Journal of Geodesy

, Volume 91, Issue 5, pp 465–484 | Cite as

A priori fully populated covariance matrices in least-squares adjustment—case study: GPS relative positioning

  • Gaël KermarrecEmail author
  • Steffen Schön
Original Article


In this contribution, using the example of the Mátern covariance matrices, we study systematically the effect of apriori fully populated variance covariance matrices (VCM) in the Gauss–Markov model, by varying both the smoothness and the correlation length of the covariance function. Based on simulations where we consider a GPS relative positioning scenario with double differences, the true VCM is exactly known. Thus, an accurate study of parameters deviations with respect to the correlation structure is possible. By means of the mean-square error difference of the estimates obtained with the correct and the assumed VCM, the loss of efficiency when the correlation structure is missspecified is considered. The bias of the variance of unit weight is moreover analysed. By acting independently on the correlation length, the smoothness, the batch length, the noise level, or the design matrix, simulations allow to draw conclusions on the influence of these different factors on the least-squares results. Thanks to an adapted version of the Kermarrec–Schön model, fully populated VCM for GPS phase observations are computed where different correlation factors are resumed in a global covariance model with an elevation dependent weighting. Based on the data of the EPN network, two studies for different baseline lengths validate the conclusions of the simulations on the influence of the Mátern covariance parameters. A precise insight into the impact of apriori correlation structures when the VCM is entirely unknown highlights that both the correlation length and the smoothness defined in the Mátern model are important to get a lower loss of efficiency as well as a better estimation of the variance of unit weight. Consecutively, correlations, if present, should not be neglected for accurate test statistics. Therefore, a proposal is made to determine a mean value of the correlation structure based on a rough estimation of the Mátern parameters via maximum likelihood estimation for some chosen time series of observations. Variations around these mean values show to have little impact on the least-squares results. At the estimates level, the effect of varying the parameters of the fully populated VCM around these approximated values was confirmed to be nearly negligible (i.e. a mm level for strong correlations and a submm level otherwise).


Mátern covariance family Correlations, smoothness GPS Double difference Design matrix Loss of efficiency Mean-square errors Variance of unit weight 



The authors gratefully acknowledge the EPN network and corresponding agencies for providing freely the data. Three anonymous reviewers are thanked for pointing out additional references as well as for their valuable comments which helped improve the original manuscript.


  1. Abramowitz M, Segun IA (1972) Handbook of mathematical functions. Dover, New YorkGoogle Scholar
  2. Amiri-Simkooei AR (2007) Least-squares variance component estimation: theory and applications. PhD thesis, Delft University of Technology, DelftGoogle Scholar
  3. Amiri-Simkooei AR, Teunissen PJG, Tiberius C (2009) Application of least-squares variance component estimation to GPS observables. J Surv Eng 135(4):149–160CrossRefGoogle Scholar
  4. Amiri-Simkooei AR, Zangeneh-Nejad F, Asgari J (2013) Least-squares variance component estimation applied to GPS geometry based observation model. J Surv Eng 139(4):176–187CrossRefGoogle Scholar
  5. Amiri-Simkooei AR, Jazaeri S, Zangeneh-Nejad F, Asgari J (2016) Role of stochastic model on GPS integer ambiguity resolution success rate. GPS Solutions 20(1):51–61CrossRefGoogle Scholar
  6. Beutler G, Bauersima I, Gurtner W, Rothacher M (1987) Correlations between simultaneous GPS double difference carrier phase observations in the multistation mode: Implementation considerations and first experiences. Manuscr Geod 12:40–44Google Scholar
  7. Bischoff W, Heck B, Howind J, Teusch A (2005) A procedure for testing the assumption of homoscedasticity in least-squares residuals: a case study of GPS carrier-phase observations. JoG 78:397–404Google Scholar
  8. Bona P (2000) Precision, cross correlation, and time correlation of GPS phase and code observations. GPS Solutions 4(2):3–13CrossRefGoogle Scholar
  9. Bruyninx C, Habrich H, Söhne W, Kenyeres A, Stangl G, Völksen C (2012) Enhancement of the EUREF permanent network services and products. Geod Planet Earth IAG Symp Ser 136:27–35CrossRefGoogle Scholar
  10. Chatfield C (1989) The analysis of time series: an introduction, 4th edn. Chapman and Hall, LondonGoogle Scholar
  11. Cressie N (1993) Statistics for spatial data. Wiley, New YorkGoogle Scholar
  12. Dach R, Hugentobler U, Fridez P, Meindl M, Schildknecht T (2007) Bernese GPS Software Version 5.0. Astronomical Institute, University of Bern. SpringerGoogle Scholar
  13. Dufour JM (1989) Non linear hypotheses, inequality restrictions and non-nested hypotheses: exact simultaneous tests in linear regression. Econometrica 57:335–355CrossRefGoogle Scholar
  14. El-Rabbany A (1994) The effect of physical correlations on the ambiguity resolution and accuracy estimation in GPS differential positioning. PhD thesis, Department of Geodesy and Geomatics Engineering, University of New Brunswick, CanadaGoogle Scholar
  15. Euler HJ, Goad CC (1991) On optimal filtering of GPS dual frequency observations without using orbit information. Bull Geod 65(2):130–143CrossRefGoogle Scholar
  16. Fuentes M (2002) Spectral methods for nonstationary processes. Biometrika 89:197–210CrossRefGoogle Scholar
  17. Grafarend EW, Awange J (2012) Applications of linear and nonlinear models. Springer, BerlinCrossRefGoogle Scholar
  18. Greene WH (2003) Econometric analysis, 5th edn. Prentice Hall, Upper Saddle RiverGoogle Scholar
  19. Guttorp P, Gneiting T (2005) On the Whittle–Mátern correlation family. NRCSE technical report series no 80Google Scholar
  20. Handcock MS, Wallis JR (1994) An approach to statistical spatial-temporal modeling of meteorological fields. J Am Stat Assoc 89(426):368–378CrossRefGoogle Scholar
  21. Hannan EJ (1970) Multiple time series. Wiley, New YorkCrossRefGoogle Scholar
  22. Hahn M, Van Mierlo J (1987) Abhängigkeit der Ausgleichungsergebnisse von der Genauigkeitsänderung einer Beobachtung. Zeitschr Vermess 3:105–115Google Scholar
  23. Hoffmann-Wellenhof B, Lichtenegger H, Collins J (2001) GPS theory and practice, 5th revised edn. Springer, New YorkGoogle Scholar
  24. Howind J, Kutterer H, Heck B (1999) Impact of temporal correlations on GPS-derived relative point positions. J Geod 73(5):246–258CrossRefGoogle Scholar
  25. Journel AG, Huifbregts CJ (1978) Mining geostatistics. Academic Press, New YorkGoogle Scholar
  26. Kermarrec G, Schön S (2014) On the Mátern covariance family: a proposal for modeling temporal correlations based on turbulence theory. J Geod 88(11):1061–1079CrossRefGoogle Scholar
  27. Kermarrec G, Schön S (2016) Taking correlations in GPS least squares adjustments into account with a diagonal covariance matrix. J Geod 90(9):793–805Google Scholar
  28. Koch KR (1999) Parameter estimation and hypothesis testing in linear models. Springer, BerlinCrossRefGoogle Scholar
  29. Koch KR, Kuhlmann H, Schuh WD (2010) Approximating covariance matrices estimated in multivariate models by estimated auto- and cross-covariances. J Geod 84(6):383–397Google Scholar
  30. Koivunen AC, Kostinski AB (1999) The feasibility of data whitening to improve performance of weather radar. J Appl Meteor 38:741–749CrossRefGoogle Scholar
  31. Kutterer H (1999) On the sensitivity of the results of least-squares adjustments concerning the stochastic model. J Geod 73:350–361CrossRefGoogle Scholar
  32. Li B, Shen Y, Lou L (2011) Efficient estimation of variance and covariance components: a case study for GPS stochastic model evaluation. IEEE Trans Geosci Remote Sens 49(1):203–210CrossRefGoogle Scholar
  33. Li B, Lou L, Shen Y (2016) GNSS elevation-dependent stochastic modeling and its impacts on the statistic testing. J Surv Eng 142(2):04015012CrossRefGoogle Scholar
  34. Luo X, Mayer M, Heck B (2012) Analysing time series of GNSS residuals by means of ARIMA processes. Int Assoc Geod Symp 137:129–134CrossRefGoogle Scholar
  35. Luo X (2012) Extending the GPS stochastic model by means of signal quality measures and ARMA processes. PhD, Karlsruhe Institute of TechnologyGoogle Scholar
  36. Luo X, Mayer M, Heck B, Awange JL (2013) A realistic and easy-to-implement weighting model for GNSS phase observations. IEEE Trans Geosci Remote SensGoogle Scholar
  37. Mardia KV, Watkins AJ (1989) On multimodality of the likelihood estimation of models for residual covariance in spatial regression. Biometrika 71(11):134–146Google Scholar
  38. Mátern B (1960) Spatial variation: stochastic models and their application to some problems in forest surveys and other sampling investigation. Medd Statens Skogsforskningsinstitut 49(5):1–144Google Scholar
  39. Meier S (1981) Planar geodetic covariance functions. Rev Geophys Space Phys 19(4):673–686CrossRefGoogle Scholar
  40. Puntanen S, Styan GPH (1986) The equality of the ordinary least squares estimator and the best linear unbiased estimator. Am Stat 43(3):153–161Google Scholar
  41. Radovanovic RS (2001) Variance–covariance modeling of carrier phase errors for rigorous adjustment of local area networks. IAG 2001 Scientific Assembly. Budapest, Hungary, 2-7 Sept 2001Google Scholar
  42. Rao C (1967) Least squares theory using an estimated dispersion matrix and its application to measurement of signals. Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, vol 1. University of California Press, Berkeley, p 355–372Google Scholar
  43. Rao C, Toutenburg H (1999) Linear models. Least-squares and alternatives, Springer, New York Second EditionGoogle Scholar
  44. Rasmussen CE, Williams C (2006) Gaussian processes for machine learning. MIT Press, CambridgeGoogle Scholar
  45. Santos MC, Vanicek P, Langley RB (1997) Effect of mathematical correlation on GPS network computation. J Surv Eng 123(3):101–112CrossRefGoogle Scholar
  46. Satirapod C, Wang J, Rizos C (2003) Comparing different GPS data processing techniques for modelling residual systematic errors. J Surv Eng 129(4):129–135CrossRefGoogle Scholar
  47. Schön S, Brunner FK (2008) Atmospheric turbulence theory applied to GPS carrier-phase data. J Geod 1:47–57CrossRefGoogle Scholar
  48. Spöck G, Pilz J (2008) Non-spatial modeling using harmonic analysis. VIII Int. Geostatistics congress, Santiago, p 1–10Google Scholar
  49. Stein ML (1999) Interpolation of spatial data: some theory for kriging. Springer, New YorkCrossRefGoogle Scholar
  50. Strand ON (1974) Coefficient errors caused by using the wrong covariance matrix in the general linear model. Ann Stat 2(5):935–949CrossRefGoogle Scholar
  51. Teunissen PJG, Jonkman NF, Tiberius CCJM (1998) Weighting GPS dual frequency observations: bearing the cross of cross correlation. GPS Solutions 2(2):28–37CrossRefGoogle Scholar
  52. Teunissen PJG (2000) Testing theory and introduction. Series on mathematical geodesy and positioning. Delft University Press, The NetherlandsGoogle Scholar
  53. Teunissen PJG, Amiri-Simkooei AR (2008) Least-squares variance component estimation. J Geod 82:65–82Google Scholar
  54. Tiberius C, Kenselaar F (2003) Variance component estimation and precise GPS positioning: case study. J Surv Eng 129(1):11–18CrossRefGoogle Scholar
  55. Vermeer M (1997) The precision of geodetic GPS and one way of improving it. J Geod 71:240–245CrossRefGoogle Scholar
  56. Dennis Wackerly, William Mendenhall, Scheaffer Richard L (2008) Mathematical statistics with applications, 7th edn. Thomson Higher Education, BelmontGoogle Scholar
  57. Wang J, Stewart MP, Tsakiri M (1998) Stochastic modelling for GPS static baseline data processing. J Surv Eng 124:171–181CrossRefGoogle Scholar
  58. Wang J, Stewart P, Tsakiri M (2000) A comparative study of integer ambiguity validation procedures. Earth Planet Space 52:813–817Google Scholar
  59. Wang J, Satirapod C, Rizos C (2002) Stochastic assessment of GPS carrier phase measurements for precise static relative positioning. J Geod 76(2):95–104CrossRefGoogle Scholar
  60. Wang J, Lee HK, Musa T, Rizos C (2005) Online stochastic modelling for network based GPS real-time kinematic positioning. J Glob Position Syst 4(1–2):113–119CrossRefGoogle Scholar
  61. Watson GS (1955) Serial correlation in regression analysis. Biometrika 42(3/4):327–341CrossRefGoogle Scholar
  62. Watson GS (1967) Linear least-squares regression. Ann Math. Stat 38:1679–1699Google Scholar
  63. Wheelon AD (2001) Electromagnetic scintillation part I geometrical optics. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  64. Whittle P (1954) On stationary processes in the plane. Biometrika 41:434–449CrossRefGoogle Scholar
  65. Wieser A, Brunner FK (2000) An extended weight model for GPS phase observations. Earth Planet Space 52:777–782CrossRefGoogle Scholar
  66. Wolf H (1961) Der Einfluss von Gewichtsänderungen auf die Ausgleichungsergebnisse. Z Vermess 86:361–362Google Scholar
  67. Xu P (1991) Least squares collocation with incorrect prior information. Zeitschr Vermess 116:266–273Google Scholar
  68. Xu P, Liu Y, Shen Y, Fukuda Y (2007) Estimability analysis of variance and covariance components. J Geod 81:593–602CrossRefGoogle Scholar
  69. Xu P (2013) The effect of incorrect weights on estimating the variance of unit weigth. Stud Geophy Geod 57:339–352CrossRefGoogle Scholar
  70. Yaglom AM (1987) Correlation theory of stationary and related random functions. Springer series in statistics. Springer, New YorkGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Institut für Erdmessung (IfE)Leibniz Universität HannoverHannoverGermany

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