Advertisement

Mathematical Geosciences

, Volume 47, Issue 6, pp 627–646 | Cite as

Modeling Geodetic Processes with Levy \(\alpha \)-Stable Distribution and FARIMA

  • Jean-Philippe Montillet
  • Kegen YuEmail author
Article

Abstract

Over the last years the scientific community has been using the autoregressive moving average (ARMA) model in the modeling of the noise in global positioning system (GPS) time series (daily solution). This work starts with the investigation of the limit of the ARMA model which is widely used in signal processing when the measurement noise is white. Since a typical GPS time series consists of geophysical signals (e.g., seasonal signal) and stochastic processes (e.g., coloured and white noise), the ARMA model may be inappropriate. Therefore, the application of the fractional auto-regressive integrated moving average (FARIMA) model is investigated. The simulation results using simulated time series as well as real GPS time series from a few selected stations around Australia show that the FARIMA model fits the time series better than other models when the coloured noise is larger than the white noise. The second fold of this work focuses on fitting the GPS time series with the family of Levy \(\alpha \)-stable distributions. Using this distribution, a hypothesis test is developed to eliminate effectively coarse outliers from GPS time series, achieving better performance than using the rule of thumb of \(n\) standard deviations (with \(n\) chosen empirically).

Keywords

GPS Time series analysis Coloured noise Hurst parameter Fractional ARIMA ARMA Hypothesis test  Outliers 

Notes

Acknowledgments

This research is supported by the Australian Research Council Grant Number DP0877381. J.-P. Montillet thanks the Geodesy group at the Australian National University (in particular Dr. Paul Tregoning) for the discussions. The authors also acknowledge the comments of the anonymous reviewers to improve the manuscript.

References

  1. Amiri-Simkooei AR, Tiberius CC, Teunissen PJG (2007) Assessment of noise in GPS coordinate time series: methodology and results. J Geophys Res 112(B07413). doi: 10.1029/2006JB004913
  2. Blewitt G, Lavallée D (2002) Effect of annual signals on geodetic velocity. J Geophys Res 107. doi: 10.1029/2001JB000570
  3. Boehm J, Werl B, Schuh H (2006) Troposphere mapping functions for GPS and very long baseline interferometry from European Centre for medium-range weather forecasts operational analysis data. J Geophys Res 111(B02406). doi: 10.1029/2005JB003629
  4. Bos MS, Fernandes RM, Williams SDP, Bastos L (2012) Fast error analysis of continuous GNSS observations with missing data. J Geod. doi: 10.1007/s00190-012-0605-0
  5. Calais E (1999) Continuous GPS measurements across the Western Alps, 1996–1998. Geophys J Int 138:221–230CrossRefGoogle Scholar
  6. Cheung YW, Lai KS (1995) A search for long memory in international stock market returns. J Int Money Financ 14:597–615CrossRefGoogle Scholar
  7. Cochrane JH (1988) How big is the random walk in GNP? J Polit Econ 96:893–920CrossRefGoogle Scholar
  8. Contreras-Reyes JE, Palma W (2013) Statistical analysis of autoregressive fractionally integrated moving average models in R. Comput Stat 28(5):2309–2331. doi: 10.1007/s00180-013-0408-7 CrossRefGoogle Scholar
  9. Darbeheshti N, Zhou L, Tregoning P, McClusky S, Purcell A (2013) The ANU GRACE visualisation web portal. Comput Geosci 52:227–233. doi: 10.1016/j.cageo.2012.10.005 CrossRefGoogle Scholar
  10. Davis JL, Wernicke BP, Tamisiea ME (2012) On seasonal signals in geodetic time series. J Geophys Res 117(B01403). doi: 10.1029/2011JB008690
  11. Ding ZW (2009) ARMA model Of ARFIMA model and its bayesian statistical inference and empirical analysis. Master Dissertation. http://globethesis.com/?t=2190360278969346
  12. Doblic V, Scansaroli DJ, Storer RH (2004) New estimators of the hurst index for fractional Brownian motion. Report 11T–004. http://www.lehigh.edu/ise/documents/11t_004
  13. Granger CW, Joyeux R (1980) An introduction to long-memory time series models and fractional differencing. J Time Ser Anal 1:15–29CrossRefGoogle Scholar
  14. Haykin S (2002) Adaptive filter theory, 4th edn. Prentice Hall, Upper Saddle River, chap 1, p 52Google Scholar
  15. Herring TA, King RW, McClusky SC (2010) Introduction to GAMIT/GLOBK, report. MIT, Cambridge. available at: http://www-gpsg.mit.edu/~simon/gtgk/docs.htm
  16. Hurst HE, Black R, Sinaika YM (1965) Long term-storage: an experimental study. Constable, LondonGoogle Scholar
  17. Kay SM (1993) Fundamentals of statistical signal processing: estimation theory. Prentice Hall, Englewood CliffsGoogle Scholar
  18. Koutrouvelis IA (1980) Regression-type estimation of the parameters of stable laws. J Am Stat Assoc 75:918–928CrossRefGoogle Scholar
  19. Li Q, Tao B (1982) Theory ofprobability and statistics, and its applications to geodesy. Surv Mapp 321Google Scholar
  20. Li J, Miyashita K, Kato T, Miyazaki S (2000) GPS time series modeling by autoregressive moving average method: application to the crustal deformation in central Japan. Earth Planet Space 52:155–162CrossRefGoogle Scholar
  21. Li B, Shen Y, Lou L (2011) Efficient estimation of variance and covariance components: a case study for GPS stochastic model evaluation. IEEE Trans Geosci Remote Sens 49(1):203–210CrossRefGoogle Scholar
  22. Mandelbrodt B, Van Ness JW (1968) Fractional Brownian motions. fractional noises and applications. SIAM Rev 10(4):422–437CrossRefGoogle Scholar
  23. Mao A, Harrison CG, Dixon TH (1999) Noise in GPS coordinate time series. J Geophys Res 104(B2):2797–2816. doi: 10.1029/1998JB900033 CrossRefGoogle Scholar
  24. Montillet J-P, Yu K (2011) Leaky LMS algorithm and fractional Brownian motion model for GNSS receiver position estimation. In: Proceedings of the IEEE vehicular technology conference (VTC’11 fall). doi: 10.1109/VETECF.2011.6092850
  25. Montillet J-P, Tregoning P, McClusky S, Yu K (2013) Extracting white noise statistics in GPS coordinate time series. IEEE Geosci Remote Sens Lett 10(3):563–567. doi: 10.1109/LGRS.2012.2213576 CrossRefGoogle Scholar
  26. Montillet J-P, McClusky S, Yu K (2013) Extracting colored noise statistics in time series via Negentropy. IEEE Signal Process Lett 20(9):857–860. doi: 10.1109/LSP.2013.2271241 CrossRefGoogle Scholar
  27. Nikias CL, Shao M (1995) Signal processing with alpha-stable distributions and applications. Wiley edition, New YorkGoogle Scholar
  28. Nolan JP, Gonzalez JG, Nunez RC (2010) Stable filters: A robust signal processing framework for heavy-tailed noise. In: Proceedings of the IEEE radar conference, 2010. pp 470–473Google Scholar
  29. Olivares G, Teferle FN (2013) A Bayesian Monte Carlo Markov chain method for parameter estimation of fractional differenced Gaussian processes. IEEE Trans Signal Process 61(9):2405–2412. doi: 10.1109/TSP.2013.2245658 CrossRefGoogle Scholar
  30. Panas E (2001) Estimating fractal dimension using stable distributions and exploring long memory through ARFIMA models in Athens Stock Exchange. Appl Fin Econ 11(4):395–402CrossRefGoogle Scholar
  31. Perfetti N (2006) Detection of station coordinate discontinuities within the Italian GPS Fiducial network. J Geod 80:381–396CrossRefGoogle Scholar
  32. Sowell F (1991) Modeling long-run behavior with the fractional ARIMA model. J Monet Econ 29:277–302CrossRefGoogle Scholar
  33. Srinivasan R, Rangaswamy M (2005) Fast estimation of false alarm probabilities of STAP detectors—the AMF. In: Proceedings of the IEEE international radar conference. pp 413–418. doi: 10.1109/RADAR.2005.1435860
  34. Stoev S, Taqqu MS (2003) Wavelet estimation for the Hurst parameter in stable processes. Lect Notes Phys 621:61–87CrossRefGoogle Scholar
  35. Tregoning P, Watson C (2009) Atmospheric effects and spurious signals in GPS analyses. J Geophys Res 114(B09403). doi: 10.1029/2009JB006344
  36. Wang C, Liao M, Li X (2008) Ship detection in SAR image based on the alpha-stable distribution. Sensors 8:4948–4960. doi: 10.3390/s8084948 CrossRefGoogle Scholar
  37. Wang J, Knight NL (2012) New outlier separability test and its application in GNSS positioning. J Global Position Syst 11(1):46–57. doi: 10.5081/jgps.11.1.46 CrossRefGoogle Scholar
  38. Williams SDP (2003) The effect of coloured noise on the uncertainties of rates estimated from geodetic time series. J Geod 76:483–494CrossRefGoogle Scholar
  39. 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
  40. Wooldridge JM (2010) Econometric analysis of cross section and panel data, 1st edn. MIT Press, Cambridge. ISBN:13:9780262232586Google Scholar

Copyright information

© International Association for Mathematical Geosciences 2014

Authors and Affiliations

  1. 1.Australian National UniversityCanberraAustralia
  2. 2.Cascadia Hazards InstituteEllensburgUSA
  3. 3.School of Geodesy and GeomaticsWuhan UniversityWuhanChina

Personalised recommendations