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Nonlinear Processes in Geophysics An interactive open-access journal of the European Geosciences Union
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© Author(s) 2019. This work is distributed under
the Creative Commons Attribution 4.0 License.
© Author(s) 2019. This work is distributed under
the Creative Commons Attribution 4.0 License.

Submitted as: research article 21 Oct 2019

Submitted as: research article | 21 Oct 2019

Review status
This discussion paper is a preprint. It is a manuscript under review for the journal Nonlinear Processes in Geophysics (NPG).

Application of Levy Processes in Modelling (Geodetic) Time Series With Mixed Spectra

Jean-Philippe Montillet1,2, Xiaoxing He3, and Kegen Yu4 Jean-Philippe Montillet et al.
  • 1Institute of Earth Surface Dynamics (IDYST), University of Lausanne, Lausanne, Switzerland
  • 2Space and Earth Geodetic Analysis laboratory (SEGAL), University Beira Interior, Covhila, Portugal
  • 3School of Civil Engineering and Architecture, East China Jiaotong University, Nan Chang, China
  • 4School of Environmental Science and Spatial Informatics, China University of Mining and Technology, Xuzhou, China

Abstract. Recently, various models have been developed, including the fractional Brownian motion (fBm), to analyse the stochastic properties of geodetic time series, together with the extraction of geophysical signals. The noise spectrum of these time series is generally modeled as a mixed spectrum, with a sum of white and coloured noise. Here, we are interested in modelling the residual time series, after deterministically subtracting geophysical signals from the observations. This residual time series is then assumed to be a sum of three random variables (r.v.), with the last r.v. belonging to the family of Levy processes. This stochastic term models the remaining residual signals and other correlated processes. Via simulations and real time series, we identify three classes of Levy processes: Gaussian, fractional and stable. In the first case, residuals are predominantly constituted of short-memory processes. Fractional Levy process can be an alternative model to the fBm in the presence of long-term correlations and self-similarity property. Stable process is characterized by a large variance, which can be satisfied in the case of heavy-tailed distributions. The application to geodetic time series implies potential anxiety in the functional model selection where missing geophysical information can generate such residual time series.

Jean-Philippe Montillet et al.
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Jean-Philippe Montillet et al.
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Short summary
Geodetic time series, series of observations measured from various satellites, must be modelled carefully to extract accurate information about geophysical processes. These models take into account the properties of the noise in these time series, which are generally a mixed of several kinds of noise. This work proposes a model based on the family of Levy processes (Gaussian, fractional and stable) as an alternative with real and simulated data.
Geodetic time series, series of observations measured from various satellites, must be modelled...