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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

Abstract

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).

Keywords

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

Notes

Acknowledgements

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.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

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

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