Journal of Geodesy

, Volume 86, Issue 9, pp 775–783

Estimating rate uncertainty with maximum likelihood: differences between power-law and flicker–random-walk models

Open Access
Original Article


Recent studies have documented that global positioning system (GPS) time series of position estimates have temporal correlations which have been modeled as a combination of power-law and white noise processes. When estimating quantities such as a constant rate from GPS time series data, the estimated uncertainties on these quantities are more realistic when using a noise model that includes temporal correlations than simply assuming temporally uncorrelated noise. However, the choice of the specific representation of correlated noise can affect the estimate of uncertainty. For many GPS time series, the background noise can be represented by either: (1) a sum of flicker and random-walk noise or, (2) as a power-law noise model that represents an average of the flicker and random-walk noise. For instance, if the underlying noise model is a combination of flicker and random-walk noise, then incorrectly choosing the power-law model could underestimate the rate uncertainty by a factor of two. Distinguishing between the two alternate noise models is difficult since the flicker component can dominate the assessment of the noise properties because it is spread over a significant portion of the measurable frequency band. But, although not necessarily detectable, the random-walk component can be a major constituent of the estimated rate uncertainty. None the less, it is possible to determine the upper bound on the random-walk noise.


Rate Uncertainty Bias Least squares 


  1. Agnew D (1992) The time domain behavior of power law noises. Geophys Res Lett 19(4): 333–336. doi:10.1029/91GL02832 CrossRefGoogle Scholar
  2. Amiri-Simkooei AR, Tiberius CCJM, Teunissen PJG (2007) Assessment of noise in GPS coordinate time series: methodology and results. J Geophys Res 112. doi:10.1029/2006JB004913
  3. Beavan J (2005) Noise properties of continuous GPS data from concrete pillar geodetic monuments in New Zealand and comparison with data from U.S. deep drilled braced monuments. J Geophys Res 110: B08410. doi:10.1029/2005JB003642 CrossRefGoogle Scholar
  4. Calais E, Stein S (2009) Time-variable deformation in the New Madrid Seismic zone. Science 323: 1442CrossRefGoogle Scholar
  5. Davis JL, Wernicke BP, Tamisiea ME (2012) On seasonal signals in geodetic time series. J Geophys Res 117: B01403. doi:10.1029/2011JB008690 CrossRefGoogle Scholar
  6. Frankel A, Smalley R (2011) Significant motion between GPS sites in the New Madrid region; implications for seismic hazard. Bull Seismol Soc Am (submitted)Google Scholar
  7. Hill EM, Davis JL, Elósegui P, Wernicke BP, Malikowski E, Niemi NA (2009) Characterization of site-specific GPS errors using short-baseline of braced monuments at Yucca Mountain, southern Nevada. J Geophys Res 114: B11402. doi:10.1029/2008JB006027 CrossRefGoogle Scholar
  8. Hough SE, Page M (2011) Toward a consistent model for strain accrual and release for the New Madrid Seismic Zone, Central United States. J Geophys Res 116. doi:10.1029/2010JB007783
  9. Johnston MJS, Linde AT (2002) Implications of crustal strain during conventional, slow, and silent earthquakes. In: Lee WHK (ed) et al International Handbook of Earthquake Engineering and Seismology, vol 81A, chap 36. Int Assoc Seismol Phys Earths Interior, pp 589–605Google Scholar
  10. King MA, Williams SDP (2009) Apparent stability of GPS monumentation from short-baseline time series. J Geophys Res 114: B01403CrossRefGoogle Scholar
  11. Khodabandeh A, Amiri-Simkooei AR, Sharifi MA (2011) GPS position time-series analysis based on asymptotic normality of M-estimation. J Geod 85 doi:10.1007/s00190-011-0489-4
  12. Langbein J, Breckenridge K, Quilty E (1993) Sensitivity of crustal deformation instruments to changes in secular rate. Geophys Res Lett 20(2): 85–88CrossRefGoogle Scholar
  13. Langbein J, Johnson H (1997) Correlated error in geodetic time series: implications for time-dependent deformation. J Geophys Res 102: 591–604CrossRefGoogle Scholar
  14. Langbein J (2004) Noise in two-color electronic distance meter measurements revisited. J Geophys Res 109: B04406. doi:10.1029/2003JB002819 CrossRefGoogle Scholar
  15. Langbein J (2008) Noise in GPS displacement measurements from Southern California and Southern Nevada. J Geophys Res 113: B05405. doi:10.1029/2007JB005247 CrossRefGoogle Scholar
  16. Mao A, Harrison CGA, Dixon TH (1999) Noise in GPS coordinate time series. J Geophys Res 104: 2797–2816CrossRefGoogle Scholar
  17. Santamaría-Gómez A, Bouin MN, Collilieux X, Wöppelmann G (2011) Correlated errors in GPS position time series: implications for velocity estimates. J Geophys Res 116. doi:10.1029/2010JB007701
  18. Wdowinski S, Bock Y, Zhang J, Fang P, Genrich J (1997) Southern California permanent GPS geodetic array: spatial filtering of daily positions for estimating coseismic and postseismic displacements induced by the 1992 Landers earthquake. J Geophys Res 102(B8): 18057–18070CrossRefGoogle Scholar
  19. Williams SDP (2003) The effect of coloured noise on the uncertainties of rates estimated from geodetic time series. J Geod 76: 483–494. doi:10.1007/s00190-002-0283-4 CrossRefGoogle Scholar
  20. 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. doi:10.1029/2003JB002741
  21. Wyatt F (1982) Displacement of surface monuments: horizontal motion. J Geophys Res 87: 979–989. doi:10.1029/JB087iB02p00979 CrossRefGoogle Scholar
  22. Wyatt FK (1989) Displacement of surface monuments: vertical motion. J Geophys Res 94: 1655–1664. doi:10.1029/JB094iB02p01655 CrossRefGoogle Scholar
  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): 18035–18055. doi:10.1029/97JB01380 CrossRefGoogle Scholar

Copyright information

© The Author(s) 2012

Authors and Affiliations

  1. 1.US Geological SurveyEarthquake Science CenterMenlo ParkUSA

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