Using Modified Allan Variance for Time Series Analysis

  • Z. MalkinEmail author
Conference paper
Part of the International Association of Geodesy Symposia book series (IAG SYMPOSIA, volume 138)


Allan Variance (AVAR) was introduced more than 40 years ago as an estimator of the stability of frequency standards. Now it is also used for investigations of time series in astronomy and geodesy. However, there are several issues with this method that need special consideration. First, unlike frequency measurements, astronomical and geodetic time series usually consist of data points with unequal uncertainties. Thus one needs to apply data weighting during statistical analysis. Second, some sets of scalar time series naturally form multidimensional vector series. For example, Cartesian station coordinates form the 3D station position vector. The original AVAR definition does not allow one to process unevenly weighted and/or multidimensional data. To overcome these deficiencies, AVAR modifications were proposed in Malkin (2008. On the accuracy assessment of celestial reference frame realizations. J Geodesy 82: 325–329). In this paper, we give some examples of processing geodetic and astrometric time series using the classical and the modified AVAR approaches, and compare the results.


Allan variance Time series analysis 



This work has made use of data and products provided by the International VLBI Service for Geodesy and Astrometry (IVS, Schlueter and Behrend 2007) and European Permanent GPS Network (EPN, Bruyninx and Roosbeek 2006). The author is grateful to two anonymous reviewers for careful reading of the manuscript and helpful comments and suggestions.


  1. Allan DW (1966) Statistics of atomic frequency standards. Proc IEEE 54:221–230CrossRefGoogle Scholar
  2. Bregni S, Primerano L (2005) Using the modified allan variance for accurate estimation of the hurst parameter of long-range dependent traffic. In: arXiv: cs/0510006, MilanoGoogle Scholar
  3. Bruyninx C, Roosbeek F (2006) The EUREF permanent network: recent achievements. EUREF Publication No. 16, Mitteilungen des Bundesamtes für Kartographie und Geodäsie, Band, vol 40, pp 105–112Google Scholar
  4. Feissel M, Gontier A-M, Eubanks TM (2000) Spatial variability of compact extragalactic radiosources. Astron Astr 359:1201–1204Google Scholar
  5. Feissel-Vernier M (2003) Selecting stable extragalactic compact radio sources from the permanent astrogeodetic VLBI program. Astron Astr 403:105–110. doi: 10.1051/0004-6361:20030348 CrossRefGoogle Scholar
  6. Feissel-Vernier M, Le Bail K, Berio P et al (2006) Geocentre motion measured with DORIS and SLR, and predicted by geophysical models. J Geodesy 80:637–648. doi: 10.1007/s00190-006-0079-z CrossRefGoogle Scholar
  7. Feissel-Vernier M, de Viron O, Le Bail K (2007) Stability of VLBI, SLR, DORIS, and GPS positioning. Earth Planets Space 59:475–497Google Scholar
  8. Fey AL, Ma C, Arias EF et al (2004) The second extension of the international celestial reference frame: ICRF-Ext.2. Astron J 127:3587–3608. doi: 10.1086/420998 CrossRefGoogle Scholar
  9. Gambis D (2002) Allan variance in earth rotation time series analysis. Adv Space Res 30:207–212. doi: 10.1016/S0273-1177(02)00286-7 CrossRefGoogle Scholar
  10. Gontier A-M, Le Bail K, Feissel M, Eubanks T-M (2001) Stability of the extragalactic VLBI reference frame. Astron Astr 375:661–669. doi: 10.1051/0004-6361:20010707 CrossRefGoogle Scholar
  11. Gordon D, Ma C, Gipson J, Petrov L, MacMillan D (2008) On selection of “defining” sources for ICRF2. In: Finkelstein A, Behrend D (eds) Proceedings of fifth IVS general meeting, pp 261–264Google Scholar
  12. Le Bail K (2006) Estimating the noise in space-geodetic positioning: the case of DORIS. J Geodesy 80:541–565. doi: 10.1007/s00190-006-0088-y CrossRefGoogle Scholar
  13. Le Bail K, Feissel-Vernier M (2003) Time series statistics of the DORIS and GPS colocated observations. Geophysical Reseaech Abstracts, EGS-AGU-EUG Joint Assembly, vol 5, Nice, 6–11 April 2003, p 04078Google Scholar
  14. Ma C, Arias EF, Eubanks TM et al (1998) The international celestial reference frame as realized by very long baseline interferometry. Astron J 116:516–546CrossRefGoogle Scholar
  15. Ma C, Arias EF, Bianko G et al. (2009) The second realization of the international celestial reference frame by very long baseline interferometry. In: Fey A, Gordon D, Jacobs CS (eds) Presented on behalf of the IERS/IVS working group, IERS technical note no. 35, Frankfurt am Main, Verlag des Bundesamts für Kartographie und GeodäsieGoogle Scholar
  16. Malkin ZM (2007) Empiric models of the Earth’s free core nutation. Solar Syst Res 41:492–497. doi: 10.1134/S0038094607060044 CrossRefGoogle Scholar
  17. Malkin Z (2008a) On the accuracy assessment of celestial reference frame realizations. J Geodesy 82:325–329. doi: 10.1007/s00190-007-0181-x CrossRefGoogle Scholar
  18. Malkin Z (2008b) On construction of ICRF-2. In: Finkelstein A, Behrend D (eds) Proceedings of fifth IVS general meeting, pp 256–260Google Scholar
  19. Malkin Z (2009) Some results of analysis of source position time series. IVS Memorandum 2009-001v01.
  20. Malkin ZM (2011) Study of astronomical and geodetic series using the allan variance. Kinemat Phys Celest Bodies 27:42–49. doi: 10.3103/S0884591311010053 CrossRefGoogle Scholar
  21. Malkin ZM, Voinov AV (2001) Preliminary results of processing EUREF network observations using a non-fiducial strategy. Phys Chem Earth (A) 26:579–583. doi: 10.1016/S1464-1895(01)00104-1 CrossRefGoogle Scholar
  22. Schlueter W, Behrend D (2007) The international VLBI service for geodesy and astrometry (IVS): current capabilities and future prospects. J Geodesy 81:379–387. doi: 10.1007/s00190-006-0131-z CrossRefGoogle Scholar
  23. Sokolova J, Malkin Z (2007) On comparison and combination of catalogues of radio source positions. Astron Astr 474:665–670. doi: 10.1051/0004-6361:20077450 CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Pulkovo ObservatorySt. PetersburgRussia

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