Erweiterung des stochastischen Modells von GPS-Beobachtungen durch Modellierung physikalischer Korrelationen

• Applicant: Professor Dr.-Ing. Bernhard Heck
• Subject Area: Geophysics and Geodesy
• Term: from 2006 to 2012
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 30246717

• Final Report Year: 2011

Final Report Abstract

Accompanying the modernisation of the Global Positioning System (GPS), increased positioning accuracy and reliability are required for a wide range of geodetic applications in the industrial, commercial, and cadastral sectors. To meet the rising demands on accurate position determination and realistic quality evaluation, continuous improvements, not only in hardware developments but also in mathematical models, are necessary. The mathematical models applied within GPS data analysis consist of functional and stochastic components. In contrast to the intensively investigated functional model, the stochastic model characterising the observations' precisions and correlations is still under development. One of its main deficiencies arises from neglecting temporal correlations between GPS measurements, which leads to unreliable parameter estimates and over-optimistic accuracy measures. In the framework of this research project, temporal correlation properties of GPS observations were analysed and characterised in a mathematically rigorous way. Based on time series of observation residuals from GPS data processing, the proposed analysis approach is essentially realised in two steps: 1) noise extraction by conducting residual decomposition, 2) noise characterisation by means of autoregressive moving average (ARMA) processes. The main objective of the first step is to reduce the residual systematic signals induced, for example, by unmodelled multipath effects and the remaining atmospheric delays. Thereby, the long-periodic trends and quasi-periodic signals are handled by Vondrak filtering and sidereal stacking, respectively. The associated important issues such as outlier detection, Vondrak filter parameters, and satellite orbit repeat lags are thoroughly considered. For each decomposed noise series, the best-fitting ARMA model is automatically identified using the free MATLABR Toolbox ARMASA. The efficiency of residual decomposition and ARMA modelling is verified by performing continuous wavelet transforms (CWT) and suitable statistical tests for normality, trend, stationarity, and uncorrelatedness. By incorporating surface meteorological data such as wind speed (WS) and relative humidity (RH), the determined noise temporal correlation properties are physically interpreted. Relying upon the ARMA model autocorrelation functions (ACF), the GPS stochastic model is extended in the case of precise point positioning (PPP). The proposed temporal correlation analysis was applied to a large entity of representative residual data sets with respect to sampling rate, positioning technique, baseline length, multipath impact, and atmospheric conditions. After sufficient removal of residual systematic signals, the noise temporal correlation was found to be statistically significant, showing a mean zero-crossing correlation length and lage-1 correlation level of about 8 min and 0.4, respectively. The degree of correlation appeared to decrease with increasing WS and to increase with increasing RH and satellite elevation angle. While baseline length seemed to marginally affect the noise temporal correlation, strong multipath decreased the correlation time by up to 50%. Compared with first-order autoregressive (AR(1)) processes, higher-order AR and ARMA models turned out to be more suitable for noise characterisation, where AR processes illustrated better fitting results in the presence of strong multipath impact. Moreover, higher ARMA orders were selected when analysing residuals of observations from low-elevation satellites and severe multipath environments. Based on the scalograms from CWT and the results of statistical tests, the efficiency of residual decomposition and ARMA modelling could be verified. The long-periodic trend was found to play a dominant role in residual deviations from normality and stationarity. Using the statistically valid ARMA estimates to model the physically interpretable noise temporal correlation, the extended PPP stochastic model reflected variations in multipath effects and atmospheric conditions. In view of practical application and knowledge transfer, the obtained numerical results like noise temporal correlation properties and ARMA order magnitudes may serve as references and contribute to further studies in this field. The employed mathematical methods such as outlier detection, ARMA modelling, wavelet transforms, and statistical tests are universally applicable.

Publications

• (2007). Quantifizierung verschiedener Einflussfaktoren in GNSS-Residuen. Zeitschrift fur Geodasie, Geoinformation und Landmanagement, Vol. 132.2007, Issue 2, pp. 97-107.
  Luo, X., Mayer, M., and Heck, B.
  
• (2008). Automatisiertes GNSS-basiertes Bewegungsmonitoring am Black Forest Observatory (BFO) in Nahezu-Echtzeit. Zeitschrift für Geodäsie, Geoinformation und Landmanagement, Vol. 133. 2008, Issue 5, pp.283-294.
  Luo, X. and Mayer, M.
  
• (2008). Erweiterung des stochastischen Modells von GNSSBeobachtungen unter Verwendung der Signalqualitat. Zeitschrift fur Geodasie, Geoinformation und Landmanagement, Vol. 133. 2008, Issue 2, pp. 98-107.
  Luo, X., Mayer, M., and Heck, B.
  
• (2008). Impact of various factors on the quality of site specific neutrospheric parameters within GNSS data processing - A case study. Boletim de Ciencias Geodesicas, Vol. 14. 2008, Issue 4, pp. 461-481.
  Luo, X., Mayer, M., and Heck, B.
  
• (2009). Improving the stochastic model of GNSS observations by means of SNR-based weighting. In: M. G. Sideris (ed.), Observing our Changing Earth, Proceedings of the 2007 IAG General Assembly, Perugia, Italy, July 2-13, 2007, Springer-Vlg, International Association of Geodesy Symposia, Vol. 133. 2009, pp 725-734.
  Luo, X., Mayer, M., and Heck, B.
  (See online at https://dx.doi.org/10.1007/978-3-540-85426-5_83)
• (2011). Verification of ARMA identication for modelling temporal correlations of GNSS observations using the ARMASA toolbox. Studia Geophysica et Geodaetica, Vol. 55. 2011, Issue 3, pp 537-556.
  Luo, X., Mayer, M., and Heck, B.
  (See online at https://dx.doi.org/10.1007/s11200-011-0033-2)
• On the probability distribution of GNSS carrier phase observations. GPS Solutions, Vol. 15. 2011, Issue 4, pp 369-379.
  Luo, X., Mayer, M., and Heck, B.
  (See online at https://dx.doi.org/10.1007/s10291-010-0196-2)
• (2012). GPS Stochastic Modelling. Signal Quality Measures and ARMA Processes.(Submission to the Faculty of Civil Engineering, Geo and Environmental Sciences, Karlsruhe Institute of Technology (KIT), 2011)Springer Theses, 2012, ISBN 978-3-642-34835-8.
  Luo, X.
  
• Analysing time series of GNSS residuals by means of AR(I)MA processes. In: N. Sneeuw et al. (eds.), VII Hotine-Marussi Symposium on Mathematical Geodesy, Proceedings of the VII Hotine-Marussi Symposium, Rome, Italy, July 6-10, 2009, Intern. Ass. of Geodesy Symposia, vol. 137.2012, pp. 129-134, Springer-Vlg.
  Luo, X., Mayer, M., and Heck, B.
  (See online at https://dx.doi.org/10.1007/978-3-642-22078-4_19)