Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

On modelling GPS phase correlations: a parametric model

On modelling GPS phase correlations: a parametric model Abstract Least-squares estimates are unbiased with minimal variance if the correct stochastic model is used. However, due to computational burden, diagonal variance covariance matrices (VCM) are often preferred where only the elevation dependency of the variance of GPS observations is described. This simplification that neglects correlations between measurements leads to a less efficient least-squares solution. In this contribution, an improved stochastic model based on a simple parametric function to model correlations between GPS phase observations is presented. Built on an adapted and flexible Mátern function accounting for spatiotemporal variabilities, its parameters are fixed thanks to maximum likelihood estimation. Consecutively, fully populated VCM can be computed that both model the correlations of one satellite with itself as well as the correlations between one satellite and other ones. The whitening of the observations thanks to such matrices is particularly effective, allowing a more homogeneous Fourier amplitude spectrum with respect to the one obtained by using diagonal VCM. Wrong Mátern parameters—as for instance too long correlation or too low smoothness—are shown to skew the least-squares solution impacting principally results of test statistics such as the apriori cofactor matrix of the estimates or the aposteriori variance factor. The effects at the estimates level are minimal as long as the correlation structure is not strongly wrongly estimated. Thus, taking correlations into account in least-squares adjustment for positioning leads to a more realistic precision and better distributed test statistics such as the overall model test and should not be neglected. Our simple proposal shows an improvement in that direction with respect to often empirical used model. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Geodaetica et Geophysica Springer Journals

On modelling GPS phase correlations: a parametric model

Kermarrec, Gael; Schön, Steffen
Acta Geodaetica et Geophysica , Volume 53 (1): 18 – Mar 1, 2018
Download PDF
18 pages

Loading next page...
 
/lp/springer-journals/on-modelling-gps-phase-correlations-a-parametric-model-rQaRsaHLCx
Publisher
Springer Journals
Copyright
2017 Akadémiai Kiadó
ISSN
2213-5812
eISSN
2213-5820
DOI
10.1007/s40328-017-0209-5
Publisher site
See Article on Publisher Site

Abstract

Abstract Least-squares estimates are unbiased with minimal variance if the correct stochastic model is used. However, due to computational burden, diagonal variance covariance matrices (VCM) are often preferred where only the elevation dependency of the variance of GPS observations is described. This simplification that neglects correlations between measurements leads to a less efficient least-squares solution. In this contribution, an improved stochastic model based on a simple parametric function to model correlations between GPS phase observations is presented. Built on an adapted and flexible Mátern function accounting for spatiotemporal variabilities, its parameters are fixed thanks to maximum likelihood estimation. Consecutively, fully populated VCM can be computed that both model the correlations of one satellite with itself as well as the correlations between one satellite and other ones. The whitening of the observations thanks to such matrices is particularly effective, allowing a more homogeneous Fourier amplitude spectrum with respect to the one obtained by using diagonal VCM. Wrong Mátern parameters—as for instance too long correlation or too low smoothness—are shown to skew the least-squares solution impacting principally results of test statistics such as the apriori cofactor matrix of the estimates or the aposteriori variance factor. The effects at the estimates level are minimal as long as the correlation structure is not strongly wrongly estimated. Thus, taking correlations into account in least-squares adjustment for positioning leads to a more realistic precision and better distributed test statistics such as the overall model test and should not be neglected. Our simple proposal shows an improvement in that direction with respect to often empirical used model.

Journal

Acta Geodaetica et Geophysica – Springer Journals

Published: Mar 1, 2018

Keywords: geophysics/geodesy

References

  • Handbook of mathematical functions
    IA Segun
  • Application of least-squares variance component estimation to GPS observables
    AR Amiri-Simkooei, PJG Teunissen, C Tiberius
  • Role of stochastic model on GPS integer ambiguity resolution success rate
    AR Amiri-Simkooei, S Jazaeri, F Zangeneh-Nejad, J Asgari
  • A procedure for testing the assumption of homoscedasticity in least squares residuals: a case study of GPS carrier-phase observations
    W Bischoff, B Heck, J Howind, A Teusch
  • Precision, cross correlation, and time correlation of GPS phase and code observations
    P Bona
  • Enhancement of the EUREF permanent network services and products
    C Bruyninx, H Habrich, W Söhne, A Kenyeres, G Stangl, C Völksen
  • The effect of physical correlations on the ambiguity resolution and accuracy estimation in GPS differential positioning
    El-Rabbany A
  • Handbook of spatial statistics
    P Guttorp
  • Applications of linear and nonlinear models
    J Awange
  • Econometric analysis
    WH Greene
  • An approach to statistical spatial-temporal modeling of meteorological fields
    MS Handcock, JR Wallis
  • Impact of temporal correlations on GPS-derived relative point positions
    J Howind, H Kutterer, B Heck
  • Effects of physical correlations on long-distance GPS positioning and zenith tropospheric delay estimates
    SG Jin, O Luo, C Ren
  • Mining geostatistics
    CJ Huifbregts
  • On the Mátern covariance family: a proposal for modeling temporal correlations based on turbulence theory
    G Kermarrec, S Schön
  • Taking correlations into account with a diagonal covariance matrix
    G Kermarrec, S Schön
  • Apriori fully populated covariance matrices in least-squares adjustment—case study: GPS relative positioning
    G Kermarrec, S Schön
  • Parameter estimation and hypothesis testing in linear models
    KR Koch
  • On the sensitivity of the results of least-squares adjustments concerning the stochastic model
    H Kutterer
  • An empirical approach for the estimation of GPS covariance matrix of observations.
    Leandro R, Santos M, Cove K
  • Stochastic modeling of triple-frequency BeiDou signals: estimation, assessment and impact analysis
    B Li
  • Assessment of stochastic models for GPS measurements with different types of receivers
    B Li, Y Shen, L Lou
  • GNSS elevation-dependent stochastic modelling and its impacts on the statistic testing
    B Li, L Lou, Y Shen
  • Extending the GPS stochastic model y means of signal quality measures and ARMA processes
    Luo X
  • Spatial variation-Stochastic models and their application to some problems in forest surveys and other sampling investigation
    Mátern B
  • Planar geodetic covariance functions
    S Meier
  • Global positioning system
    Misra P, Enge P
  • Noise characteristics of high-frequency, short-duration GPS records from analysis of identical, collocated instruments
    F Moschas, S Stiros
  • Variance-covariance modeling of carrier phase errors for rigorous adjustment of local area networks.
    Radovanovic RS
  • Linear models, least-squares and alternatives
    H Toutenburg
  • Gaussian processes for machine learning
    C Williams
  • Effect of mathematical correlation on GPS network computation
    MC Santos, P Vanicek, RB Langley
  • Comparing different GPS data processing techniques for modelling residual systematic errors
    C Satirapod, J Wang, C Rizos
  • A proposal for modeling physical correlations of GPS phase observations
    S Schön, FK Brunner
  • Satellite geodesy
    G Seeber
  • Interpolation of spatial data. Some theory for kriging
    ML Stein
  • Testing theory and introduction. Series on Mathematical Geodesy and Positioning
    PJG Teunissen
  • Least-squares variance component estimation
    PJG Teunissen, AR Amiri-Simkooei
  • Estimation of the stochastic model for GPS code and phase observables
    C Tiberius, F Kenselaar
  • Numerical methods for the solution of Ill-posed problems
    AG Yagola
  • Stochastic assessment of GPS carrier phase measurements for precise static relative positioning
    J Wang, C Satirapod, C Rizos
  • Electromagnetic scintillation part I GeomETRICAL OPTICS
    AD Wheelon
  • On stationary processes in the plane
    P Whittle
  • Estimating the ionosphere using one or more dual frequency GPS receivers.
    Wild U, Beutler G, Gurtner W, Rothacher M
  • The effect of incorrect weights on estimating the variance of unit weigth
    P Xu
  • Correlation theory of stationary and related random functions I basic results
    AM Yaglom
  • Positive definite functions depending on a norm
    VP Zastavnyi