# Taking correlations into account: a diagonal correlation model

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## 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

Variance model Weighting GPS Least-squares Exponential model## Notes

### Acknowledgements

The authors gratefully acknowledge the EPN and corresponding agencies for providing the data freely. Three reviewers are warmly acknowledged for their constructive comments.

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