Skip to main content
Log in

Fast error analysis of continuous GPS observations

  • Original Article
  • Published:
Journal of Geodesy Aims and scope Submit manuscript


It has been generally accepted that the noise in continuous GPS observations can be well described by a power-law plus white noise model. Using maximum likelihood estimation (MLE) the numerical values of the noise model can be estimated. Current methods require calculating the data covariance matrix and inverting it, which is a significant computational burden. Analysing 10 years of daily GPS solutions of a single station can take around 2 h on a regular computer such as a PC with an AMD AthlonTM 64 X2 dual core processor. When one analyses large networks with hundreds of stations or when one analyses hourly instead of daily solutions, the long computation times becomes a problem. In case the signal only contains power-law noise, the MLE computations can be simplified to a \({\mathcal{O}}(N\log N)\) process where N is the number of observations. For the general case of power-law plus white noise, we present a modification of the MLE equations that allows us to reduce the number of computations within the algorithm from a cubic to a quadratic function of the number of observations when there are no data gaps. For time-series of three and eight years, this means in practise a reduction factor of around 35 and 84 in computation time without loss of accuracy. In addition, this modification removes the implicit assumption that there is no environment noise before the first observation. Finally, we present an analytical expression for the uncertainty of the estimated trend if the data only contains power-law noise.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  • Abramowitz M, Stegun IA (1965) Handbook of mathematical functions with formulas, graphs and mathematical tables. National Bureau of standards applied mathematics series 55. US Government Printing Office, Washington DC

  • Agnew DC (1992) The time-domain behaviour of power-law noises. Geophys Res Lett 19(4):333–336

    Article  Google Scholar 

  • Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J, Du Croz J, Greenbaum A, Hammarling S, McKenney A, Sorensen D (1999) LAPACK users’ guide, 3rd edn. Society for Industrial and Applied Mathematics, Philadelphia

  • Bojanczyk AW, Brent RP, de Hoog FR, Sweet DR (1995) On the stability of the Bareiss and related Toeplitz factorization algorithms. SIAM J Matrix Anal Appl 16(1):40–57

    Article  Google Scholar 

  • Hosking JRM (1981) Fractional differencing. Biometrika 68(1):165–176

    Article  Google Scholar 

  • Johnson HO, Agnew DC (1995) Monument motion and measurements of crustal velocities. Geophys Res Lett 22(21):2905–2908

    Article  Google Scholar 

  • Kasdin NJ (1995) Discrete simulation of colored noise and stochastic processes and 1/f α power law noise generation. Proc IEEE 83(5):802–827

    Article  Google Scholar 

  • Langbein J (2004) Noise in two-color electronic distance meter measurements revisited. J Geophys Res 109, B04406, doi:10.1029/ 2003JB002819

  • Langbein J, Johnson H (1997) Correlated errors in geodetic time series: Implications for time-dependent deformation. J Geophys Res 102(B1):591–603

    Article  Google Scholar 

  • Mao A, Harrison CGA, Dixon TH (1999) Noise in GPS coordinate time series. J Geophys Res 104(B2):2797–2816

    Article  Google Scholar 

  • Press WH, Flannery BP, Teukolsky SA, Vetterling WT (1988) Numerical recipes in C. 2nd edn. University Press, Cambridge

    Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

  • 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):18035–18055

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to M. S. Bos.

Electronic supplementary material

Below are the electronic supplementary materials.

ESM 1 (TXT 3.38 kb)

ESM 2 (TXT 2.77 kb)

ESM 3 (TXT 2.38 kb)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bos, M.S., Fernandes, R.M.S., Williams, S.D.P. et al. Fast error analysis of continuous GPS observations. J Geod 82, 157–166 (2008).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: