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Journal of Geodesy

, Volume 82, Issue 3, pp 157–166 | Cite as

Fast error analysis of continuous GPS observations

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

Abstract

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.

Keywords

GPS Power-law Correlated noise Time-series analysis 

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Supplementary material

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References

  1. 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 DCGoogle Scholar
  2. Agnew DC (1992) The time-domain behaviour of power-law noises. Geophys Res Lett 19(4):333–336CrossRefGoogle Scholar
  3. 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, PhiladelphiaGoogle Scholar
  4. 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):40–57CrossRefGoogle Scholar
  5. Hosking JRM (1981) Fractional differencing. Biometrika 68(1):165–176CrossRefGoogle Scholar
  6. Johnson HO, Agnew DC (1995) Monument motion and measurements of crustal velocities. Geophys Res Lett 22(21):2905–2908CrossRefGoogle Scholar
  7. Kasdin NJ (1995) Discrete simulation of colored noise and stochastic processes and 1/f α power law noise generation. Proc IEEE 83(5):802–827CrossRefGoogle Scholar
  8. Langbein J (2004) Noise in two-color electronic distance meter measurements revisited. J Geophys Res 109, B04406, doi:10.1029/ 2003JB002819Google Scholar
  9. Langbein J, Johnson H (1997) Correlated errors in geodetic time series: Implications for time-dependent deformation. J Geophys Res 102(B1):591–603CrossRefGoogle Scholar
  10. Mao A, Harrison CGA, Dixon TH (1999) Noise in GPS coordinate time series. J Geophys Res 104(B2):2797–2816CrossRefGoogle Scholar
  11. Press WH, Flannery BP, Teukolsky SA, Vetterling WT (1988) Numerical recipes in C. 2nd edn. University Press, CambridgeGoogle Scholar
  12. 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/s00190-002-0283-4CrossRefGoogle Scholar
  13. 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:10.1029/2003JB002741Google Scholar
  14. 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):18035–18055CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • M. S. Bos
    • 1
    Email author
  • R. M. S. Fernandes
    • 2
    • 3
  • S. D. P. Williams
    • 4
  • L. Bastos
    • 1
  1. 1.University of Porto, Monte da VirgemVila Nova de GaiaPortugal
  2. 2.University of Beira Interior, CGUL, IDLCovilhãPortugal
  3. 3.Delft University of Technology, DEOSDelftThe Netherlands
  4. 4.Proudman Oceanographic LaboratoryLiverpoolUK

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