## Details

Original language | English |
---|---|

Pages (from-to) | 139-156 |

Number of pages | 18 |

Journal | Acta geodaetica et geophysica |

Volume | 53 |

Issue number | 1 |

Early online date | 31 Oct 2017 |

Publication status | Published - Mar 2018 |

## 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.

## Keywords

- Correlation, GPS, Mátern covariance function, Realistic stochastic model

## ASJC Scopus subject areas

- Engineering(all)
**Building and Construction**- Earth and Planetary Sciences(all)
**Geophysics**- Earth and Planetary Sciences(all)
**Geology**

## Cite this

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- Harvard
- Apa
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- BibTeX
- RIS

**On modelling GPS phase correlations: a parametric model.**/ Kermarrec, Gael; Schön, Steffen.

In: Acta geodaetica et geophysica, Vol. 53, No. 1, 03.2018, p. 139-156.

Research output: Contribution to journal › Article › Research › peer review

*Acta geodaetica et geophysica*, vol. 53, no. 1, pp. 139-156. https://doi.org/10.1007/s40328-017-0209-5

*Acta geodaetica et geophysica*,

*53*(1), 139-156. https://doi.org/10.1007/s40328-017-0209-5

}

TY - JOUR

T1 - On modelling GPS phase correlations: a parametric model

AU - Kermarrec, Gael

AU - Schön, Steffen

PY - 2018/3

Y1 - 2018/3

N2 - 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.

AB - 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.

KW - Correlation

KW - GPS

KW - Mátern covariance function

KW - Realistic stochastic model

UR - http://www.scopus.com/inward/record.url?scp=85042296092&partnerID=8YFLogxK

U2 - 10.1007/s40328-017-0209-5

DO - 10.1007/s40328-017-0209-5

M3 - Article

AN - SCOPUS:85042296092

VL - 53

SP - 139

EP - 156

JO - Acta geodaetica et geophysica

JF - Acta geodaetica et geophysica

SN - 2213-5812

IS - 1

ER -