# On modelling GPS phase correlations: a parametric model

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## 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

Mátern covariance function Correlation GPS Realistic stochastic model## Notes

### Acknowledgements

The authors gratefully acknowledge the EPN network and corresponding agencies for providing freely the data. Anonymous reviewers are warmly thanks for their valuable comments which helped improve the original manuscript.

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