Theory and Realization of Reference Systems

  • Athanasios DermanisEmail author
Living reference work entry
Part of the Springer Reference Naturwissenschaften book series (SRN)


After a short introduction on the basics of reference system theory and its application for the description of earth rotation, the problem of establishing a reference system for the discrete stations of a geodetic network is studied, from both a theoretical and a practical – implementation point of view.

First the case of rigid networks is examined, which covers also the case of deformable networks with data collected within a time span, small enough for the network shape to remain practically unaltered. The problem of how to analyze observations, which are invariant under particular changes of the reference system, is examined within the framework of least squares estimation theory, with a rank deficiency in the design matrix. The complete theory is presented, including all necessary proofs. Not only of the usual statistical results for the rank deficient linear Gauss-Markov model, but also those of the rich geodetic theory are presented, based on the fact that the physical cause of the rank deficiency is known to be the lack of definition of the reference system. The additional geodetic results are based on the fact that one can easily construct a matrix with columns that are a basis of the null space of the design matrix. Insights are presented into the geometric characteristics of the problem and its relation to the theory of generalized inverses. Passing into deformable networks, a deterministic mathematical model is presented, based of the concept of geodesic lines which are the shortest between linear shape manifolds, associated with the network shape at each instant. Reference system optimality for a discrete network is related to the relevant ideas of Tisserand, developed for the continuum of the earth masses.

The practical problem of choosing a reference system for a coordinate time series is examined, for the case where a linear-in-time model is adopted for the temporal variation of coordinates. The choice of reference system is related to the choice of minimal constraints for obtaining one out of the infinitely many least squares solutions, corresponding to descriptions in different reference systems of the same sequence of network shapes. The a-posteriori change of the reference system is examined, where one moves from one least squares solution to another one, satisfying particular minimal constraints. Kinematic minimal constraints are also introduced, leading to coordinates that demonstrate the minimum coordinate variation and are thus connected to the ideas of Tisserand for reference system optimality. It is also shown how to convert a reference system of a geodetic network to one for the whole continuous earth, or at least the lithosphere, utilizing additional geophysical information.

The last item is the combination of data from four space techniques (VLBI, SLR, GPS, DORIS) in order to establish a global reference system realized though a number of parameters that constitute the International Terrestrial Reference Frame. After a theoretical exposition of the basics of data combination, the various methods of spatial data combination are presented, for both coordinate and Earth Orientation Parameter time series, while alternatives are presented for the choice of the origin (geocenter) and the network scale from the scale of VLBI and SLR. Finally, existing and new methodologies are presented for building post linear models, describing the temporal variation of station coordinates.


Reference systems Rank deficient linear model Minimal constraints Inner constraints Kinematic constraints Tisserand reference system Coordinate tine series Earth orientation parameters Combination of geodetic space techniques International Terrestrial Reference Frame (ITRF) 


  1. 1.
    Altamimi, Z., Dermanis, A.: The choice of reference system in ITRF formulation. In: Sneeuw, N., et al. (eds.) VII Hotine-Marussi Symposium on Mathematical Geodesy, International Association of Geodesy, Symposia, vol. 137, pp. 329–334. Springer, Berlin (2009)CrossRefGoogle Scholar
  2. 2.
    Altamimi, Z., Dermanis, A.: Theoretical foundations of ITRF determination. The algebraic and the kinematic approach. In: Katsampalos, K.V., Rossikopoulos, D., Spatalas, S., Tokmakidis, K. (eds.) On Measurements of Lands and Constructions. Volume in honor of Prof. Dimitios G. Vlachos. Publication of the School of Rural & Surveying Engineering, Aristotle University of Thessaloniki, pp. 331–359 (2013)Google Scholar
  3. 3.
    Altamimi, Z., Sillard, P., Boucher, C.: ITRF2000: a new release of the international terrestrial reference frame for earth science applications. J. Geophys. Res. 107(B10), 2214 (2002)CrossRefGoogle Scholar
  4. 4.
    Altamimi, Z., Sillard, P., Boucher, C.: ITRF2000: from theory to implementation. In: Sansò, F. (ed.) V Hotine–Marussi Symposium on Mathematical Geodesy. IAG Symposia, vol. 127, pp. 157–163. Springer, Berlin (2004)CrossRefGoogle Scholar
  5. 5.
    Altamimi, Z., Collilieux, X., Legrand, J., Garayt, B., Boucher, C.: ITRF2005: a new release of the international terrestrial reference frame based on time series of station positions and earth orientation parameters. J. Geophys. Res. 112, B09401 (2007)CrossRefGoogle Scholar
  6. 6.
    Altamimi, Z., Collilieux, X., Métivier, L.: ITRF2008: an improved solution of the international terrestrial reference frame. J. Geod. 85, 457–473 (2011)CrossRefGoogle Scholar
  7. 7.
    Altamimi, Z., Rebischung, P., Métivier, L., Collilieux, X.: ITRF2014: a new release of the international terrestrial reference frame modeling nonlinear station motions. J. Geophys. Res. Solid Earth 121, 6109–6131 (2016)CrossRefGoogle Scholar
  8. 8.
    Angermann, D., Drewes, H., Krügel, M., Meisel, B., Gerstl, M., Kelm, R., Müller, H., Seemüller, W., Tesmer, V.: ITRS Combination Center at DGFI: A Terrestrial Feference Frame Realization 2003. Deutsche Geodätische Kommission Reihe B Nr. 313, München (2004)Google Scholar
  9. 9.
    Angermann, D., Drewes, H., Gerstl, M., Krügel, M., Meisel, B.: DGFI combination methodology for ITRF2005 computation. In: Drewes, H. (ed.) Geodetic Reference Frames. IAG Symposia, vol. 134, pp. 11–16. Springer, Berlin (2009)CrossRefGoogle Scholar
  10. 10.
    Artz, T., Bernhard, L., Nothnagel, A., Steigenberger, P., Tesmer, S.: Methodology for the combination of sub-daily Earth rotation from GPS and VLBI observations. J. Geod. 86, 221–239 (2012)CrossRefGoogle Scholar
  11. 11.
    Baarda, W.: S-Transformations and Criterion Matrices. Netherlands Geodetic Commission, Publ in Geodesy, New Series, vol. 5, no. 1, Delft (1973).
  12. 12.
    Baarda, W.: Linking up spatial models in geodesy. Extended S-Transformations. Netherlands Geodetic Commission, Publ. in Geodesy, New Series, no. 41, Delft (1995).
  13. 13.
    Biagi, L., Sanso, F.: Sistemi di riferimento in geodesia: algebra e geometria die minimi quadrati per un modello con deficienza di rango. Bollettino di Geodesia e Scienze Affini. Parte prima: Anno LXII, N. 4, 261–284. Parte seconda: Anno LXIII, N. 1, 29–52. Parte terza: Anno LXIII, N. 2, 129–149 (2003)Google Scholar
  14. 14.
    Bjerhammar, A.: Rectangular reciprocal matrices with special emphasis to geodetic calculations. Bulletin Géodésique 52, 188–220 (1951)CrossRefGoogle Scholar
  15. 15.
    Blaha, G.: Inner adjustment constraints with emphasis on range observations. Department of Geodetic Science, Report 148, The Ohio State University, Columbus (1971)Google Scholar
  16. 16.
    Blaha, G.: Free networks: minimum norm solution as obtained by the inner adjustment constraint method. Bull Géodésique 56, 209–219 (1982)CrossRefGoogle Scholar
  17. 17.
    Bolotin, S., Bizouard, C., Loyer, S., Capitaine, N.: High frequency variations of the earth’s instantaneous angular velocity vector. Determination by VLBI data analysis. Astron. Astrophys. 317, 601–609 (1997)Google Scholar
  18. 18.
    Capitaine, N., Guinod, B., Souchay, J.: A non-rotating origin of the instantaneous equator: definition, properties and use. Cel. Mech. 39, 283–307 (1986)CrossRefGoogle Scholar
  19. 19.
    Chatzinikos, M., Dermanis, A.: A comparison of existing and new methods for the analysis of nonlinear variations in coordinate time series. In: IUGG 2015, Prague, 22 June–3 July 2015. Available at:
  20. 20.
    Chatzinikos, M., Dermanis, A.: A coordinate-invariant model for deforming geodetic networks: understanding rank deficiencies, non-estimability of parameters, and the effect of the choice of minimal constraints. J. Geod. 91, 375–396 (2017)CrossRefGoogle Scholar
  21. 21.
    Chatzinikos, M., Dermanis, A.: Interpretation of numerically detected rank defects in GNSS data analysis problems in terms of deficiencies in reference system definition. GPS Solutions 21, 1239–1250 (2017)CrossRefGoogle Scholar
  22. 22.
    Chen, Q., van Dam, T., Sneeuw, N., Collilieux, X., Weigelt, M., Rebischung, P.: Singular spectrum analysis for modeling seasonal signals from GPS time series. J. Geodyn. 72, 25–35 (2013)CrossRefGoogle Scholar
  23. 23.
    Dermanis, A.: The Non-Linear and the Space-Time Datum problem. Paper presented at the Meeting “Mathematische Methoden der Geodaesie”, Mathematisches Forschungsinstitut Oberwolfach, 1–7 Oct 1995. Available at:,
  24. 24.
    Dermanis, A.: Generalized inverses of nonlinear mappings and the nonlinear geodetic datum problem. J. Geod. 72(2), 71–100 (1998)CrossRefGoogle Scholar
  25. 25.
    Dermanis, A.: Establishing global reference frames. Nonlinear, temporal, geophysical and stochastic aspects. Invited paper presented at the IAG international symposium Banff, Alberta, 31 July–4 Aug 2000 (2000). In: Sideris, M.G. (ed) Gravity, Geoid and Geodynamics”, IAG Symposia, vol. 123, pp. 35–42. Springer, Berlin (2002)Google Scholar
  26. 26.
    Dermanis, A.: Global reference frames: connecting observation to theory and geodesy to geophysics. In: IAG 2001 Scientific Assembly “Vistas for Geodesy in the New Milennium”, Budapest, 2–8 Sept 2001. Available at,
  27. 27.
    Dermanis, A.: Some remarks on the description of earth rotation according to the IAU 2000 resolutions. From Stars to Earth and Culture. In honor of the memory of Professor Alexandros Tsioumis, pp. 280–291. School of Rural & Surveying Engineering, The Aristotle University of Thessaloniki (2003)Google Scholar
  28. 28.
    Dermanis, A.: The rank deficiency in estimation theory and the definition of reference frames. In: Sansò, F. (ed.) V Hotine-Marussi Symposium on Mathematical Geodesy, Matera, 17–21 June 2003. International Association of Geodesy Symposia, vol. 127, pp. 145–156. Springer, Heidelberg (2003)Google Scholar
  29. 29.
    Dermanis, A.: Coordinates and Reference Systems. Ziti Publications, Thessaloniki (2005)Google Scholar
  30. 30.
    Dermanis, A.: Compatibility of the IERS earth rotation representation and its relation to the NRO conditions. Proceedings, Journées 2005 Systèmes de Référence Spatio-Temporels “Earth dynamics and reference systems: five years after the adoption of the IAU 2000 Resolutions”, Warsaw, 19–21 Sept 2005, pp. 109–112 (2005)Google Scholar
  31. 31.
    Dermanis, A.: The ITRF beyond the “Linear” model. Choices and challenges. In: Xu, P., Liu, J., Dermanis, A.: (eds.) VI Hotine-Marussi Symposium on Theoretical and Computational Geodesy. International Association of Geodesy Symposia, vol. 132, pp. 111–118. Springer (2006) (Invited presentation at the VI Hotine-Marussi Symposium, Wuhan, 29 May–2 June 2006)Google Scholar
  32. 32.
    Dermanis, A.: On the alternative approaches to IITRF formulation. A theoretical comparison. IUGG General Assembly, Melbourne. In: Rizos, C., Willis, P. (eds.) Earth on the Edge: Science for a Sustainable Planet, International Association of Geodesy Symposia, vol. 139, pp. 223–229. Springer, Berlin/Heidelberg (2014)CrossRefGoogle Scholar
  33. 33.
    Dermanis, A.: Global reference systems: theory and open questions. Invited paper at the Academia dei Lincei Session, VIII Hotine-Marussi Symposium on Mathematical Geodesy, Rome, 17–21 June 2013. In: Sneeuw, N., Novák, P., Crespi, M., Sansò, F. (eds.) VIII Hotine-Marussi Symposium on Mathematical Geodesy, IAG Symposia, vol. 142, pp. 9–16. Springer International Publishing, Switzerland (2016)Google Scholar
  34. 34.
    Dermanis, A., Sansò, F.: Different equivalent approaches to the geodetic reference system. Rendiconti della Accademia dei Lincei, Scienze fisiche e naturali. On-Line-First (volume in print) (2018)Google Scholar
  35. 35.
    Dow, J., Neilan, R.E., Rizos, C.: The international GNSS service in a changing landscape of global navigation satellite systems. J. Geod. 83(3–4), 191–198 (2009). CrossRefGoogle Scholar
  36. 36.
    Elsner, J.B., Tsonis, A.A.: Singular Spectrum Analysis. A New Tool in Time Series Analysis. New York, Plenum Press (1996)CrossRefGoogle Scholar
  37. 37.
    Golyandina, N., Zhigljavsky, A.: Singular Spectrum Analysis for Time Series. Springer Briefs in Statistics. Springer (2013). ISBN:978-3-642-34912-6CrossRefGoogle Scholar
  38. 38.
    Grafarend, E., Schaffrin, B.: Unbiased free net adjustment. Surv. Rev. 22(171), 200–218 (1974)CrossRefGoogle Scholar
  39. 39.
    Grafarend, E., Schaffrin, B.: Equivalence of estimable quantities and invariants in geodetic networks. Zeitschrift für Vemessungswesen 101(11), 485–491 (1976)Google Scholar
  40. 40.
    Gross, J.: The general Gauss-Markov model with possibly singular dispersion matrix. J. Stat. Pap. 45, 311–336 (2004)CrossRefGoogle Scholar
  41. 41.
    Koch, K.-R.: Parameter estimation and hypothesis testing in linear models, 2nd edn. Springer, Berlin (1999)CrossRefGoogle Scholar
  42. 42.
    Kotsakis, C.: Generalized inner constraints for geodetic network densification problems. J. Geodesy 87, 661–673 (2013)CrossRefGoogle Scholar
  43. 43.
    Lavallée, D.A., van Dam, T., Blewitt, G., Clarke, P.J.: Geocenter motions from GPS: a unified observation model. J. Geophys. Res. Solid Earth 111(B5) (2006). CrossRefGoogle Scholar
  44. 44.
    Meindl, M., Beutler, G., Thaller, D., Dach, R., Jäggi, A.: Geocenter coordinates estimated from GNSS data as viewed by perturbation theory. Adv. Space Res. 51(7), 1047 (2013)CrossRefGoogle Scholar
  45. 45.
    Meissl, P.: Die innere Genauigkeit eines Punkthaufens. Österreichers Zeitschrift für Vermessungswesen 50, 159–165 and 186–194 (1962)Google Scholar
  46. 46.
    Meissl, P.: Uber die innere Genauigkeit dreidimensionalern Punkthaufen. Zeitschrift für Vermessungswesen, 1965, 90. Jahrgang, Heft 4, 109–118 (1965)Google Scholar
  47. 47.
    Meissl, P.: Zusammenfassung und Ausbau der inneren Fehlertheorie eines Punkthaufens. Deutsche Geodätische Kommission, Reihe A, Nr. 61, 8–21 (1969)Google Scholar
  48. 48.
    Moore, E.H.: On the reciprocal of the general algebraic matrix. Bull. Am. Math. Soc. 26(9), 394–95 (1920)CrossRefGoogle Scholar
  49. 49.
    Munk, W.H., MacDonald, G.J.F.: The Rotation of the Earth. Cambridge University Press, London (1960)Google Scholar
  50. 50.
    Penrose, R.: A generalized inverse for matrices. Proc. Cambridge Philos. Soc. 51, 406–413 (1955)CrossRefGoogle Scholar
  51. 51.
    Pearlman, M.R., Degnan, J.J., Bosworth, J.M.: The international laser ranging service. Adv. Space Res. 30(2), 135–143 (2002)CrossRefGoogle Scholar
  52. 52.
    Petit, G., Luzum, B.: IERS Conventions. IERS Technical Note No. 36, Verlag des Bundesamts für Kartographie und Geodäsie, Frankfurt am Main 2010. Working version under continuous updating is available at (2010)
  53. 53.
    Rangelova, E., van der Wal, W., Sideris, M.G., Wu, P.: Spatiotemporal analysis of the GRACE-derived mass variations in North America by means of multi-channel singular spectrum analysis. In: Mertikas, S.P. (ed.) Gravity, Geoid and Earth Observation, International Association of Geodesy Symposia, vol. 135, pp. 539–546. Springer, Berlin/Heidelberg (2010)CrossRefGoogle Scholar
  54. 54.
    Rao, C.R.: Unified Theory of Linear Estimation. Sankhya, Series A, vol. 33, pp. 371–394 (1971). Corrigenda. Sankhya, Series A, Springer, vol. 34, pp. 194, 477 (1972)Google Scholar
  55. 55.
    Rao, C.R.: Unified theory of least squares. Commun. Stat. Theory Methods 1(1), 1–8 (1973)Google Scholar
  56. 56.
    Rao, C.R.: Linear Statistical Inference and Its Applications, 2nd edn. Wiley, New York (1973)CrossRefGoogle Scholar
  57. 57.
    Rebischung, P., Altamimi, Z., Springer, T.: A colinearity diagnosis of the GNSS geocenter determination. J. Geod. 88(1), 65–85 (2014). CrossRefGoogle Scholar
  58. 58.
    Rothacher, M., Angermann, D., Artz, T., Bosch, W., Drewes, H., Gerstl, M., Kelm, R., König, D., König, R., Meisel, B., Müller, H., Nothnagel, A., Panafidina, N., Richter, B., Rudenko, S., Schwegmann, W., Seitz, M., Steigenberger, P., Tesmer, S., Tesmer, V., Thaller, D.: GGOS-D: homogeneous reprocessing and rigorous combination of space geodetic observations. J. Geod. 85, 679–705 (2011)CrossRefGoogle Scholar
  59. 59.
    Schuh, H., Behrend, D.: VLBI: a fascinating technique for geodesy and astrometry. J. Geodyn. 61, 68–80 (2012). CrossRefGoogle Scholar
  60. 60.
    Seitz, M., Angermann, D., Blossfeld, M., Drewes, H., Gerstl, M.: The 2008 DGFI realization of the ITRS: DTRF2008. J. Geod. 86, 1097–1123 (2012)CrossRefGoogle Scholar
  61. 61.
    Tisserand, F.: Traité de Mécanique Céleste. Gauthieu-Villars, Paris (1889)Google Scholar
  62. 62.
    Willis, P., Fagard, H., Ferraged, P., Lemoinee, F.G., Noll, C.E., Noomen, R., Otten, M., Ries, J.C., Rothacher, M., Soudarin, L., Tavernier, G., Valette, J.-J.: The international DORIS service: toward maturity. Adv. Space Res. 45(12), 1408–1420 (2010). CrossRefGoogle Scholar
  63. 63.
    Zhu, S.-Y., Mueller, I.I.: Effects of adopting new precession, nutation and equinox corrections on the terrestrial reference frame. Bull. Geod. 57(1983), 29–42 (1983)CrossRefGoogle Scholar

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© Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Geodesy and Surveying (DGS)Aristotle University of ThessalonikiThessalonikiGreece

Section editors and affiliations

  • Willi Freeden
    • 1
  1. 1.Geomathematics GroupUniversity of KaiserslauternKaiserslauternGermany

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