Approximations of the GOCE error variance-covariance matrix for least-squares estimation of height datum offsets

Ch. Gerlach 1  and Th. Fecher 2
  • 1 Commission for Geodesy and Glaciology, Bavarian Academy of Sciences and Humanities, Alfons-Goppel-Strasse 11, 80539 Munich, Germany
  • 2 Institute of Astronomical and Physical Geodesy, Technische Universität München, Arcisstrasse 21, 80333 Munich, Germany


One main geodetic objective of the European Space Agency’s satellite mission GOCE (gravity field and steady-state ocean circulation explorer) is the contribution to global height system unification. This can be achieved by applying the Geodetic Boundary Value Problem (GBVP) approach. Thereby one estimates the unknown datum offsets between different height networks (datum zones) by comparing the physical (e.g. orthometric) height values H of benchmarks in different datum zones to the corresponding values derived from the difference between ellipsoidal heights h (e.g. determined by means of global navigation satellite systems) and geoid heights N. In the ideal case, i.e. neglecting data errors, the misfit between H and (h − N) is constant inside one datum zone and represents the datum offset. In practise, the accuracy of the offset estimation depends on the accuracy of the three quantitiesH, h andN, where the latter can be computed from the combination of a GOCE-derived Global Potential Model (GPM) for the long to medium wavelength and terrestrial data for the short wavelength content. Providing an optimum estimation of the datum offsets along with realistic error estimates, theoretically, requires propagation of the full error variance and covariance information of the GOCE spherical harmonic coefficients to geoid heights, respectively geoid height differences. From a numerical point of view, this is a very demanding task which cannot simply be run on a single PC. Therefore it is worthwhile to investigate on different levels of approximation of the full variance-covariance matrix (VCM) with the aim of minimizing the numerical effort. In this paper, we compare the estimation error based on three levels of approximation, namely (1) using the full VCM, (2) using only elements of the dominant m-block structure of the VCM and (3) using only the main diagonal of the VCM, i.e. neglecting all error covariances between the spherical harmonic coefficients. We show that the m-block approximation gives almost the same result as provided by the full VCM. The diagonal approximation however over- or underestimates the geoid height error, depending on the geographic location and therefore is not regarded to be a suitable approximation.

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  • ESA, 1999, Gravity Field and Steady-State Ocean Circulation Mission, Report for mission selection of the four candidate earth explorer core missions, ESA.

  • Gerlach Ch. and Rummel R., 2012, Global height system unification with GOCE: a simulation study on the indirect bias term in the GBVP approach, J. Geod., DOI: 10.1007/s00190-012-0579-y.

  • Gruber Th., Gerlach Ch. and Haagmans R.H.N., 2012, Intercontinental height datum connection with GOCE and GPS-levelling data, J. Geod. Sci., 2, 4, 270-280.

  • Haagmans R.H.N. and van Gelderen M., 1991, Error Variances-Covariances of GEM-T1: Their Characteristics and Implications in Geoid Computation, J. Geophys. Res., 96(B12), 20011-20022, doi:10.1029/91JB01971.

  • Heiskanen W.A. and Moritz H., 1967, Physical Geodesy, W.H. Freeman and Company, San Francisco.

  • Marsh J.G. et al., 1988, A new gravitational model for the Earth from satellite tracking data: GEM-T1, J. Geophys. Res., 93(B6), 6169-6215, doi:10.1029/JB093iB06p06169.

  • Mayer-Gürr T., 2006, Gravitationsfeldbestimmung aus der Analyse kurzer Bahnbögen am Beispiel der Satellitenmissionen CHAMP und GRACE, Dissertation, University of Bonn.

  • Pail R., Goiginger H., Mayrhofer R., Schuh W.-D., Brockmann J.M., Krasbutter I., Höck E. and Fecher Th., 2010a, GOCE gravity field model derived from orbit and gradiometry data applying the time-wise method, In: Lacoste-Francis H. (ed.) Proceedings of the ESA Living Planet Symposium, ESA Publication SP-686, ESA/ESTEC, ISBN (Online) 978-92-9221-250-6, ISSN 1609-042X.

  • Pail R., Goiginger H., Schuh W.-D., Höck E., Brockmann J.M., Fecher Th., Gruber Th., Mayer-Gürr T., Kusche J., Jäggi A. and Rieser D., 2010b, Combined satellite gravity field model GOCO01S derived from GOCE and GRACE, Geophys. Res. Lett., 37, EID L20314, American Geophysical Union, ISSN 0094-8276, DOI: 10.1029/2010GL044906, 2010.

  • Rummel R., 2001, Global unification of height systems and GOCE, In: Sideris M.G. (ed.) Gravity, Geoid and Geodynamics 2000, Springer, Berlin, pp. 13-20.

  • Rummel R. and Teunissen P.J.G., 1988, Height Datum Definition, Height Datum Connection and the Role of the Geodetic Boundary Value Problem, Bull. Geod., 62, 477-498.

  • Sacher M., Ihde J. and Seeger H., 1999, Preliminary Transformation Relations between National European Height Systems and the United European Levelling Network, In: Gubler, Torres, Hornik (eds.) Report on the Symposium of the IAG Subcommission for Europe (EUREF) held in Prague, 2-5 June, 1999. Veröffentlichungen der Bayerischen Kommisssion für die Internationale Erdmessung, 60, München.

  • Sneeuw N., 2000, A semi-analytical approach to gravity field analysis from satellite observations. DGK series C, 527, Bavarian Academy of Sciences and Humanities, Munich.

  • Swenson S. and Wahr J., 2006, Post-processing removal of correlated errors in GRACE data. Geophys. Res. Lett., 33, L08402, doi:10.1029/2005GL025285.

  • Tscherning C.C. and Rapp R.H., 1974, Closed Covariance Expressions for Gravity Anomalies, Geoid Undulations, and Deflections of the Vertical Implied by Anomaly Degree Variance Models, Rep. 208, Department of Geodetic Science, The Ohio State University.

  • Wong L. and Gore R., 1969, Accuracy of geoid heights from modified Stokes kernels, Geophys. J. Roy. Astro. Soc., 18.

  • Xu P., 1992, A quality investigation of global vertical datum connection, Geophys. J. Int., 110, 361-370, doi:10.1111/j.1365-246X.1992.tb00880.x.

  • Xu P. and Rummel R., 1991, A quality investigation of global vertical datum connection, Publications on Geodesy, 34, Netherlands Geodetic Commission.


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