Surveys in Geophysics

, Volume 38, Issue 3, pp 571–590 | Cite as

GOCO05c: A New Combined Gravity Field Model Based on Full Normal Equations and Regionally Varying Weighting

  • T. Fecher
  • R. Pail
  • T. Gruber
  • the GOCO Consortium


GOCO05c is a gravity field model computed as a combined solution of a satellite-only model and a global data set of gravity anomalies. It is resolved up to degree and order 720. It is the first model applying regionally varying weighting. Since this causes strong correlations among all gravity field parameters, the resulting full normal equation system with a size of 2 TB had to be solved rigorously by applying high-performance computing. GOCO05c is the first combined gravity field model independent of EGM2008 that contains GOCE data of the whole mission period. The performance of GOCO05c is externally validated by GNSS–levelling comparisons, orbit tests, and computation of the mean dynamic topography, achieving at least the quality of existing high-resolution models. Results show that the additional GOCE information is highly beneficial in insufficiently observed areas, and that due to the weighting scheme of individual data the spectral and spatial consistency of the model is significantly improved. Due to usage of fill-in data in specific regions, the model cannot be used for physical interpretations in these regions.


Gravity Combined gravity field model Full normal equation systems High-performance computing Stochastic model 



We acknowledge the provision of regional terrestrial and altimetric data sets by the institutions mentioned in Sect. 2, and the provision of extensive supercomputing resources by the Leibniz Supercomputing Centre (LRZ; Address: Boltzmannstraße 1, 85748 Garching bei München, Germany). We also thank two anonymous reviewers; their comments and suggestions helped to improve the paper significantly.


  1. Andersen OB, Knudsen P, Berry PAM (2010) The DNSC08GRA global marine gravity field from double retracked satellite altimetry. J Geod 84(3):191–199. doi: 10.1007/s00190-009-0355-9 CrossRefGoogle Scholar
  2. Andersen OB, Knudsen P, Kenyon SC, Factor JK, Holmes S (2013) The DTU13 Global marine gravity field—first evaluation. OSTST Meeting, Boulder, CO.
  3. Anderson OB, Knudsen P, Stenseng L (2015) The DTU13 MSS (Mean Sea Surface) and MDT (Mean Dynamic Topography) from 20 years of satellite altimetry. In: International association of geodesy symposia. Springer, pp 1–10. doi: 10.1007/1345-2015_1
  4. Bingham RJ, Haines K, Hughes CW (2008) Calculating the ocean’s mean dynamic topography from a mean sea surface and a geoid. J Atmos Ocean Technol 25(10):1808–1822. doi: 10.1175/2008JTECHO568.1 CrossRefGoogle Scholar
  5. Bingham RJ, Knudsen P, Andersen O, Pail R (2011) An initial estimate of the North Atlantic steady-state geostrophic circulation from GOCE. Geophys Res Lett. doi: 10.1029/2010GL045633 Google Scholar
  6. Brockmann JM, Zehentner N, Höck E, Pail R, Loth I, Mayer-Gürr T, Schuh W-D (2014) EGM_TIM_RL05: an independent geoid with centimeter accuracy purely based on the GOCE mission. Geophys Res Lett 41:8089–8099. doi: 10.1002/2014GL061904 CrossRefGoogle Scholar
  7. Colombo O (1981) Numerical methods for harmonic analysis on the sphere. Department of Geodetic Science, Report No. 310, The Ohio State University, Columbus, OhioGoogle Scholar
  8. Denker H, Barriot JP, Barzaghi R, Fairhead D, Forsberg R, Ihde J (2008) A new European gravimetric quasigeoid EGG2008. In: International symposium on ‘gravity, geoid and earth observation 2008’, Chania, GreeceGoogle Scholar
  9. Drinkwater MR, Floberghagen R, Haagmans R, Muzi D, Popescu A (2003) GOCE: ESA’s first Earth Explorer Core mission. In: Beutler G et al (eds) Earth gravity field from space—from sensors to earth science, space sciences series of ISSI, vol 18. Kluwer Academic Publishers, Dordrecht, pp 419–432. ISBN 1-4020-1408-2CrossRefGoogle Scholar
  10. Ekman M (1989) Impacts of geodynamic phenomena on systems for height and gravity. Bull Geod 63(3):281–296. doi: 10.1007/BF02520477 CrossRefGoogle Scholar
  11. Featherstone WE, Kirby JF, Hirt C, Filmer MS, Claessens SJ, Brown NJ, Hu G, Johnston GM (2011) The AUSGeoid09 model of the Australian Height Datum. J Geod 85(3):133–150. doi: 10.1007/s00190-010-0422-2 CrossRefGoogle Scholar
  12. Fecher T (2015) Globale kombinierte Schwerefeldmodellierung auf Basis voller Normalgleichungssysteme. Dissertation, Technical University of Munich, Munich, GermanyGoogle Scholar
  13. Fecher T, Pail R, Gruber T (2015) Global gravity field modeling based on GOCE and complementary gravity data. Int J Appl Earth Obs 35:120–127. doi: 10.1016/j.jag.2013.10.005 CrossRefGoogle Scholar
  14. Forsberg R, Kenyon SC (2004) Gravity and geoid in the Arctic region—The Northern Polar Gap now filled. In: GOCE, the geoid and oceanography. ESA/ESRIN, Frascati, ItalyGoogle Scholar
  15. Förste C, Bruinsma SL, Abrikosov O, Lemoine JM, Schaller T, Götze H-J, Ebbing J, Marty JC, Flechtner F, Balmino G, Biancale R (2014) EIGEN-6C4 The latest combined global gravity field model including GOCE data up to degree and order 2190 of GFZ Potsdam and GRGS Toulouse. In: The 5th GOCE user workshop, 25–28 Nov 2014, Paris, FranceGoogle Scholar
  16. Grombein T, Seitz K, Heck B (2013) Topographic-isostatic reduction of GOCE gravity gradients. In: Proceedings of the XXV general assembly of the international union of geodesy and geophysics. Melbourne, AustraliaGoogle Scholar
  17. Gruber T (2001) High-resolution gravity field modeling with full variance-covariance matrices. J Geod 75(9):505–514. doi: 10.1007/s001900100202 CrossRefGoogle Scholar
  18. Gruber T, Visser PNAM, Ackermann C, Hosse M (2011) Validation of GOCE gravity field models by means of orbit residuals and geoid comparisons. J Geod 85(11):845–860. doi: 10.1007/s00190-011-0486-7 CrossRefGoogle Scholar
  19. Gruber T, Gerlach C, Haagmans R (2012) Intercontinental height datum connection with GOCE and GPS-levelling data. J Geod Sci. doi: 10.2478/v10156-012-0001-y Google Scholar
  20. Hirt C, Kuhn M (2014) Band-limited topographic mass distribution generates full-spectrum gravity field: gravity forward modeling in the spectral and spatial domains revisited. J Geophy Res Solid Earth 119(4):3646–3661. doi: 10.1002/2013JB010900 CrossRefGoogle Scholar
  21. Kenyon SC, Pavlis NK (1996) The development of a global surface gravity data base to be used in the joint DMA/GSFC geopotential model. In: Proceedings of IAG symposium No. 116, global gravity field and its temporal variations. Springer, BerlinGoogle Scholar
  22. Knudsen P, Bingham R, Andersen O, Rio M-H (2011) A global mean dynamic topography and ocean circulation estimation using a preliminary GOCE gravity model. J Geod 85(11):861–879. doi: 10.1007/s00190-011-0485-8 CrossRefGoogle Scholar
  23. Koch K-R, Kusche J (2002) Regularization of geopotential determination from satellite data by variance components. J Geod 76(5):259–268. doi: 10.1007/s00190-002-0245-x CrossRefGoogle Scholar
  24. Lemoine FG, Kenyon SC, Factor JK, Trimmer RG, Pavlis NK, Chinn DS, Cox CM, Klosko SM, Luthcke SB, Torrence MH, Wang YM, Williamson RG, Pavlis EC, Rapp RH, Olson TR (1998) The development of the joint NASA GSFC and the National Imaery and Mapping Agency (NIMA) geopotential model EGM96. Greenbelt, MDGoogle Scholar
  25. Maximenko N, Niiler P, Rio M-H, Melnichenko O, Centurioni L, Chambers D, Zlotnicki V, Galperin B (2009) Mean dynamic topography of the ocean derived from satellite and drifting buoy data using three different techniques. J Atmos Ocean Tech 26(9):1910–1919CrossRefGoogle Scholar
  26. Mayer-Gürr T (2006) Gravitationsfeldbestimmung aus der Analyse kurzer Bahnbögen am Beispiel der Satellitenmissionen CHAMP und GRACE. Dissertation, Universität Bonn, Bonn, GermanyGoogle Scholar
  27. Mayer-Gürr T, the GOCO Team (2015) The combined satellite gravity field model GOCO05s. EGU2015-12364, EGU General Assembly, Vienna, AustriaGoogle Scholar
  28. Mayer-Gürr T, Zehentner N, Klinger B, Kvas A (2014) ITSG-Grace2014: a new GRACE gravity field release computed in Graz. GRACE Science Team Meeting (GSTM), Potsdam, GermanyGoogle Scholar
  29. McKenzie D, Yi W, Rummel R (2014) Estimates of Te from GOCE data. Earth Planet Sci Lett 399:116–127. doi: 10.1016/j.epsl.2014.05.003 CrossRefGoogle Scholar
  30. Moritz H (1980) Advanced physical geodesy. Wichmann, Karlsruhe. ISBN 3879071063Google Scholar
  31. Pail R, Goiginger H, Schuh W-D, Höck E, Brockmann JM, Fecher T, Gruber T, Mayer-Gürr T, Kusche J, Jäggi A, Rieser D (2010) Combined satellite gravity field model GOCO01S derived from GOCE and GRACE. Geophys Res Lett. doi: 10.1029/2010GL044906 Google Scholar
  32. Pail R, Bruinsma S, Migliaccio F, Förste C, Goiginger H, Schuh W-D, Höck E, Reguzzoni M, Brockmann JM, Abrikosov O, Veicherts M, Fecher T, Mayrhofer R, Krasbutter I, Sansò F, Tscherning CC (2011) First GOCE gravity field models derived by three different approaches. J Geod 85(11):819–843. doi: 10.1007/s00190-011-0467-x CrossRefGoogle Scholar
  33. Pavlis NK (1988) Modeling and estimation of a low degree geopotential model from terrestrial gravity data. OSU Report No. 386, The Ohio State University, Columbus, OhioGoogle Scholar
  34. Pavlis NK, Holmes SA, Kenyon SC, Factor JK (2012) The development and evaluation of the Earth Gravitational Model 2008 (EGM2008). J Geophys Res. doi: 10.1029/2011JB008916 Google Scholar
  35. Pavlis NK, Holmes SA, Kenyon SC, Factor JK (2013) The development and evaluation of the Earth Gravitational Model 2008 (EGM2008)—erratum. J Geophys Res. doi: 10.1002/jgrb.50167 Google Scholar
  36. Rapp RH, Pavlis NK (1990) The development and analysis of geopotential coefficient models to spherical harmonic degree 360. J Geophys Res 95(B13):21885–21911. doi: 10.1029/JB095iB13p21885 CrossRefGoogle Scholar
  37. Rummel R (2012) Height unification using GOCE. J Geod Sci. doi: 10.2478/v10156-011-0047-2 Google Scholar
  38. Schneider M (1967) Lösungsvorschlag zum Bahnbestimmungsproblem. Forschungsbericht FB W 67-35, Deutsche Gesellschaft für FlugwissenschaftenGoogle Scholar
  39. Siegismund F (2013) Assessment of optimally filtered recent geodetic mean dynamic topographies. J Geophy Res Oceans 118(1):108–117. doi: 10.1029/2012JC008149 CrossRefGoogle Scholar
  40. Tapley BD, Bettadpur S, Watkins M, Reigber C (2004) The gravity recovery and climate experiment: mission overview and early results. Geophys Res Lett. doi: 10.1029/2004GL019920 Google Scholar
  41. Torge W, Müller J (2012) Geodesy, 4th edn. De Gruyter, Berlin. ISBN 978-3-11-025000-8CrossRefGoogle Scholar
  42. van der Meijde M, Julià J, Assumpção M (2013) Gravity derived Moho for South America. Tectonophysics 609:456–467. doi: 10.1016/j.tecto.2013.03.023 CrossRefGoogle Scholar
  43. Wenzel HG (1985) Hochauflösende Kugelfunktionsmodelle für das Gravitationspotential der Erde. Universität Hannover, HannoverGoogle Scholar
  44. Woodworth PL, Hughes CW, Bingham RJ, Gruber T (2012) Towards worldwide height system unification using ocean information. J Geod Sci. doi: 10.2478/v10156-012-0004-8 Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  • T. Fecher
    • 1
  • R. Pail
    • 1
  • T. Gruber
    • 1
  • the GOCO Consortium
  1. 1.Institute of Astronomical and Physical GeodesyTechnical University of MunichMunichGermany

Personalised recommendations