Recently, various models have been developed, including the fractional Brownian motion (fBm), to analyse the stochastic properties of geodetic time series together with the estimated geophysical signals. The noise spectrum of these time series is generally modelled 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 stochastic processes, including the family of Lévy processes. The introduction of a third stochastic term models the remaining residual signals and other correlated processes. Via simulations and real time series, we identify three classes of Lévy processes, namely Gaussian, fractional and stable. In the first case, residuals are predominantly constituted of short-memory processes. The fractional Lévy process can be an alternative model to the fBm in the presence of long-term correlations and self-similarity properties. The stable process is here restrained to the special case of infinite variance, which can be only satisfied in the case of heavy-tailed distributions in the application to geodetic time series. Therefore, the model implies potential anxiety in the functional model selection, where missing geophysical information can generate such residual time series.

Among the geodetic data, Global Navigation Satellite System (GNSS) time series have been of particular interest for the study of geophysical phenomena at regional and global scales

In addition, recent studies

This work discusses several statistical assumptions (i.e. stationary properties and presence of long-term correlations) on the underlying processes in the GNSS time series, justifying the application of the fBm and the family of Lévy

Here, the statistical modelling is applied to residual time series following

The next section starts with the statistical inference on the residual geodetic time series, including the application of the fBm model and the relationship with the fractional autoregressive integrated moving average (FARIMA) model. Section

Let us model the GNSS observations and residual time series as an additive model as follows:

We jointly estimate the functional and stochastic models in order to produce

This function must be maximized. Assuming that the covariance matrix

In the modelling of the GNSS time series, a strong assumption is the so-called Gauss–Markov hypothesis

Moreover, if the probability density function of the noise is Gaussian or has a different density function with a finite value of variance, its fractal properties can be described by the Hurst parameter (

Long-memory processes are modelled with a specific class of autoregressive moving average (ARMA) models called FARIMA by allowing for non-integer differentiation. A comprehensive literature on the application of FARIMA can be found in financial analysis

The fBm and the fractional Lévy distribution are well known in statistics

A random variable

Now, restricting the focus to our case study, we assume that if the stochastic process exhibits a self-similar property, then it can be modelled by the fBm. Following

The residual time series is now modelled as a sum of three stochastic processes, namely it is the sum of a white noise, a coloured noise, and a third process. It is a similar approach to that used in previous works

Gaussian Lévy – the Lévy process is a Gaussian Lévy process if the process follows the properties of a pure Brownian motion, which is also called a Wiener process

Fractional Lévy – the residual time series exhibits self-similarity with possible long-term correlations. The fractional Lévy process is described by the model of the fLsm for the specific case reduced to the fBm. The long-term correlation process is mostly due to the presence of coloured noise

Stable Lévy – the Lévy process is a Lévy

To recall Sect.

Furthermore, our method is based on varying the length of the time series, resulting in the variations in the stochastic and functional models, which should allow classifying the type of Lévy process. The variations in the length of the time series should take into account that the coloured noise is a non-stationary signal (around the mean – see the Supplement), and thus the properties (i.e.

Let us call the geodetic time series

Let us describe the method for the first, second, and

To recall the assumptions in Sect.

Assumptions on the functional model and the stochastic parameters estimated via

Furthermore, the fitting of the

This section deals with the application of the

We have restrained our simulations to the stochastic model corresponding to the flicker noise (with white noise –

We simulate a 10-year long time series, fixing

low value (i.e.

intermediate value (i.e.

high value (i.e.

Percentage of variations in the estimated parameters included in the stochastic and functional models when varying the length of the time series. The letters A, B, and C refer to the various scenarios with different coloured noise amplitudes.

Scenarios A, B, and C in Fig.

The first result, which is common to all three figures, is that the variations in terms of variance in the functional model increases faster than for the results associated with the stochastic model. Previous studies have shown that there is a part of the noise amplitude absorbed in the functional model

Statistics on the error when fitting the ARMA and FARIMA model to the residual time series following the three scenarios.

Correlation between the distribution of the residuals and the normal (corr. normal) distribution or the Lévy

Now, Table

Finally, those three scenarios support the theory where, in the case of small-amplitude coloured noise, the stochastic noise properties are dominated by the Gaussian noise and, hence, define a third process as a Gaussian Lévy. However, the increase in the coloured noise amplitude shows that it is much more difficult to discriminate between the fractional Lévy and the stable Lévy. The results indicate that the third process can be modelled as a stable Lévy process when mostly the FARIMA fits the residuals due to large-amplitude long-memory processes and, hence, creates a heavy-tail distribution. This result is restrictive for the application to geodetic time series.

We process the daily position time series of the three GNSS stations, namely DRAO, ASCO, and ALBH retrieved from the SOPAC and PANGA websites. The functional model includes the tectonic rate, the first and second harmonic of the seasonal signal, and the occurrence time of the offsets. This occurrence time is obtained from the log file of each station. However, ALBH is known to record slow-slip events from the Cascadia subduction zone

Similar to the previous section, Fig.

Percentage of variations in the estimated parameters included in the stochastic and functional models when varying the length of the daily position GNSS time series corresponding to the stations DRAO, ASCO, and ALBH. The statistics are estimated over the eastern and northern coordinates.

Looking at Fig.

The second result is the large variations in the functional model compared with the stochastic model. To recall the simulation results, the functional model partially absorbs the variations in the noise, i.e. the tectonic rate partially fits into the power-law noise. In addition, to some extent at ASCO, the sudden increase in the functional model variations at 0.5 year may be explained by the absorption of some of the noise with the second harmonic of the seasonal signal.

When comparing the variations in the stochastic and functional models with amplitude below 20 % for DRAO and ASCO, the results agree with the definition of the fractional Lévy process defined in Table

Statistics on the error when fitting the ARMA and FARIMA model to the residual time series for each coordinate of the stations ALBH, DRAO and ASCO based on the

Furthermore, Table

GNSS time series for the ASCO station (eastern coordinate) with the

In

To recall Sect.

The closed-form solution of the variance

We have investigated the statistical assumptions behind using the fBm and the family of Lévy

In order to check our model, we have simulated mixed spectra time series with various levels of coloured noise. We have then developed an

The discussion on the limits of modelling the stochastic properties of the residuals with the stable Lévy process underlines that the infinite variance property can only be satisfied in the case of heavy-tailed distributions resulting from (1) the presence of a large-amplitude random walk (e.g. temporal aggregation in financial time series), (2) an important misfit between the models (i.e. functional and stochastic) and the observations, which means that there is anxiety in the choice of the functional model (e.g. unmodelled large jumps, large outliers). With longer and longer time series, one may be able to statistically characterize the third stochastic process more precisely. Finally, future work should investigate the autoregressive conditional heteroscedasticity (ARCH) model applied to GNSS time series in order to model differently the stochastic properties (e.g. non-stationary beyond the mean).

Here, we model the offsets in the time series as Heaviside step functions according to

In the presence of small (undetectable) offsets (

The code is a collection of routine in MATLAB (MathWorks) and is available upon request to the corresponding author.

All the GNSS data are available in public repositories such as the SOPAC archive at

The supplement related to this article is available online at:

JPM, XH, and KY developed the paper and the analysis. CX worked on the code.

The authors declare that they have no conflict of interest.

We would like to thank Machiel S. Bos from SEGAL (Space and Earth Geodetic Analysis Laboratory, University of Beira Interior) for multiple discussions on the stochastic properties of the GNSS time series. He also developed the Hector software used in this study. Xiaoxing He has been supported by the Natural Science Foundation of Jiangxi Province, the Science and Technology Research Project of Education Department of Jiangxi Province, and the National Natural Science Foundation of China. Finally, we would like to thank the reviewers for the constructive comments that helped to improve this article.

This research has been supported by the Natural Science Foundation of Jiangxi Province (grant nos. 20202BAB214029 and 20202BABL213033), the Science and Technology Research Project of Education Department of Jiangxi Province (grant no. GJJ200639), and the National Natural Science Foundation of China (grant nos. 42061077, 41904171, and 41904031).

This paper was edited by Stéphane Vannitsem and reviewed by Reik Donner and one anonymous referee.