Problem. “Downward Continuation” ist stets ein schlechtgestelltes Problem, nicht nur wegen der numerischen Instabilitten, sondern vor allem wegen der Nichteindeutigkeit der Bestimmung von Massenschichtung aus Potentialwerten.

Als Konsequenz fokussiert sich der Beitrag auf neuere Resultate aus dem Bereich geodätischer Randwertprobleme und zum anderen auf das inverse Gravimetrieproblem. Dabei machen wir den Versuch, die Bedeutung von mathematischer Theorie für numerische Anwendungen heraus zu streichen.

Das Paper ist vom mathematischem Anspruch her schlicht gehalten. Dazu bedienen wir uns oftmals der Rückführung auf sphärische Beispiele. Der größte Teil des Materials ist bereits in der Literatur vorhanden, bis auf Teile für die globalen Modelle und das inverse Gravimetrieproblem für Schichtungen.


Upward continuation Downward continuation Geodetic boundary value problem Inverse gravimetric problem 


The knowledge of the gravity field has widespread applications in geosciences, in particular in Geodesy and Geophysics. The point of view of the paper is to describe the properties of the propagation of the potential, or of its relevant functionals, while moving upward or downward. The upward propagation is always a properly posed problem, in fact a smoothing and somehow related to the Newton integral and to the solution of boundary value problems (BVP). The downward propagation is always improperly posed, not only due to its intrinsic numerical instability but also because of the non-uniqueness that is created as soon as we penetrate layers of unknown mass density.

So the paper focuses on recent results on the Geodetic Boundary Value Problems on the one side and on the inverse gravimetric problem on the other, trying to highlight the significance of mathematical theory to numerical applications. Hence, on the one hand we examine the application of the BVP theory to the construction of global gravity models, on the other hand the inverse gravimetric problem is studied for layers together with proper regularization techniques.

The level of the mathematics employed in the paper is willingly kept at medium level, often recursing to spherical examples in support to the theory. Most of the material is already present in literature but for a few parts concerning global models and the inverse gravimetric problem for layers.


  1. 1.
    Andersen, O.B.: Marine gravity and geoid from satellite altimetry. In: Sanso’, F., Sideris, M. (eds.) Geoid Determination: Theory and Methods, Chapter 9. LNESS. Springer, Berlin/Heidelberg (2013)Google Scholar
  2. 2.
    Ballani, L., Stromeyer, D.: The inverse gravimetric problem: a Hilbert space approach. In: Proceedings of International Symposium on Figure of the Earth, the Moon, and Other Planets, Prague, pp. 359–373 (1982)Google Scholar
  3. 3.
    Barzaghi, R., Sansó, F.: Remarks on the inverse gravimetric problem. Bollettino di geodesia e scienze affini 45(2), 203–216 (1986)Google Scholar
  4. 4.
    Bucha, B., Hirt, C., Kuhn, M.: Runge Krarup type gravity field solutions to avoid divergence in traditional external spherical harmonic modelling. J. Geod. (2018, under review)Google Scholar
  5. 5.
    Flechtner, F., Sneeuw, N., Schuh, W.D.: Observation of the System Earth from Space: CHAMP, GRACE, GOCE and Future Missions. Springer, Berlin/Heidelberg (2014)CrossRefGoogle Scholar
  6. 6.
    Freeden, W., Michel, V.: Multiscale Potential Theory: With Applications to Geoscience. Springer, Berlin/Heidelberg (2012)Google Scholar
  7. 7.
    Freeden, W., Nashed, M.Z.: Operator-theoretic and regularization approaches to ill-posed problems. Int. J. Geomath. Vol 9(1) pp. 1–115 (2018)CrossRefGoogle Scholar
  8. 8.
    Hörmander, L.: The boundary problems of physical geodesy. Arch. Ration. Mech. Anal. 62(1), 1–52 (1976)CrossRefGoogle Scholar
  9. 9.
    Isakov, V.: Inverse Source Problems, vol. 34. A.M.S., Providence (1990)Google Scholar
  10. 10.
    Krarup, T.: Letters on Molodewsky’s problem. I the single Molodewsky’s problem. In: Borre, K.(ed.) Mathematical Foundation of Geodesy. Selected papers of Torben Krarup. Springer, Berlin/Heidelberg (2006)Google Scholar
  11. 11.
    Krarup, T.: “On Potential Theory” in Mathematical Foundation of Geodesy: Selected Papers of Torben Krarup. Selected Papers of Torben Krarup. Springer, Berlin/Heidelberg (2006)Google Scholar
  12. 12.
    Lauricella. G.: Sulla distribuzione di massa all’interno dei pianeti. Rend. Acc. Lincei. XII, 18–21 (1912)Google Scholar
  13. 13.
    Leliwa-Ropystynski, J.: Distribution of selected physical parameters in the Earth and planetary interiors. In: Leliwa-Ropystynski, J., Teisseyre, R. (eds.) Constitution of the Earth’s Interior. Elsevier, New York (1984)Google Scholar
  14. 14.
    McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge/New York (2000)Google Scholar
  15. 15.
    Migliaccio, F., Reguzzoni, M., Gatti, A., Sansò, F., Herceg, M.: A GOCE-only global gravity field model by the space-wise approach. In: Proceedings of the 4th International GOCE User Workshop. ESA SP-696 (2011)Google Scholar
  16. 16.
    Miranda, C.: Partial Differential Equations of Elliptic Type. Springer, Berlin/Heidelberg (1970)Google Scholar
  17. 17.
    Moritz, H.: The Figure of the Earth: Theoretical Geodesy and the Earth’s Interior. H. Wichmann, Karlsruhe (1990)Google Scholar
  18. 18.
    Moritz, H., Heiskanen, W.A.: Physical Geodesy. W.H. Freeman, San Francisco (1967)Google Scholar
  19. 19.
    Pavlis, N.: Global gravitational models. In: Sanso’, F., Sideris, M. (eds.) Geoid Determination: Theory and Methods, Chapter 9. LNESS. Springer, Berlin/Heidelberg (2013)Google Scholar
  20. 20.
    Pizzetti, P.: Intorno alle possibili distribuzioni della massa nell’interno della Terra. Annali di Mat. Milano XVII, 225–258 (1910)CrossRefGoogle Scholar
  21. 21.
    Richter, M.: Inverse Problems. Lecture Notes in Geosystems Mathematics and Computing. Birkhäusen, Cham (2016)CrossRefGoogle Scholar
  22. 22.
    Riesz, F., Nagy, B.S.: Lecons d’Analyse Fonctionnelle. Gauthier-Villars, Paris (1968)Google Scholar
  23. 23.
    Sansó, F.: Internal Collocation. Memorie dell’Accademia dei Lincei, vol. 16, Rome (1980)Google Scholar
  24. 24.
    Sansó, F.: Theory on GBVP’s applied to the analysis of altimetric data. In: Rummel, R., Sanso’, F. (eds.) Satellite Altimetry in Geodesy and Oceanography. LNESS, vol. 50. Springer, Berlin/Heidelberg (1993)Google Scholar
  25. 25.
    Sansò, F.: The long road from measurements to boundary value problems in physical geodesy. Manuscripta geodaetica 20(5), 326–344 (1995)Google Scholar
  26. 26.
    Sansó, F.: The Analysis of the GBVP: State and Perspectives: Handbook of Math-Geodesy, pp. 463–493. Springer, Berlin/Heidelberg (2017)Google Scholar
  27. 27.
    Sansò, F., Sideris, M.G.: Geoid Determination: Theory and Methods. LNESS. Springer, Berlin/Heidelberg (2013)CrossRefGoogle Scholar
  28. 28.
    Sansò, F., Sideris, M.G.: Geodetic Boundary Value Problem: The Equivalence Between Molodensky’s and Helmert’s Solutions. Springer, Cham, Switzerland (2016)Google Scholar
  29. 29.
    Sansó, F.: On the regular decomposition of the inverse gravimetric problem in non-L 2 spaces. Int. J. Geomath. 5(1), 33–61 (2014)CrossRefGoogle Scholar
  30. 30.
    Vaníc̆ek, P., Kleusberg, A.: What an external gravitational potential can really tell us about mass distribution. Boll. Geof. Teor. ed Appl. 102(108), 243–250 (1985)Google Scholar
  31. 31.
    Wermer, J.: Potential Theory. Lecture Notes in March 2008. Springer, Berlin/Heidelberg (1974)Google Scholar
  32. 32.
    Yosida, K.: Functional Analysis. Springer, Berlin/Heidelberg (1995)CrossRefGoogle Scholar
  33. 33.
    Zidarov, D.P.: Inverse Gravimetric Problem in Geoprospecting and Geodesy. Public House of Bulg. Ac. of Sc. Elsevier, Sofia (1990)Google Scholar

Copyright information

© Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Civil and Environmental EngineeringPolitecnico di MilanoMilanItaly
  2. 2.Department of Civil, Constructional and Environmental EngineeringUniversità di Roma La SapienzaRomeItaly
  3. 3.Department of Civil and Environmental EngineeringPolitecnico di MilanoMilanItaly
  4. 4.Geomatics Research & Development s.r.l.ComoItaly

Section editors and affiliations

  • Willi Freeden
    • 1
  1. 1.Geomathematics GroupUniversity of KaiserslauternKaiserslauternGermany

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