Journal of Geodesy

, Volume 87, Issue 4, pp 351–360 | Cite as

Fast error analysis of continuous GNSS observations with missing data

  • M. S. Bos
  • R. M. S. Fernandes
  • S. D. P. Williams
  • L. Bastos
Original Article


One of the most widely used method for the time-series analysis of continuous Global Navigation Satellite System (GNSS) observations is Maximum Likelihood Estimation (MLE) which in most implementations requires \(\mathcal{O }(n^3)\) operations for \(n\) observations. Previous research by the authors has shown that this amount of operations can be reduced to \(\mathcal{O }(n^2)\) for observations without missing data. In the current research we present a reformulation of the equations that preserves this low amount of operations, even in the common situation of having some missing data.Our reformulation assumes that the noise is stationary to ensure a Toeplitz covariance matrix. However, most GNSS time-series exhibit power-law noise which is weakly non-stationary. To overcome this problem, we present a Toeplitz covariance matrix that provides an approximation for power-law noise that is accurate for most GNSS time-series.Numerical results are given for a set of synthetic data and a set of International GNSS Service (IGS) stations, demonstrating a reduction in computation time of a factor of 10–100 compared to the standard MLE method, depending on the length of the time-series and the amount of missing data.


GNSS Error Power-law 



The authors would like to thank the assistant editor Jeff Freymueller and three anonymous reviewers for their thorough and constructive review of our manuscript. This work was funded by national funds through FCT in the scope of the Project PesTC/ Mar/LA0015/2011.


  1. Agnew DC (1992) The time-domain behaviour of power-law noises. Geophys Res Lett 19(4):333–336CrossRefGoogle Scholar
  2. Altamimi Z, Collilieux X, Mtivier L (2011) ITRF2008: an improved solution of the international terrestrial reference frame. J Geod 85(8):457–473. doi: 10.1007/s00190-011-0444-4 CrossRefGoogle Scholar
  3. Amiri-Simkooei AR (2009) Noise in multivariate GPS position time-series. J Geod 83:175–187. doi: 10.1007/s00190-008-0251-8 CrossRefGoogle Scholar
  4. Ammar GS, Gragg WB (1988) Superfast solution of real positive definite Toeplitz systems. SIAM J Matrix Anal Appl 9:61–76CrossRefGoogle Scholar
  5. Bos MS, Fernandes RMS, Williams SDP, Bastos L (2008) Fast error analysis of continuous GPS observations. J Geod 82:157–166. doi: 10.1007/s00190-007-0165-x Google Scholar
  6. Brualdi RA, Schneider H (1983) Determinantal identities: Gauss, Schur, Cauchy, Sylvester, Kronecker, Jacobi, Binet, Laplace, Muir, and Cayley. Linear Algebra Appl 52(53):769–791Google Scholar
  7. Dow JM, Neilan RE, Rizos C (2009) The International GNSS Service in a changing landscape of Global Navigation Satellite Systems. J Geod 83:191–198. doi: 10.1007/s00190-008-0300-3 CrossRefGoogle Scholar
  8. Gohberg IC, Semencul AA (1972) On the inversion of finite Toeplitz matrices and their continuous analogs. Mat Issled 7(2):201–223Google Scholar
  9. Hackl M, Malservisi R, Hugentobler U, Wonnacott R (2011) Estimation of velocity uncertainties from GPS time series: examples from the analysis of the South African TrigNet network. J Geophys Res 116(B15):B11404Google Scholar
  10. Hosking JRM (1981) Fractional differencing. Biometrika 68:165–176CrossRefGoogle Scholar
  11. Johnson HO, Agnew DC (1995) Monument motion and measurements of crustal velocities. Geophys Res Lett 22(21):2905–2908CrossRefGoogle Scholar
  12. Kasdin NJ (1995) Discrete simulation of colored noise and stochastic processes and \(1/f^\alpha \) power-law noise generation. Proc IEEE 83(5):802–827CrossRefGoogle Scholar
  13. Langbein J (2004) Noise in two-color electronic distance meter measurements revisited. J Geophys Res 109(B04406). doi: 10.1029/2003JB002819
  14. Langbein J (2010) Computer algorithm for analyzing and processing borehole strainmeter data. Comput Geosci 36(5):611–619. doi: 10.1016/j.cageo.2009.08.011 CrossRefGoogle Scholar
  15. Langbein J (2012) Estimating rate uncertainty with maximum likelihood: differences between power-law and flicker-random-walk models. J Geod 86:775–783. doi: 10.1007/s00190-012-0556-5 CrossRefGoogle Scholar
  16. Mao A, Harrison CGA, Dixon TH (1999) Noise in GPS coordinate time series. J Geophys Res 104(B2):2797–2816. doi: 10.1029/1998JB900033 Google Scholar
  17. Santamaría-Góme A, Bouin MN, Collilieux X, Wöppelmann G (2011) Correlated errors in GPS position time series: implications for velocity estimates. J Geophys Res 116(B15):B01405Google Scholar
  18. Trench WF (1964) An algorithm for the inversion of finite Toeplitz matrices. J Soc Indust Appl Math 12:515–522CrossRefGoogle Scholar
  19. Webb FH, Zumberge JF (1995) An introduction to GIPSY/OASIS-II. Technical report JPL D-11088, California Institute of Technology, Pasadena, CAGoogle Scholar
  20. Williams SDP (2003) The effect of coloured noise on the uncertainties of rates from geodetic time series. J Geod 76(9–10):483–494. doi: 10.1007/s00190-002-0283-4 CrossRefGoogle Scholar
  21. Williams SDP (2008) CATS: GPS coordinate time series analysis software. GPS Solutions 12(2):147–153. doi: 10.1007/s10291-007-0086-4
  22. Williams SDP, Bock Y, Fang P, Jamason P, Nikolaidis RM, Prawirodirdjo L, Miller M, Johnson DJ (2004) Error analysis of continuous GPS position time series. J Geophys Res 109(B03412). doi: 10.1029/2003JB002741
  23. Zhang J, Bock Y, Johnson H, Fang P, Williams S, Genrich J, Wdowinski S, Behr J (1997) Southern California Permanent GPS Geodetic Array: error analysis of daily position estimates and site velocities. J Geophys Res 102(B8):18035–18055Google Scholar
  24. Zumberge JF, Heflin MB, Jefferson DC, Watkins MM, Webb FH (1997) Precise point positioning for the efficient and robust analysis of GPS data from large networks. J Geophys Res 102(B3):5005–5018. doi: 10.1029/96JB03860 Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • M. S. Bos
    • 1
  • R. M. S. Fernandes
    • 2
    • 3
  • S. D. P. Williams
    • 4
  • L. Bastos
    • 1
    • 5
  1. 1.CIIMAR/CIMAR, University of PortoPortoPortugal
  2. 2.University of Beira Interior, IDLCovilhãPortugal
  3. 3.Delft Earth-Oriented Space Research (DEOS), DUTDelftThe Netherlands
  4. 4.National Oceanography Centre, LiverpoolLiverpoolUK
  5. 5.Faculty of Science, DGAOTUniversity of PortoPortoPortugal

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