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


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).


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



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.


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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

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