## Details

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

Pages (from-to) | 1895-1906 |

Number of pages | 12 |

Journal | GPS solutions |

Volume | 21 |

Issue number | 4 |

Publication status | Published - 6 Sept 2017 |

## Abstract

The true covariance matrix of the GPS phase observations is unknown and has to be assumed or estimated. The variance of the least-squares residuals was empirically shown to have an elevation dependency and is often expressed as a sum of a constant and an exponential function. Disregarding correlations that are for instance due to atmospheric effects, the variance covariance matrices are diagonal. This simplification leads to errors in the estimates, including the float ambiguity vector, as well as to an overoptimistic precision. Thus, results of test statistics such as the outlier or the overall model test are impacted. For the particular case of GPS positioning, an innovative proposal was made to take correlations into account easily, condensed in an equivalent diagonal matrix. However, the a posteriori variance factor obtained with this simplification is strongly underestimated and in most cases the inversion of fully populated matrices has anyway to be carried out. In this contribution, we propose an alternative diagonal correlation model based on a simple exponential function to approximate the developed equivalent model. This way, correlations can be included in a diagonal variance covariance matrix without computation burden. A case study with an 80-km baseline where the ambiguities are estimated together with the coordinates in the least-squares adjustment demonstrates the potential of the model. It leads to a proposal based on the autocorrelation coefficient for fixing its parameters.

## Keywords

- Exponential model, GPS, Least-squares, Variance model, Weighting

## ASJC Scopus subject areas

## Cite this

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**Taking correlations into account: a diagonal correlation model.**/ Kermarrec, Gaël; Schön, Steffen.

In: GPS solutions, Vol. 21, No. 4, 06.09.2017, p. 1895-1906.

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

*GPS solutions*, vol. 21, no. 4, pp. 1895-1906. https://doi.org/10.1007/s10291-017-0665-y

*GPS solutions*,

*21*(4), 1895-1906. https://doi.org/10.1007/s10291-017-0665-y

}

TY - JOUR

T1 - Taking correlations into account

T2 - a diagonal correlation model

AU - Kermarrec, Gaël

AU - Schön, Steffen

PY - 2017/9/6

Y1 - 2017/9/6

N2 - The true covariance matrix of the GPS phase observations is unknown and has to be assumed or estimated. The variance of the least-squares residuals was empirically shown to have an elevation dependency and is often expressed as a sum of a constant and an exponential function. Disregarding correlations that are for instance due to atmospheric effects, the variance covariance matrices are diagonal. This simplification leads to errors in the estimates, including the float ambiguity vector, as well as to an overoptimistic precision. Thus, results of test statistics such as the outlier or the overall model test are impacted. For the particular case of GPS positioning, an innovative proposal was made to take correlations into account easily, condensed in an equivalent diagonal matrix. However, the a posteriori variance factor obtained with this simplification is strongly underestimated and in most cases the inversion of fully populated matrices has anyway to be carried out. In this contribution, we propose an alternative diagonal correlation model based on a simple exponential function to approximate the developed equivalent model. This way, correlations can be included in a diagonal variance covariance matrix without computation burden. A case study with an 80-km baseline where the ambiguities are estimated together with the coordinates in the least-squares adjustment demonstrates the potential of the model. It leads to a proposal based on the autocorrelation coefficient for fixing its parameters.

AB - The true covariance matrix of the GPS phase observations is unknown and has to be assumed or estimated. The variance of the least-squares residuals was empirically shown to have an elevation dependency and is often expressed as a sum of a constant and an exponential function. Disregarding correlations that are for instance due to atmospheric effects, the variance covariance matrices are diagonal. This simplification leads to errors in the estimates, including the float ambiguity vector, as well as to an overoptimistic precision. Thus, results of test statistics such as the outlier or the overall model test are impacted. For the particular case of GPS positioning, an innovative proposal was made to take correlations into account easily, condensed in an equivalent diagonal matrix. However, the a posteriori variance factor obtained with this simplification is strongly underestimated and in most cases the inversion of fully populated matrices has anyway to be carried out. In this contribution, we propose an alternative diagonal correlation model based on a simple exponential function to approximate the developed equivalent model. This way, correlations can be included in a diagonal variance covariance matrix without computation burden. A case study with an 80-km baseline where the ambiguities are estimated together with the coordinates in the least-squares adjustment demonstrates the potential of the model. It leads to a proposal based on the autocorrelation coefficient for fixing its parameters.

KW - Exponential model

KW - GPS

KW - Least-squares

KW - Variance model

KW - Weighting

UR - http://www.scopus.com/inward/record.url?scp=85028826975&partnerID=8YFLogxK

U2 - 10.1007/s10291-017-0665-y

DO - 10.1007/s10291-017-0665-y

M3 - Article

AN - SCOPUS:85028826975

VL - 21

SP - 1895

EP - 1906

JO - GPS solutions

JF - GPS solutions

SN - 1080-5370

IS - 4

ER -