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

TL;DR

This paper explores the use of Levy processes to model residual geodetic time series with mixed spectra, offering alternatives to traditional models like fractional Brownian motion, and demonstrates their applicability through simulations and real data.

Contribution

It introduces a novel framework for modeling residual geodetic time series using Levy processes, including Gaussian, fractional, and stable types, enhancing the understanding of their stochastic properties.

Findings

Fractional Levy processes can model long-term correlations.

Stable Levy processes capture heavy-tailed residuals.

Residuals are often short-memory or heavy-tailed depending on the Levy process.

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 imply potential anxiety in the functional model selection where missing geophysical information can generate such residual time series.