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Journal of Geodesy

, Volume 88, Issue 2, pp 179–197 | Cite as

Spherical integral formulas for upward/downward continuation of gravitational gradients onto gravitational gradients

  • Michal ŠprlákEmail author
  • Josef Sebera
  • Miloš Val’ko
  • Pavel Novák
Original Article

Abstract

New integral formulas for upward/downward continuation of gravitational gradients onto gravitational gradients are derived in this article. They provide more options for continuation of gravitational gradient combinations and extend available mathematical apparatus formulated for this purpose up to now. The starting point represents the analytical solution of the spherical gradiometric boundary value problem in the spatial domain. Applying corresponding differential operators on the analytical solution of the spherical gradiometric boundary value problem, a total of 18 integral formulas are provided. Spatial and spectral forms of isotropic kernels are given and their behaviour for parameters of a GOCE-like satellite is investigated. Correctness of the new integral formulas and the isotropic kernels is tested in a closed-loop simulation. The derived integral formulas and the isotropic kernels form a theoretical basis for validation purposes and geophysical applications of satellite gradiometric data as provided currently by the GOCE mission. They also extend the well-known Meissl scheme.

Keywords

GOCE Gravitational gradient Integral equation  Meissl scheme Satellite gradiometry 

Notes

Acknowledgments

This work is supported in the framework of ESA’s project GOCE-GDC AO/1-6367/10/NL/AF. M. Šprlák and J. Sebera were supported by the project EXLIZ - CZ.1.07/2.3.00/30.0013, which is co-financed by the European Social Fund and the state budget of the Czech Republic. P. Novák was supported by the project 209/12/1929 of the Czech Science Foundation. Constructive comments of the three anonymous reviewers are gratefully acknowledged. Thanks are extended to the editor-in-chief Prof. Roland Klees and the responsible editor Prof. Wolfgang Keller for handling our manuscript.

References

  1. Arfken G (1968) Mathematical methods for physicists. Academic Press, New YorkGoogle Scholar
  2. Braitenberg C, Mariani P, Ebbing J, Šprlák M (2011) The enigmatic Chad lineament revisited with global gravity and gravity gradient fields. In: van Hinsbergen DJJ, Buiter SJH, Torsvik TH, Gaina C, Webb SJ (eds) The formation and evolution of Africa: a synopsis of 3.8 Ga of earth history, vol 357. Geological Society, Special Publications, London, pp 329–341Google Scholar
  3. Bölling C, Grafarend EW (2005) Ellipsoidal spectral properties of the Earth’s gravitational potential and its first and second derivatives. J Geodesy 79:300–330CrossRefGoogle Scholar
  4. Denker H (2003) Computation of gravity gradients for Europe for calibration/validation of GOCE data. In: Tziavos IN (ed) Gravity and Geoid 2002, 3rd Meeting of the IGGC, Ziti Editions, pp 287–292Google Scholar
  5. EGG-C (2010) GOCE L2 product data handbook. Issue 4, Revision 3, GO-MA-HPF-GS-0110. The European GOCE Gravity Consortium EGG-CGoogle Scholar
  6. ESA (1999) Gravity field and steady-state ocean circulation mission. ESA SP-1233(1), Report for mission selection of the four candidate earth explorer missions, ESA Publication DivisionGoogle Scholar
  7. Eshagh M (2008) Non-singular expressions for the vector and the gradient tensor of gravitation in a geocentric spherical frame. Comput Geosci 34:1762–1768CrossRefGoogle Scholar
  8. Eshagh M (2011a) On integral approach to regional gravity field modelling from satellite gradiometric data. Acta Geophysica 59:29–54CrossRefGoogle Scholar
  9. Eshagh M (2011b) The effect of spatial truncation error on the integral inversion of satellite gravity gradiometry data. Adv Space Res 47:1238–1247CrossRefGoogle Scholar
  10. Grafarend EW (2001) The spherical horizontal and spherical vertical boundary value problem—vertical deflections and geoid undulations—the completed Meissl diagram. J Geodesy 75:363–390CrossRefGoogle Scholar
  11. Haagmans R, Prijatna K, Omang OCD (2003) An alternative concept for validation of GOCE gradiometry results based on regional gravity. In: Tziavos IN (ed) Gravity and Geoid 2002, 3rd Meeting of the IGGC. Ziti Editions, pp 281–286Google Scholar
  12. Heck B (1979) Zur lokalen Geoidbestimmung aus terrestrischen Messungen vertikaler Schweregradienten. Deutsche Geodätische Kommission, Reihe C, No. 259, München, GermanyGoogle Scholar
  13. Heiskanen WA, Moritz H (1967) Physical geodesy. Freeman and Co., San FranciscoGoogle Scholar
  14. Hirt C, Featherstone WE, Claessens SJ (2011) On the accurate numerical evaluation of geodetic convolution integrals. J Geodesy 85:519–538CrossRefGoogle Scholar
  15. Ilk KH (1983) Ein Beitrag zur Dynamik ausgedehnter Körper - Gravitationswechelswirknung. Deutsche Geodätische Kommission, Reihe C, No. 288, München, GermanyGoogle Scholar
  16. Janák J, Fukuda Y, Xu P (2009) Application of the GOCE data for regional gravity field modeling. Earth Planets Space 61:835–843Google Scholar
  17. Kern M, Haagmans R (2005) Determination of gravity gradients from terrestrial gravity data for calibration and validation of gradiometric data. In: Jekeli C, Bastos L, Fernandes L (eds) Gravity, Geoid and Space Missions, IAG Symposia, vol 129. Springer, Berlin, pp 95–100Google Scholar
  18. Kern M, Schwarz KP, Sneeuw N (2003) A study on the combination of satellite, airborne, and terrestrial gravity data. J Geodesy 77:217–225Google Scholar
  19. Koop R (1993) Global gravity field modelling using satellite gravity gradiometry. Publications on Geodesy, Netherlands Geodetic Commission, No. 38, Delft, The NetherlandsGoogle Scholar
  20. Li J (2002) A formula for computing the gravity disturbance from the second radial derivative of the disturbing potential. J Geodesy 76:226–231 Google Scholar
  21. Li J (2005) Integral formulas for computing the disturbing potential, gravity anomaly and the deflection of the vertical from the second-order radial gradient of the disturbing potential. J Geodesy 79:64–70Google Scholar
  22. Martinec Z (2003) Green’s function solution to spherical gradiometric boundary-value problems. J Geodesy 77:41–49Google Scholar
  23. Moritz H (2000) Geodetic reference system 1980. J Geodesy 74:128–133CrossRefGoogle Scholar
  24. 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):38Google Scholar
  25. Petrovskaya MS, Zielinski JB (1997) Determination of the global and regional gravitational fields from satellite and balloon gradiometry missions. Adv Space Res 19:1723–1728CrossRefGoogle Scholar
  26. Pick M, Pícha J, Vyskočil V (1973) Theory of the Earth’s gravity field. Elsevier, AmsterdamGoogle Scholar
  27. Reed GB (1973) Application of kinematical geodesy for determining the short wavelength components of the gravity field by satellite gradiometry. Report No. 201, Ohio State University, Department of Geodetic Sciences, Columbus, USAGoogle Scholar
  28. Rummel R, van Gelderen M (1995) Meissl scheme—spectral characteristics of physical geodesy. Manuscripta Geodaetica 20:379–385Google Scholar
  29. Thalhammer M (1995) Regionale Gravitationsfeldbestimmung mit zukünftigen Satellitenmissionen (SST und Gradiometrie). Deutsche Geodätische Kommission, Reihe C, Nr. 437, München, GermanyGoogle Scholar
  30. Tóth G (2003) The Eötvös spherical horizontal gradiometric boundary value problem - gravity anomalies from gravity gradients of the torsion balance. In: Tziavos IN (ed) Gravity and Geoid 2002, 3rd Meeting of the IGGC, Ziti Editions, pp. 102–107Google Scholar
  31. Tóth G, Rózsa S, Ádám J, Tziavos IN (2002) Gravity field modeling by torsion balance data—a case study in Hungary. In: Ádám J, Schwarz KP (eds) Vistas for Geodesy in the New Millenium, IAG Symposia, vol 125. Springer, Berlin, pp 193–198Google Scholar
  32. Tóth G, Ádám J, Földváry L, Tziavos IN, Denker H (2005) Calibration/validation of GOCE data by terrestrial torsion balance observations. In: Sansó F (ed) A Window on the Future Geodesy, IAG Symposia, vol 128. Springer, Berlin, pp 214–219CrossRefGoogle Scholar
  33. Tóth G, Földváry L, Tziavos IN, Ádám J (2006) Upward/downward continuation of gravity gradients for precise geoid determination. Acta Geodaetica et Geophysica Hungarica 41:21–30CrossRefGoogle Scholar
  34. van Gelderen M, Rummel R (2001) The solution of the general geodetic boundary value problem by least squares. J Geodesy 75:1–11CrossRefGoogle Scholar
  35. van Gelderen M, Rummel R (2002) Corrections to “The solution of the general geodetic boundary value problem by least squares”. J Geodesy 76:121–122CrossRefGoogle Scholar
  36. Winch DE, Roberts PH (1995) Derivatives of addition theorem for Legendre functions. J Aust Math Soc Ser B Appl Math 37:212–234CrossRefGoogle Scholar
  37. Wolf KI (2007) Kombination globaler Potentialmodelle mit terrestrische Schweredaten für die Berechnung der zweiten Ableitungen des Gravitationspotentials in Satelitenbahnhöhe. Deutsche Geodätische Kommission, Reihe C, No. 603, München, GermanyGoogle Scholar
  38. Wolf KI, Denker H (2005) Upward continuation of ground data for GOCE calibration. In: Jekeli C, Bastos L, Fernandes L (eds) Gravity, Geoid and Space Missions, IAG Symposia, vol 129. Springer, Berlin, pp 60–65CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Michal Šprlák
    • 1
    Email author
  • Josef Sebera
    • 1
  • Miloš Val’ko
    • 1
  • Pavel Novák
    • 1
  1. 1.New Technologies for the Information Society, Faculty of Applied SciencesUniversity of West BohemiaPlzeňCzech Republic

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