Journal of Geodesy

, Volume 90, Issue 8, pp 727–739 | Cite as

Spherical gravitational curvature boundary-value problem

  • Michal ŠprlákEmail author
  • Pavel Novák
Original Article


Values of scalar, vector and second-order tensor parameters of the Earth’s gravitational field have been collected by various sensors in geodesy and geophysics. Such observables have been widely exploited in different parametrization methods for the gravitational field modelling. Moreover, theoretical aspects of these quantities have extensively been studied and well understood. On the other hand, new sensors for observing gravitational curvatures, i.e., components of the third-order gravitational tensor, are currently under development. As the gravitational curvatures represent new types of observables, their exploitation for modelling of the Earth’s gravitational field is a subject of this study. Firstly, the gravitational curvature tensor is decomposed into six parts which are expanded in terms of third-order tensor spherical harmonics. Secondly, gravitational curvature boundary-value problems defined for four combinations of the gravitational curvatures are formulated and solved in spectral and spatial domains. Thirdly, properties of the corresponding sub-integral kernels are investigated. The presented mathematical formulations reveal some important properties of the gravitational curvatures and extend the so-called Meissl scheme, i.e., an important theoretical framework that relates various parameters of the Earth’s gravitational field.


Boundary-value problem Differential operator Gravitational curvature Integral transformation  Tensor spherical harmonics 



The authors were supported by the project No. GA15-08045S of the Czech Science Foundation. Thoughtful and constructive comments of the three anonymous reviewers are gratefully acknowledged. Thanks are also extended to the editor-in-chief J. Kusche and the responsible editor W. Keller for handling our manuscript.

Supplementary material

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  1. Abramowitz M, Stegun IA (1972) Handbook of mathematical functions with formulas, graphs, and mathematical tables. In: Tenth printing, National Bureau of Standards, Department of Commerce, Washington, DCGoogle Scholar
  2. Ardalan AA, Safari A (2005) Global height datum unification: a new approach in gravity potential space. J Geod 79:512–523CrossRefGoogle Scholar
  3. Ardestani VE, Martinec Z (2001) Ellipsoidal Stokes boundary-value problem with ellipsoidal corrections in the boundary condition. Stud Geophys Geod 45:109–126CrossRefGoogle Scholar
  4. Balakin AB, Daishev RA, Murzakhanov ZG, Skochilov AF (1997) Laser-interferometric detector of the first, second and third derivatives of the potential of the Earth gravitational field. Izv Vysshikh Uchebnykh Zaved Seriya Geol Razved 1:101–107Google Scholar
  5. Bjerhammar A, Svensson L (1983) On the geodetic boundary value problem for a fixed boundary surface—a satellite approach. Bull Géod 57:382–393CrossRefGoogle Scholar
  6. 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
  7. Brieden P, Müller J, Flury J, Heinzel G (2010) The mission OPTIMA—novelties and benefit. In: Geotechnologien, science report No. 17, Potsdam, pp 134–139Google Scholar
  8. Brovelli M, Sansó F (1990) Gradiometry: the study of the \(V_{yy}\) component in the BVP approach. Manuscr Geod 15:240–248Google Scholar
  9. Casotto S, Fantino E (2009) Gravitational gradients by tensor analysis with application to spherical coordinates. J Geod 83:621–634CrossRefGoogle Scholar
  10. Chauvenet W (1875) A treatise on plane and spherical trigonometry, 9th edn. JB Lippincott, PhiladelphiaGoogle Scholar
  11. Claessens SJ (2006) Solutions to ellipsoidal boundary value problems for gravity field modelling. PhD. Dissertation, Curtin University of Technology, PerthGoogle Scholar
  12. DiFrancesco D, Meyer T, Christensen A, FitzGerald D (2009) Gravity gradiometry—today and tomorrow. In: 11th SAGA Biennial technical meeting and exhibition, 13–18 September 2009, Swaziland, pp 80–83Google Scholar
  13. ESA (1999) Gravity field and steady-state ocean circulation mission. In: Reports for mission selection, ESA SP-1233(1)—the four candidate earth explorer core missions, ESA Publication Division, ESTEC, NoordwijkGoogle Scholar
  14. Fei Z (2000) Refinements of geodetic boundary value problem solutions. PhD Thesis, Department of Geomatics Engineering, University of CalgaryGoogle Scholar
  15. Freeden W, Gerhards C (2013) Geomathematically oriented potential theory. Pure and applied mathematics, A series of monographs and textbooks, Taylor & Francis, New York 452 pGoogle Scholar
  16. Freeden W, Michel V (2001) Basic aspects of geopotential field approximation from satellite-to-satellite tracking data. Math Methods Appl Sci 24:827–846CrossRefGoogle Scholar
  17. Freeden W, Nutz H (2011) Satellite gravity gradiometry as tensorial inverse problem. Int J Geomath 2:177–218CrossRefGoogle Scholar
  18. Freeden W, Schreiner M (2009) Spherical functions of mathematical geosciences. A scalar, vectorial, and tensorial setup. In: Advances in geophysical and environmental mechanics and mathematics. Springer, BerlinGoogle Scholar
  19. Freeden W, Gervens T, Schreiner M (1994) Tensor spherical harmonics and tensor spherical splines. Manuscr Geod 19:70–100Google Scholar
  20. Freeden W, Glockner O, Thalhammer M (1999) Multiscale gravitational field recovery from GPS-satellite-to-satellite tracking. Stud Geophys Geod 43:229–264CrossRefGoogle Scholar
  21. Freeden W, Michel V, Nutz H (2002) Satellite-to-satellite and satellite gravity gradiometry (advanced techniques for high-resolution geopotential field determination). J Eng Math 43:19–56CrossRefGoogle Scholar
  22. 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
  23. Grafarend EW, Heck B, Knickmeyer EH (1985) The free versus fixed geodetic boundary value problem for different combinations of geodetic observables. Bull Géod 59:11–32CrossRefGoogle Scholar
  24. Haagmans R, de Min E, van Gelderen M (1993) Fast evaluation of convolution integral on the sphere using 1D-FFT and a comparison with existing methods for Stokes integral. Manuscr Geod 18:227–241Google Scholar
  25. Heck B (1989) On the non-linear geodetic boundary value problem for a fixed boundary surface. Bull Géod 63:57–67CrossRefGoogle Scholar
  26. Heck B, Seitz K (2003) Solution of the linearized geodetic boundary value problem for an ellipsoidal boundary to order \(e^3\). J Geod 77:182–192CrossRefGoogle Scholar
  27. Heiskanen WA, Moritz H (1967) Physical geodesy. Freeman, San FranciscoGoogle Scholar
  28. Holota P (1983a) The altimetry gravimetry boundary value problem I: linearization, Friedrich’s inequality. Boll Geod Sci Affini 42:14–32Google Scholar
  29. Holota P (1983b) The altimetry gravimetry boundary value problem II: weak solution, V-ellipticity. Boll Geod Sci Affini 42:70–84Google Scholar
  30. Holota P (1995) Boundary and initial value problems in airborne gravimetry. In: Proceedings of IAG symposium on airborne gravity field determination, IUGG XXI general assembly, Boulder, 2–14 July 1995. Special report No. 60010, Department of Geomatics Engineering, University of Calgary, Calgary, pp 67–71Google Scholar
  31. Holota P, Nesvadba O (2014) Reproducing kernel and Neumann’s function for the exterior of an oblate ellipsoid of revolution: application in gravity field studies. Stud Geophys Geod 58:505–535CrossRefGoogle Scholar
  32. Hotine M (1969) Mathematical geodesy. In: ESSA monograph No. 2, US Department of Commerce, Washington, DCGoogle Scholar
  33. Hörmander L (1976) The boundary problems of physical geodesy. Arch Rational Mech Anal 62:1–52CrossRefGoogle Scholar
  34. Huang J, Vaníček P, Novák P (2000) An alternative algorithm to FFT for the numerical evaluation of Stokes’s integral. Stud Geophys Geod 44:374–380CrossRefGoogle Scholar
  35. Jekeli C (2007) Potential theory and static gravity field of the Earth. In: Schubert G (ed) Treatise on geophysics, vol 3. Elsevier, Oxford, pp 11–42CrossRefGoogle Scholar
  36. Keller W (1996) On the scalar fixed altimetry gravimetry boundary value problem of physical geodesy. J Geod 70:459–469Google Scholar
  37. Keller W, Hirch M (1994) A boundary value approach to downward continuation. Manuscr Geod 19:101–118Google Scholar
  38. Kellogg OD (1929) Foundations of potential theory. Verlag von Julius Springer, Berlin 384 pCrossRefGoogle Scholar
  39. Koch KR (1971) Die geodätische Randwertaufgabe bei bekannter Erdoberfläche. Z Vermess 96:218–224Google Scholar
  40. 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
  41. Kotsiaros S, Olsen N (2012) The geomagnetic field gradient tensor: properties and parametrization in terms of spherical harmonics. Int J Geomath 3:297–314CrossRefGoogle Scholar
  42. Lehmann R (1999) Boundary-value problems in the complex world of geodetic measurements. J Geod 73:491–500CrossRefGoogle Scholar
  43. Lehmann R (2000) Altimetry–gravimetry problems with free vertical datum. J Geod 74:327–334CrossRefGoogle Scholar
  44. Martinec Z (1997) Solution to the Stokes boundary-value problem on an ellipsoid of revolution. Stud Geophys Geod 41:103–129CrossRefGoogle Scholar
  45. Martinec Z (2003) Green’s function solution to spherical gradiometric boundary-value problems. J Geod 77:41–49CrossRefGoogle Scholar
  46. Martinec Z, Grafarend EW (1997) Construction of Green’s function to the Dirichlet boundary-value problem for the Laplace equation on an ellipsoid of revolution. J Geod 71:562–570CrossRefGoogle Scholar
  47. Mather RS (1973) A solution of the geodetic boundary value problem to order \(e^3\). In: Report No. X-592-73-11, Goddard Space Flight Center, GreenbeltGoogle Scholar
  48. Mazurova EM, Yurkina MI (2011) Use of Green’s function for determining the disturbing potential of an ellipsoidal Earth. Stud Geophys Geod 55:455–464CrossRefGoogle Scholar
  49. Meissl P (1971) A study of covariance functions related to the Earth’s disturbing potential. In: Report No. 151, Department of Geodetic Science, The Ohio State University, ColumbusGoogle Scholar
  50. Molodensky MS, Eremeev VF, Yurkina MI (1962) Methods for study of the external gravitational field and figure of the Earth. The Israel Program for Scientific Translations, Department of Commerce, Washington, DCGoogle Scholar
  51. Moritz H (1967) Kinematical geodesy. In: Report No. 92, Department of Geodetic Science, The Ohio State University, ColumbusGoogle Scholar
  52. Moritz H (1989) Advanced physical geodesy, 2nd edn. Herbert Wichmann, Karlsruhe 500 pGoogle Scholar
  53. Novák P, Heck B (2002) Downward continuation and geoid determination based on band-limited airborne gravity data. J Geod 76:269–278CrossRefGoogle Scholar
  54. Pick M, Pícha J, Vyskočil V (1973) Theory of the Earth’s gravity field. Elsevier, Amsterdam 538 pGoogle Scholar
  55. Reigber C, Luehr H, Schwintzer P (2002) CHAMP mission status. Adv Space Res 30:129–134CrossRefGoogle Scholar
  56. Ritter S (1998) The nullfield method for the ellipsoidal Stokes problem. J Geod 72:101–106CrossRefGoogle Scholar
  57. Rosi G, Cacciapuoti L, Sorrentino F, Menchetti M, Prevedelli M, Tino GM (2015) Measurements of the gravity-field curvature by atom interferometry. Phys Rev Lett 114:013001Google Scholar
  58. Rummel R (1975) Downward continuation of gravity information from satellite to satellite tracking or satellite gradiometry in local areas. In: Report No. 221, Department of Geodetic Science, The Ohio State University, ColumbusGoogle Scholar
  59. Rummel R (1997) Spherical spectral properties of the Earth’s gravitational potential and its first and second derivatives. In: Sansó F, Rummel R (eds) Lecture notes in earth sciences, vol 65., Geodetic boundary value problems in view of the one centimeter geoidSpringer, Berlin, pp 359–404Google Scholar
  60. Rummel R, van Gelderen M (1995) Meissl scheme—spectral characteristics of physical geodesy. Manuscr Geod 20:379–385Google Scholar
  61. Rummel R, Teunissen P, van Gelderen M (1989) Uniquely and overdetermined geodetic boundary value problems by least squares. Bull Géod 63:1–33Google Scholar
  62. Sacerdote F, Sansó F (1983) A contribution to the analysis of altimetry gravimetry problems. Bull Géod 57:183–201CrossRefGoogle Scholar
  63. Safari A, Ardalan AA, Grafarend EW (2005) A new ellipsoidal gravimetric, satellite altimetry and astronomic boundary value problem, a case study: The geoid of Iran. J Geodyn 39:545–568CrossRefGoogle Scholar
  64. Sansó F (1978) Molodensky’s problem in gravity space: a review of the first results. Bull Géod 52:59–70CrossRefGoogle Scholar
  65. Sansó F (1995) The long road from the measurements to boundary value problems of physical geodesy. Manuscr Geod 20:326–344Google Scholar
  66. Schwarz KP, Li Z (1997) An introduction to airborne gravimetry and its boundary value problems. In: Sansó F, Rummel R (eds) Lecture notes in earth sciences, vol 65., Geodetic boundary value problems in view of the one centimeter geoidSpringer, Berlin, pp 312–358Google Scholar
  67. Simmonds JG (1994) A brief on tensor analysis, 2nd edn., Undergraduate texts in mathematics, Springer, New York 112 pGoogle Scholar
  68. Sjöberg LE (2003) Ellipsoidal corrections to order \(e^2\) of geopotential coefficients and Stokes’ formula. J Geod 77:139–147CrossRefGoogle Scholar
  69. Stokes GG (1849) On the variation of gravity on the surface of the Earth. Trans Camb Philos Soc 8:672–695Google Scholar
  70. Svensson SL (1983) Solution of the altimetry–gravimetry problem. Bull Géod 57:332–353CrossRefGoogle Scholar
  71. Tapley BD, Bettadpur S, Watkins M, Reigber C (2004) The gravity recovery and climate experiment: mission overview and early results. Geophys Res Lett 31:L09607Google Scholar
  72. Thong NC (1993) Untersuchung zur Lösungen der fixen gravimetrischen Randwertprobleme mittels sphäroidaler und Greenscher Funktionen. In: Deutsche Geodätische Kommission, Reihe C, Nr. 399, MünchenGoogle Scholar
  73. 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 international gravity and geoid commission, 26–30 August 2002. Ziti Publishing, Thessaloniki, pp 102–107Google Scholar
  74. Tóth G (2005) The gradiometric-geodynamic boundary value problem. In: Jekeli C, Bastos L, Fernandes L (eds) Gravity, geoid and space missions, IAG symposia, vol 129. Springer, Berlin, pp 352–357CrossRefGoogle Scholar
  75. van Gelderen M, Rummel R (2001) The solution of the general geodetic boundary value problem by least squares. J Geod 75:1–11CrossRefGoogle Scholar
  76. Witsch KJ (1985) On a free boundary value problem of physical geodesy, I (uniqueness). Math Methods Appl Sci 7:269–289CrossRefGoogle Scholar
  77. Witsch KJ (1986) On a free boundary value problem of physical geodesy, II (existence). Math Methods Appl Sci 8:1–22CrossRefGoogle Scholar
  78. Yu J, Jekeli C, Zhu M (2003) Analytical solutions of the Dirichlet and Neumann boundary-value problems with an ellipsoidal boundary. J Geod 76:653–667CrossRefGoogle Scholar
  79. Zhu Z (1981) The Stokes problem for the ellipsoid using ellipsoidal kernels. In: Report No. 319, Department of Geodetic Science and Surveying, The Ohio State University, ColumbusGoogle Scholar

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Authors and Affiliations

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

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