Surveys in Geophysics

, Volume 37, Issue 3, pp 681–700 | Cite as

Spherical Harmonic Analysis of Gravitational Curvatures and Its Implications for Future Satellite Missions



In this study we assume that a gravitational curvature tensor, i.e. a tensor of third-order directional derivatives of the Earth’s gravitational potential, is observable at satellite altitudes. Such a tensor is composed of ten different components, i.e. gravitational curvatures, which may be combined into vertical–vertical–vertical, vertical–vertical–horizontal, vertical–horizontal–horizontal and horizontal–horizontal-horizontal gravitational curvatures. Firstly, we study spectral properties of the gravitational curvatures. Secondly, we derive new quadrature formulas for the spherical harmonic analysis of the four gravitational curvatures and provide their corresponding analytical error models. Thirdly, requirements for an instrument that would eventually observe gravitational curvatures by differential accelerometry are investigated. The results reveal that measuring third-order directional derivatives of the gravitational potential imposes very high requirements on the accuracy of deployed accelerometers which are beyond the limits of currently available sensors. For example, for orbital parameters and performance similar to those of the GOCE mission, observing third-order directional derivatives requires accelerometers with the noise level of \({\sim}10^{-17}\,\hbox {m}\,\hbox {s}^{-2}\) Hz\(^{-1/2}\).


Differential accelerometry Earth’s gravitational field Gravitational curvature Gravitational gradient Spherical harmonic analysis 



The authors were supported by the Project No. GA15-08045S of the Czech Science Foundation. Thoughtful and constructive comments of two anonymous reviewers are gratefully acknowledged. Thanks are also extended to the editor-in-chief Dr. Michael J. Rycroft for handling our manuscript.


  1. Armano M, Benedetti M, Bogenstahl J, Bortoluzzi D, Bosetti P, Brandt N, Cavalleri A, Ciani G, Cristofolini I, Cruise AM, Danzmann K, Diepholz I, Dixon G, Dolesi R, Fauste J, Ferraioli L, Fertin D, Fichter W, Freschi M, García A, García C, Grynagier A, Guzmán F, Fitzsimons F, Heinzel G, Hewitson M, Hollington D, Hough J, Hueller M, Hoyland D, Jennrich O, Johlander B, Killow C, Lobo A, Mance D, Mateos I, McNamara PW, Monsky A, Nicolini D, Nicolodi D, Nofrarias M, Perreur-Lloyd M, Plagnol E, Racca GD, Ramos-Castro J, Robertson D, Sanjuan J, Schulte MO, Shaul DNA, Smit M, Stagnaro L, Steier F, Sumner TJ, Tateo N, Tombolato D, Vischer G, Vitale S, Wanner G, Ward H, Waschke S, Wand V, Wass P, Weber WJ, Ziegler T, Zweifel P (2009) LISA pathfinder: the experiment and the route to LISA. Class Quantum Gravity 26:094001CrossRefGoogle Scholar
  2. 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. Izvestiya vysshikh uchebnykh zavedenii, seriya Geologiya i Razvedka 1:101–107Google 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. Brieden P, Müller J, Flury J, Heinzel G (2010) The mission OPTIMA—novelties and benefit. Geotechnologien, Science Report no. 17, Potsdam, Germany, pp 134–139Google Scholar
  5. Cai L, Zhou Z, Luo J (2015) Analytical method for error analysis of high–low satellite-to-satellite tracking missions. Stud Geophys Geod 59:380–393CrossRefGoogle Scholar
  6. Carraz O, Siemes C, Massotti L, Haagmans R, Silvestrin P (2014) A spaceborne gravity gradiometer concept based on cold atom interferometers for measuring Earth’s gravity field. Microgravity Sci Technol 26:139–145CrossRefGoogle Scholar
  7. Casotto S, Fantino E (2009) Gravitational gradients by tensor analysis with application to spherical coordinates. J Geod 83:621–634CrossRefGoogle Scholar
  8. Cesare S, Aguirre M, Allasio A, Leone B, Massotti L, Muzi D, Silvestrin P (2010) The measurement of Earth’s gravity field after the GOCE mission. Acta Astronaut 67:702–712CrossRefGoogle Scholar
  9. Colombo OL (1989) Advanced techniques for high-resolution mapping of the gravitational field. In: Sansó F, Rummel R (eds) Theory of satellite geodesy and gravity field determination. Lecture notes in Earth Sciences, vol 25, Springer, Berlin, pp 335–369Google Scholar
  10. DiFrancesco D, Meyer T, Christensen A, FitzGerald D (2009) Gravity gradiometry—today and tomorrow. In: 11th SAGA biennial technical meeting and exhibition, September 13–18, 2009, Swaziland, South Africa, pp 80–83Google Scholar
  11. Elsaka B, Raimondo JC, Brieden P, Reubelt T, Kusche J, Flechtner F, Pour SI, Sneeuw N, Müller J (2014) Comparing seven candidate mission configurations for temporal gravity field retrieval through full-scale numerical simulation. J Geod 88:31–43CrossRefGoogle Scholar
  12. ESA (1999) Gravity field and steady-state ocean circulation mission. Reports for Mission Selection, ESA SP-1233(1)—The Four Candidate Earth Explorer Core Missions, ESA Publication Division, ESTEC, Noordwijk, The Netherlands, 217 pGoogle Scholar
  13. Flechtner F, Neumayer KH, Doll B, Munder J, Reigber C, Raimondo JC (2009) GRAF-A GRACE follow-on mission feasibility study. Geophys Res Abstr 11:EGU2009-8516Google Scholar
  14. Freeden W, Schreiner M (2009) Spherical functions of mathematical geosciences. A scalar, vectorial, and tensorial setup. Advances in geophysical and environmental mechanics and mathematics. Springer, BerlinGoogle Scholar
  15. 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
  16. Heiskanen WA, Moritz H (1967) Physical geodesy. Freeman and Co., San FranciscoGoogle Scholar
  17. Jekeli C, Rapp RH (1980) Accuracy of the determination of mean anomalies and mean geoid undulations from a satellite gravity field mapping mission. Report no. 307, Department of Geodetic Science, The Ohio State University, Columbus, USA, 22 pGoogle Scholar
  18. Kaula WM (1966) Theory of satellite geodesy: applications of satellite to geodesy. Dover Publications Inc., New YorkGoogle Scholar
  19. Loomis BD, Nerem RS, Luthcke SB (2012) Simulation study of a follow-on gravity mission to GRACE. J Geod 86:319–335CrossRefGoogle Scholar
  20. Martinec Z (2003) Green’s function solution to spherical gradiometric boundary-value problems. J Geod 77:41–49CrossRefGoogle Scholar
  21. Mayrhofer R, Pail R (2012) Future satellite gravity field missions: feasibility study of post-Newtonian method. In: Kenyon S, Pacino MC, Marti U (eds) Geodesy for planet earth, IAG symposia, vol 136. Springer, Berlin, pp 231–238CrossRefGoogle Scholar
  22. 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, USA, 86 pGoogle Scholar
  23. Moritz H (1967) Kinematical geodesy. Report no. 92, Department of Geodetic Science, The Ohio State University, Columbus, USA, 65 pGoogle Scholar
  24. Panet I, Flury J, Biancale R, Gruber T, Johannessen J, van den Broeke MR, van Dam T, Gegout P, Hughes CW, Ramillien G, Sasgen I, Seoane L, Thomas M (2013) Earth system mass transport mission (e.motion): a concept for future earth gravity field measurements from space. Surv Geophys 34:141–163CrossRefGoogle Scholar
  25. 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:B04406CrossRefGoogle Scholar
  26. Petrovskaya MS, Vershkov AN (2006) Non-singular expressions for gravity gradients in the local north-oriented and orbital reference frames. J Geod 80:117–127CrossRefGoogle Scholar
  27. Petrovskaya MS, Vershkov AN (2008) Development of the second-order derivatives of the Earth’s potential in the local north-oriented reference frame in orthogonal series of modified spherical harmonics. J Geod 82:929–944CrossRefGoogle Scholar
  28. Reigber C, Luehr H, Schwintzer P (2002) CHAMP mission status. Adv Space Res 30:129–134CrossRefGoogle Scholar
  29. 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 114:013001CrossRefGoogle Scholar
  30. Rummel R (1997) Spherical spectral properties of the Earth’s gravitational potential and its first and second derivatives. In: Sansó F, Rummel R (eds) Geodetic boundary value problems in view of the one centimeter geoid, Lecture notes in earth sciences, vol 65. Springer, Berlin, Germany, pp 359–404Google Scholar
  31. Rummel R (2003) How to climb the gravity wall. Space Sci Rev 108:1–14CrossRefGoogle Scholar
  32. Rummel R (2010) GOCE: gravitational gradiometry in a satellite. In: Freeden W, Nashed MZ, Sonar T (eds) Handbook of geomathematics. Springer, Berlin, pp 93–103CrossRefGoogle Scholar
  33. Rummel R, van Gelderen M (1992) Spectral analysis of the full gravity tensor. Geophys J Int 111:159–169CrossRefGoogle Scholar
  34. Schiller S, Tino GM, Gill P, Salomon C, Sterr U, Peik E, Nevsky A, Görlitz A, Svehla D, Ferrari G, Poli N, Lusanna L, Klein H, Margolis H, Lemonde P, Laurent P, Santarelli G, Clairon A, Ertmer W, Rasel E, Müller J, Iorio L, Lämmerzahl C, Dittus H, Gill E, Rothacher M, Flechner F, Schreiber U, Flambaum V, Ni W-T, Liu L, Chen X, Chen J, Gao K, Cacciapuoti L, Holzwarth R, Hess MP, Schäfer W (2009) Einstein Gravity Explorer—a medium-class fundamental physics mission. Exp Astron 23:573–610CrossRefGoogle Scholar
  35. Silvestrin P, Aguirre M, Massotti L, Leone B, Cesare S, Kern M, Haagmans R (2012) The future of the satellite gravimetry after the GOCE mission. In: Kenyon S, Pacino MC, Marti U (eds) Geodesy for planet Earth, IAG symposia, vol 136. Springer, Berlin, pp 223–230CrossRefGoogle Scholar
  36. Simmonds JG (1994) A brief on tensor analysis. Undergraduate texts in mathematics, 2nd edn. Springer, New YorkCrossRefGoogle Scholar
  37. Sneeuw N (2000) A semi-analytical approach to gravity field analysis from satellite observations. Deutsche Geodätische Kommission, Reihe C, Nr. 527, München, Germany, 112 pGoogle Scholar
  38. Sneeuw N, Flury J, Rummel R (2005) Science requirements on future missions and simulated mission scenarios. Earth Moon Planets 94:113–142CrossRefGoogle Scholar
  39. Šprlák M, Sebera J, Vaľko M, Novák P (2014) Spherical integral formulas for upward/downward continuation of gravitational gradients onto gravitational gradients. J Geod 88:179–197CrossRefGoogle Scholar
  40. Šprlák M, Novák P (2015) Integral formulas for computing a third-order gravitational tensor from volumetric mass density, disturbing gravitational potential, gravity anomaly and gravity disturbance. J Geod 89:141–157CrossRefGoogle Scholar
  41. 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
  42. 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
  43. Wiese DN, Folkner WM, Nerem RS (2009) Alternative mission architectures for a gravity recovery satellite mission. J Geod 83:569–581CrossRefGoogle Scholar
  44. Wiese DN, Nerem RS, Lemoine FG (2012) Design considerations for a dedicated gravity recovery satellite mission consisting of two pairs of satellites. J Geod 86:81–98CrossRefGoogle Scholar
  45. 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, Germany, 155 pGoogle Scholar
  46. Zheng W, Xu HZ, Zhong M, Yun MJ (2009) Accurate and rapid error estimation on global gravitational field from current GRACE and future GRACE follow-on missions. Chin Phys B 18:3597–3604CrossRefGoogle Scholar
  47. Zheng W, Hsu HT, Zhong M, Liu CS, Yun MJ (2013a) Efficient and rapid accuracy estimation of the Earths gravitational field from next-generation GOCE follow-on by the analytical method. Chin Phys B 22:049101CrossRefGoogle Scholar
  48. Zheng W, Hsu HT, Zhong M, Liu CS, Yun MJ (2013b) Precise and rapid recovery of the Earth’s gravitational field by the next-generation four-satellite cartwheel formation system. Chin J Geophys 56:523–531CrossRefGoogle Scholar
  49. Zheng W, Xu HZ, Zhong M, Yun MJ (2014) Precise recovery of the Earth’s gravitational field by GRACE follow-on satellite gravity gradiometer. Chin J Geophys 57:1415–1423 (in Chinese)Google Scholar
  50. Zheng W, Xu HZ, Zhong M, Yun MJ (2015) Requirements analysis for future satellite gravity mission improved-GRACE. Surv Geophys 36:111–137CrossRefGoogle Scholar
  51. Zhu Z, Zhou ZB, Cai L, Bai YZ, Luo J (2013) Electrostatic gravity gradiometer design for the future mission. Adv Space Res 51:2269–2276CrossRefGoogle Scholar

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© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

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

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