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

- 766 Downloads
- 6 Citations

## 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.

## References

- Abramowitz M, Segun IA (1972) Handbook of mathematical functions. Dover, New YorkGoogle Scholar
- Amiri-Simkooei AR (2007) Least-squares variance component estimation: theory and applications. PhD thesis, Delft University of Technology, DelftGoogle Scholar
- 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
- 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
- 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
- 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
- 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
- Bona P (2000) Precision, cross correlation, and time correlation of GPS phase and code observations. GPS Solutions 4(2):3–13CrossRefGoogle Scholar
- 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
- Chatfield C (1989) The analysis of time series: an introduction, 4th edn. Chapman and Hall, LondonGoogle Scholar
- Cressie N (1993) Statistics for spatial data. Wiley, New YorkGoogle Scholar
- Dach R, Hugentobler U, Fridez P, Meindl M, Schildknecht T (2007) Bernese GPS Software Version 5.0. Astronomical Institute, University of Bern. SpringerGoogle Scholar
- Dufour JM (1989) Non linear hypotheses, inequality restrictions and non-nested hypotheses: exact simultaneous tests in linear regression. Econometrica 57:335–355CrossRefGoogle Scholar
- 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
- Euler HJ, Goad CC (1991) On optimal filtering of GPS dual frequency observations without using orbit information. Bull Geod 65(2):130–143CrossRefGoogle Scholar
- Fuentes M (2002) Spectral methods for nonstationary processes. Biometrika 89:197–210CrossRefGoogle Scholar
- Grafarend EW, Awange J (2012) Applications of linear and nonlinear models. Springer, BerlinCrossRefGoogle Scholar
- Greene WH (2003) Econometric analysis, 5th edn. Prentice Hall, Upper Saddle RiverGoogle Scholar
- Guttorp P, Gneiting T (2005) On the Whittle–Mátern correlation family. NRCSE technical report series no 80Google Scholar
- Handcock MS, Wallis JR (1994) An approach to statistical spatial-temporal modeling of meteorological fields. J Am Stat Assoc 89(426):368–378CrossRefGoogle Scholar
- Hannan EJ (1970) Multiple time series. Wiley, New YorkCrossRefGoogle Scholar
- Hahn M, Van Mierlo J (1987) Abhängigkeit der Ausgleichungsergebnisse von der Genauigkeitsänderung einer Beobachtung. Zeitschr Vermess 3:105–115Google Scholar
- Hoffmann-Wellenhof B, Lichtenegger H, Collins J (2001) GPS theory and practice, 5th revised edn. Springer, New YorkGoogle Scholar
- Howind J, Kutterer H, Heck B (1999) Impact of temporal correlations on GPS-derived relative point positions. J Geod 73(5):246–258CrossRefGoogle Scholar
- Journel AG, Huifbregts CJ (1978) Mining geostatistics. Academic Press, New YorkGoogle Scholar
- 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
- 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
- Koch KR (1999) Parameter estimation and hypothesis testing in linear models. Springer, BerlinCrossRefGoogle Scholar
- 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
- Koivunen AC, Kostinski AB (1999) The feasibility of data whitening to improve performance of weather radar. J Appl Meteor 38:741–749CrossRefGoogle Scholar
- Kutterer H (1999) On the sensitivity of the results of least-squares adjustments concerning the stochastic model. J Geod 73:350–361CrossRefGoogle Scholar
- 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
- 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
- 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
- Luo X (2012) Extending the GPS stochastic model by means of signal quality measures and ARMA processes. PhD, Karlsruhe Institute of TechnologyGoogle Scholar
- 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
- 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
- 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
- Meier S (1981) Planar geodetic covariance functions. Rev Geophys Space Phys 19(4):673–686CrossRefGoogle Scholar
- 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
- 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
- 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
- Rao C, Toutenburg H (1999) Linear models. Least-squares and alternatives, Springer, New York Second EditionGoogle Scholar
- Rasmussen CE, Williams C (2006) Gaussian processes for machine learning. MIT Press, CambridgeGoogle Scholar
- Santos MC, Vanicek P, Langley RB (1997) Effect of mathematical correlation on GPS network computation. J Surv Eng 123(3):101–112CrossRefGoogle Scholar
- 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
- Schön S, Brunner FK (2008) Atmospheric turbulence theory applied to GPS carrier-phase data. J Geod 1:47–57CrossRefGoogle Scholar
- Spöck G, Pilz J (2008) Non-spatial modeling using harmonic analysis. VIII Int. Geostatistics congress, Santiago, p 1–10Google Scholar
- Stein ML (1999) Interpolation of spatial data: some theory for kriging. Springer, New YorkCrossRefGoogle Scholar
- Strand ON (1974) Coefficient errors caused by using the wrong covariance matrix in the general linear model. Ann Stat 2(5):935–949CrossRefGoogle Scholar
- 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
- Teunissen PJG (2000) Testing theory and introduction. Series on mathematical geodesy and positioning. Delft University Press, The NetherlandsGoogle Scholar
- Teunissen PJG, Amiri-Simkooei AR (2008) Least-squares variance component estimation. J Geod 82:65–82Google Scholar
- Tiberius C, Kenselaar F (2003) Variance component estimation and precise GPS positioning: case study. J Surv Eng 129(1):11–18CrossRefGoogle Scholar
- Vermeer M (1997) The precision of geodetic GPS and one way of improving it. J Geod 71:240–245CrossRefGoogle Scholar
- Dennis Wackerly, William Mendenhall, Scheaffer Richard L (2008) Mathematical statistics with applications, 7th edn. Thomson Higher Education, BelmontGoogle Scholar
- Wang J, Stewart MP, Tsakiri M (1998) Stochastic modelling for GPS static baseline data processing. J Surv Eng 124:171–181CrossRefGoogle Scholar
- Wang J, Stewart P, Tsakiri M (2000) A comparative study of integer ambiguity validation procedures. Earth Planet Space 52:813–817Google Scholar
- 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
- 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
- Watson GS (1955) Serial correlation in regression analysis. Biometrika 42(3/4):327–341CrossRefGoogle Scholar
- Watson GS (1967) Linear least-squares regression. Ann Math. Stat 38:1679–1699Google Scholar
- Wheelon AD (2001) Electromagnetic scintillation part I geometrical optics. Cambridge University Press, CambridgeCrossRefGoogle Scholar
- Whittle P (1954) On stationary processes in the plane. Biometrika 41:434–449CrossRefGoogle Scholar
- Wieser A, Brunner FK (2000) An extended weight model for GPS phase observations. Earth Planet Space 52:777–782CrossRefGoogle Scholar
- Wolf H (1961) Der Einfluss von Gewichtsänderungen auf die Ausgleichungsergebnisse. Z Vermess 86:361–362Google Scholar
- Xu P (1991) Least squares collocation with incorrect prior information. Zeitschr Vermess 116:266–273Google Scholar
- Xu P, Liu Y, Shen Y, Fukuda Y (2007) Estimability analysis of variance and covariance components. J Geod 81:593–602CrossRefGoogle Scholar
- Xu P (2013) The effect of incorrect weights on estimating the variance of unit weigth. Stud Geophy Geod 57:339–352CrossRefGoogle Scholar
- Yaglom AM (1987) Correlation theory of stationary and related random functions. Springer series in statistics. Springer, New YorkGoogle Scholar