Journal of Geodesy

, Volume 88, Issue 4, pp 377–390 | Cite as

Integral transformations of gradiometric data onto a GRACE type of observable

  • Michal ŠprlákEmail author
  • Pavel Novák
Original Article


Integral transformations of gravitational gradients onto a Gravity Recovery And Climate Experiment (GRACE) type of observable are derived in this article. The gravitational gradients represent components of the gravitational tensor in the local north-oriented frame. The GRACE type of observable corresponds to a difference between two gravitational vectors as projected onto the line of sight between the two GRACE satellites. In total, three integral transformations relating vertical–vertical, vertical–horizontal and horizontal–horizontal gravitational gradients with the GRACE type of observable are provided. Spectral and closed forms of corresponding isotropic kernels are derived for each transformation. Special cases show that the integral transformations are general and relate gravitational gradients to many other quantities of the gravitational field, such as the gravitational vector, and its radial and tangential components. Correctness of the mathematical derivations is validated in a closed-loop simulation using synthetic data.


Gravitational field Upward/downward continuation  Satellite-to-satellite tracking Satellite gradiometry Gravitational gradients GRACE type of observable 



Pavel Novák was supported by the project 209/12/1929 of the Czech Science Foundation. Michal Šprlák was 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. Dr. Josef Sebera is appreciated for reading the draft version of the manuscript. Constructive comments of three anonymous reviewers are gratefully acknowledged.


  1. Ardalan AA, Grafarend EW (2004) High-resolution regional geoid computation without applying Stokes’s formula: a case study of the Iranian geoid. J Geod 78:138–156Google Scholar
  2. Bell RE, Anderson RN, Pratson LF (1997) Gravity gradiometry resurfaces. Lead Edge 16:55–60CrossRefGoogle Scholar
  3. Bölling C, Grafarend EW (2005) Ellipsoidal spectral properties of the Earth’s gravitational potential and its first and second derivatives. J Geod 79:300–330CrossRefGoogle Scholar
  4. Case K, Kuizinga G, Wu S-C (2010) GRACE level 1B data product user handbook. Version 1.3, March 24 2010, JPL D-22027, Jet Propulsion Laboratory, California Institute of Technology, USAGoogle Scholar
  5. 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
  6. 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
  7. 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
  8. Eshagh M (2009) Alternative expressions for gravity gradients in the local north-oriented frame and tensor spherical harmonics. Acta Geophysica 58:215–243CrossRefGoogle Scholar
  9. Eshagh M (2011) On integral approach to regional gravity field modelling from satellite gradiometric data. Acta Geophysica 59:29–54CrossRefGoogle Scholar
  10. Fuchs MJ, Bouman J (2011) Rotation of GOCE gravity gradients to local frames. Geophys J Int 187:743–753CrossRefGoogle Scholar
  11. Garcia RV (2002) Local geoid determination from GRACE mission. Report No. 460, Department of Civil and Environmental Engineering and Geodetic Science, The Ohio State University, Columbus, OH, USAGoogle Scholar
  12. Grafarend EW (2001) The spherical horizontal and spherical vertical boundary value problem—vertical deflections and geoid undulations: the completed Meissl diagram. J Geod 75:363–390CrossRefGoogle Scholar
  13. 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
  14. Heck B (1979) Zur lokalen Geoidbestimmung aus terrestrischen Messungen vertikaler Schweregradienten. Deutsche Geodätische Kommission, Reihe C, Nr. 259, München, GermanyGoogle Scholar
  15. Heiskanen WA, Moritz H (1967) Physical geodesy. Freeman and Co., San FranciscoGoogle Scholar
  16. Hirt C, Featherstone WE, Claessens SJ (2011) On the accurate numerical evaluation of geodetic convolution integrals. J Geod 85:519–538CrossRefGoogle Scholar
  17. Hotine M (1969) Mathematical geodesy. ESSA Monograph No. 2, US Department of Commerce, Washington, DC, USAGoogle Scholar
  18. Ilk KH (1983) Ein Beitrag zur Dynamik ausgedehnter Körper—Gravitationswechelswirknung. Deutsche Geodätische Kommission, Reihe C, Nr. 288, München, GermanyGoogle Scholar
  19. Janák J, Fukuda Y, Xu P (2009) Application of the GOCE data for regional gravity field modeling. Earth Planets Space 61:835–843Google Scholar
  20. Jekeli C (1993) A review of gravity gradiometer survey system data analyses. Geophysics 58:508–514CrossRefGoogle Scholar
  21. Keller W, Sharifi MA (2005) Satellite gradiometry using a satellite pair. J Geod 78:544–557CrossRefGoogle Scholar
  22. 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 series, vol 129. Springer, Berlin, pp 95–100Google Scholar
  23. Kern M, Schwarz KP, Sneeuw N (2003) A study on the combination of satellite, airborne, and terrestrial gravity data. J Geod 77:217–225CrossRefGoogle Scholar
  24. Kernighan BW, Ritchie DM (1988) The C programming language, 2nd edn. Prentice Hall, USAGoogle Scholar
  25. Li J (2002) A formula for computing the gravity disturbance from the second radial derivative of the disturbing potential. J Geod 76:226–231CrossRefGoogle Scholar
  26. 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 Geod 79:64–70CrossRefGoogle Scholar
  27. Martinec Z (2003) Green’s function solution to spherical gradiometric boundary-value problems. J Geod 77:41–49CrossRefGoogle Scholar
  28. Meissl P (1971) A study of covariance functions related to the Earth’s disturbing potential. Report No. 151, Department of Geodetic Science, The Ohio State University, Columbus, OH, USAGoogle Scholar
  29. Moritz H (2000) Geodetic reference system 1980. J Geod 74:128–133CrossRefGoogle Scholar
  30. Novák P (2007) Integral inversion of SST data of type GRACE. Studia Geophysica et Geodaetica 51:351–367CrossRefGoogle Scholar
  31. Novák P, Austen G, Sharifi MA, Grafarend EW (2006) Mapping Earth’s gravitation using GRACE data. In: Flury J, Rummel R, Reigber C, Rothacher M, Boedecker G, Schreiber U (eds) Observation of the earth system from space. Springer, Berlin, pp 149–164CrossRefGoogle Scholar
  32. 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, 38 ppGoogle Scholar
  33. 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
  34. Pick M, Pícha J, Vyskočil V (1973) Theory of the Earth’s gravity field. Elsevier, AmsterdamGoogle Scholar
  35. Reed GB (1973) Application of kinematical geodesy for determining the short wavelength components of the gravity field by satellite gradiometry. Report No. 201, Department of Geodetic Sciences, The Ohio State University, Columbus, USAGoogle Scholar
  36. Reigber C, Lühr H, Schwintzer P (2002) CHAMP mission status. Adv Space Res 30:129–134CrossRefGoogle Scholar
  37. Rummel R, van Gelderen M (1995) Meissl scheme—spectral characteristics of physical geodesy. Manuscripta Geodaetica 20:379–385Google Scholar
  38. Stokes GG (1849) On the variation of gravity on the surface of the Earth. Trans Camb Philos Soc 8:672–695Google Scholar
  39. Tapley BD, Bettadpur S, Watkins M, Reigber C (2004) The gravity recovery and climate experiment: mission overview and early results. Geophys Res Lett 31:L09607CrossRefGoogle Scholar
  40. Thalhammer M (1995) Regionale Gravitationsfeldbestimmung mit zukünftigen Satellitenmissionen (SST und Gradiometrie). Deutsche Geodätische Kommission, Reihe C, Nr. 437, München, GermanyGoogle Scholar
  41. 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
  42. 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 series, vol 125. Springer, Berlin, pp 193–198 Google Scholar
  43. 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 series, vol 128. Springer, Berlin, pp 214–219Google Scholar
  44. 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
  45. van Gelderen M, Rummel R (2001) The solution of the general geodetic boundary value problem by least squares. J Geod 75:1–11CrossRefGoogle Scholar
  46. van Gelderen M, Rummel R (2002) Corrections to “The solution of the general geodetic boundary value problem by least squares”. J Geod 76:121–122CrossRefGoogle Scholar
  47. Vening-Meinesz FA (1928) A formula expressing the deflection of the plumb-lines in the gravity anomalies and some formulae for the gravity field and the gravity potential outside the geoid. Koninklijke Nederlandsche Akademie van Wetenschappen 31:315–331Google Scholar
  48. Winch DE, Roberts PH (1995) Derivatives of addition theorem for Legendre functions. J Aust Math Soc Ser B Appl Math 37:212–234CrossRefGoogle Scholar
  49. Wolf KI (2007) Kombination globaler Potentialmodelle mit terrestrische Schweredaten für die Berechnung der zweiten Ableitungen des Gravitationspotentials in Satelitenbahnhöhe. Deutsche Geodätische Kommission, Reihe C, Nr. 603, München, GermanyGoogle Scholar
  50. 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 series, vol 129. Springer, Berlin, pp 60–65Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.New Technologies for the Information Society, Faculty of Applied SciencesUniversity of West BohemiaPlzeňCzech Republic

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