Time series analysis with special focus on GNSS station coordinates


Permanently operating sensor- and measurement systems increasingly gain importance in Geodesy. Nowadays their data is processed in full or semi automated mode. As example data of GNSS reference stations is mentioned which is processed to generate daily solutions of station coordinates.

The well-founded statistical analysis of time series, like e.g. daily solutions of a GNSS station over a large period of time, is a challenge and important task due to the inherent time correlations. This project is focused on appropriate modeling of such time correlations in time series. Depending on the measured object and applied sensor technology a time series may contain more or less obvious phenomena, which shall be identified within the analysis, their magnitude estimated, and the statistical significance assessed to finally draw conclusions on underlying physical causes.

In coordinate time series phenomena like long-term trends, periodic or cyclic variation, jumps etc. might occur. However, since every measurement technique comprises measurement errors, those phenomena are superimposed by or might even be covered by the noise in the time series. In terms of statistics a time series is a single realization of a random (stochastic) process. For each identified phenomenon respective parameters (e.g. magnitude) need to be estimated, and a proper description of the parameter's accuracy derived. Only on that basis a plausible statement about the significance of the estimated parameters, i. e. the observed phenomenon, is possible.

As an example the figure shows a time series of the east coordinate of a GNSS station over a time period of about 10 years in blue and in red the estimated time series model consisting of trend, yearly variation and single jumps due to exchanges of the station antenna.

GNSS Time Series Example
Time series of daily solutions of a GNSS station's east coordinate (blue) and the estimated model of the time series (red).

However an indispensible precondition for a plausible statement on significance is that the parameter estimation is based on an adequate stochastic model of the time series. This model must properly account for the time correlations which are usually included in the data. Ignoring this stochastic property otherwise leads to unrealistic and too optimistic accuracy values and in consequence to wrong statements on the significance of parameters. Such statements would be unfeasible for the interpretation and further use of analysis results.

Within the project mainly the following aspects are worked on:

Further topics, which will be addressed by the project in future, are analyzing multi-variate time series like e. g. three-dimensional position time series of GNSS stations.

  • Separation of deterministic and stochastic components in a time series.
  • Identification of an adequate stochastic model for the data in a time series, especially taking into account the time correlations.
  • Simultaneous estimation of deterministic and stochastic parameters by maximum-likelihood estimation.
  • Model comparison and assessment by means of likelihood-ratio tests.


Leinen, S., Becker, M. and Läufer, G. (2013): Effect of stochastic model fit-ting on the significance of CORS coordinate time series parameters. Journal of Applied Geodesy. Band 7, Heft 1, Seiten 21–37, ISSN (Online) 1862-9024, ISSN (Print) 1862-9016, DOI: 10.1515/jag-2012-0038, March 2013

Leinen, S. (2014): Zur Signifikanz von Phänomenen in GNSS-Zeitreihen – Der Weg zur statistisch plausiblen Aussage. in: 129. DVW-Seminar, Zeitabhängige Messgrößen – Ihre Daten haben (Mehr-)Wert, Seiten 129-147. Hannover, 26./27.02.2014.