Journal of Geodesy

, Volume 91, Issue 6, pp 627–652 | Cite as

Rank defect analysis and the realization of proper singularity in normal equations of geodetic networks

  • C. KotsakisEmail author
  • M. Chatzinikos
Original Article


The singularity of input normal equations (NEQ) is a crucial element for their optimal handling in the context of terrestrial reference frame (TRF) estimation under the minimal-constraint framework. However, this element is often missing in the recovered NEQ from SINEX files after the usual deconstraining based on the stated information for the stored solutions. The same setback also occurs with the original NEQ that are formed by the least-squares processing of space geodetic data due to the datum information which is carried by various modeling choices and/or software-dependent procedures. In the absence of this datum-related singularity, it is not possible to obtain genuine minimally constrained solutions because of the interference between the input NEQ’s content and the external datum conditions, a fact that may alter the geometrical information of the original measurements and can cause unwanted distortions in the estimated solution. The main goal of this paper is the formulation of a filtering scheme to enforce the proper (or desired) singularity in the input NEQ with regard to datum parameters that will be handled by the minimal-constraint setting in TRF estimation problems. The importance of this task is extensively discussed and justified with the help of several numerical examples in different GNSS networks.


NEQ Rank defect Minimal constraints TRF estimation Frame distortion Position time series 


  1. Altamimi Z (2003) Discussion on how to express a regional GPS solution in the ITRF. EUREF Publication No. 12, Verlag des Bundesamtes für Kartographie und Geodäsie, Frankfurt am Main, pp 162–167Google Scholar
  2. Altamimi Z, Dermanis A (2009) The choice of reference system in ITRF formulation. IAG Symposia Series, vol 137. Springer, Berlin, pp 329–334Google Scholar
  3. Altamimi Z, Boucher C, Sillard P (2002) New trends for the realization of the international terrestrial reference system. Adv Space Res 30(2):175–184CrossRefGoogle Scholar
  4. Angermann D, Drewes H, Krugel M, Meisel B, Gerstl M, Kelm R, Muller H, Seemuller W, Tesmer V (2004) ITRS Combination Center at DGFI: A Terrestrial Reference Frame Realization 2003. Deutsche Geodätische Kommission, Reihe B, Heft Nr. 313Google Scholar
  5. Altamimi Z, Collilieux X, Métivier L (2011) ITRF2008: an improved solution of the international terrestrial reference frame. J Geod 85(8):457–473CrossRefGoogle Scholar
  6. Altamimi Z, Rebischung P, Métivier L, Collilieux X (2016) ITRF2014: a new release of the international terrestrial reference frame modeling nonlinear station motions. J Geophys Res Solid Earth 121:6109–6131CrossRefGoogle Scholar
  7. Blaha G (1971) Inner adjustment constraints with emphasis on range observations. Department of Geodetic Science, The Ohio State University, OSU Report No. 148, Columbus, OhioGoogle Scholar
  8. Blewitt G (1998) GPS data processing methodology. In: Teunissen PJG, Kleusberg A (eds) GPS for Geodesy, 2nd edn. Springer, Berlin, pp 231–270CrossRefGoogle Scholar
  9. Bloßfeld M (2015) The key role of satellite laser ranging towards the integrated estimation of geometry, rotation and gravitational field of the Earth. PhD thesis, Technische Universität München. DGK, Reihe C, Heft Nr. 745Google Scholar
  10. Bloßfeld M, Seitz M, Angermann D, Moreaux G (2016) Quality assessment of IDS contribution to ITRF2014 performed by DGFI-TUM. Adv Space Res. doi: 10.1016/j.asr.2015.12.016 Google Scholar
  11. Dach R, Lutz S, Walser P, Fridez P (2015) User manual of the Bernese GNSS Software, Version 5.2, Astronomical Institute, University of Bern, Bern Switzerland.
  12. Davies P, Blewitt G (2000) Methodology for global geodetic time series estimation: a new tool for geodynamics. J Geophys Res 105(B5):11083–11100CrossRefGoogle Scholar
  13. Dermanis A (2003) The rank deficiency in estimation theory and the definition of reference frames. IAG Symposia Series, vol 127. Springer, Berlin, pp 145–156Google Scholar
  14. Ebner H (1975) Analysis of covariance matrices. In: Proceedings of the ISPRS commission III symposium, Stuttgart, 2–6 Sept 1974, pp 111–121, Deutsche Geodätische Kommission, Reihe B, Heft Nr. 214Google Scholar
  15. Glaser S, Fritsche M, Sosnica K, Rodriguez-Solano CJ, Wang K, Dach R, Hugentobler U, Rothacher M, Dietrich R (2015) A consistent combination of GNSS and SLR with minimum constraints. J Geod 89(12):1165–1180CrossRefGoogle Scholar
  16. Hansen PC (1998) Rank-deficient and discrete ill-posed problems. SIAM, PhiladelphiaCrossRefGoogle Scholar
  17. IERS (2006) SINEX—Solution (Software/technique) Independent exchange format. Technical document version 2.02, International Earth Rotation and Reference Systems Service.
  18. Jiang W, Li Z, van Dam T, Ding W (2013) Comparative analysis of different environmental loading methods and their impacts on the GPS height time series. J Geod 87(7):687–703CrossRefGoogle Scholar
  19. Kaniuth K, Vetter S (2005) Vertical velocities of European coastal sites derived from continuous GPS observations. GPS Solut 9:32–40CrossRefGoogle Scholar
  20. Kelm R (2003) Rank defect analysis and variance component estimation for inter-technique combination. In: Proceedings of the IERS Workshop on Combination Research and Global Geophysical Fluids. Richter B, Schwegmann W, Dick WR (eds) IERS Technical Note No. 30, pp 112–114, Verlag des Bundesamts fur Kartographie und Geodäsie, Frankfurt am MainGoogle Scholar
  21. Koch K-R (1999) Parameter estimation and hypothesis testing in linear models, 2nd edn. Springer, BerlinCrossRefGoogle Scholar
  22. Kotsakis C (2012) Reference frame stability and nonlinear distortion in minimum-constrained network adjustment. J Geod 86(9):755–774CrossRefGoogle Scholar
  23. Kotsakis C (2013) Generalized inner constraints for geodetic network densification problems. J Geod 87(7):661–673CrossRefGoogle Scholar
  24. Legrand J, Bergeot N, Bruyninx C, Woppelmann G, Bouin M-N, Altamimi Z (2010) Impact of regional reference frame definition on geodynamic interpretations. J Geodyn 49(3–4):116–122CrossRefGoogle Scholar
  25. Makinen J, Koivula H, Poutanen M, Saaranen V (2003) Vertical velocities from permanent GPS networks and from repeated precise levelling. J Geodyn 35(4–5):443–456CrossRefGoogle Scholar
  26. Meissl P (1965) Uber die innere Genauigheit dreidimensionaler Punkthaufens. Vermessungswesen 90:109–118Google Scholar
  27. Meissl P (1969) Zusammengfassung und Ausbau der inneren Fehlertheoric eines Punkthaufens. Deutsche Geodätische Kommission, Reihe A 61:8–21Google Scholar
  28. Rebischung P (2014) Can GNSS contribute to improving the ITRF definition? PhD thesis, Observatoire de Paris, Ecole Doctorale Astronomie et Astrophysique d’Ile-de-FranceGoogle Scholar
  29. Rebischung P, Altamimi Z, Ray J, Garayt B (2016) The IGS contribution to ITRF2014. J Geod. doi: 10.1007/s00190-016-0897-6 Google Scholar
  30. Seitz M, Angermann D, Bloßfeld M, Drewes H, Gerstl M (2012) The 2008 DGFI realization of the ITRS: DTRF2008. J Geod 86(12):1097–1123CrossRefGoogle Scholar
  31. Sillard P, Boucher C (2001) A review of algebraic constraints in terrestrial reference frame datum definition. J Geod 75(2):63–73CrossRefGoogle Scholar
  32. Tregoning P, Watson C (2009) Atmospheric effects and spurious signals in GPS analyses. J Geophys Res 114:B09403. doi: 10.1029/2009JB006344
  33. van Dam T, Collilieux X, Wuite J, Altamimi Z, Ray J (2012) Nontidal ocean loading: amplitudes and potential effects in GPS height time series. J Geod 86(1):1043–1057Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of Geodesy and SurveyingAristotle University of ThessalonikiThessalonikiGreece
  2. 2.GNSS Research GroupRoyal Observatory of BelgiumBrusselsBelgium

Personalised recommendations