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.
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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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Appendix: On the inverse of the covariance matrix
Appendix: On the inverse of the covariance matrix
Studying the inverse of the fully populated covariance matrix is interesting to gain a better insight on the way correlations act on the previous results, particularly on the apriori cofactor matrix of the estimates and by extension of the ambiguities.
However, except in particular cases, VCM based on Mátern model do not have an explicit formulation of their inverse. We will therefore first present the particular case of an AR(1) process and in a second step take an example of the matrices used in this article.
1.1 AR(1) and Mátern matrices
In this case, the inverse of the covariance matrix reads
where \( \rho \) is the autocorrelation function.
Thus, from a fully populated \( {\mathbf{W}}_{AR(1)}^{{}} \), only a sparse matrix remains which lines have 2 different values (1) the diagonal elements and (2) a negative value on the off-diagonal. A scaling factor \( \frac{1}{{1 - \rho^{2} }} \) depending on the correlation length can be further identified. This structure can be extended for Mátern covariance matrices, i.e. a factor depending on the Mátern parameter set chosen is mainly responsible for the scaling of apriori cofactor matrices of the estimates as shown in Sect. 4. The error ellipsoid may thus be artificially too voluminous if a wrong correlation structure is taken into account.
1.2 Taking noise matrix into account
In order to see how noise matrices are impacting the inverse of the covariance matrices, a particular case is chosen to compute the covariance function as defined in Eq. (9) corresponding to a satellite at 45° elevation. Only the first 100 epochs of the block diagonal matrix are analysed.
Fig. 6 (left) shows the corresponding line of the covariance matrix for different correlation structures. For \( \left[ {\alpha ,\nu } \right] = [0.0025,1] \) and due to the smoothness of 1, the first values of the covariance decreases slowly with time
The corresponding line of the inverse of the covariance matrix for the different cases are depicted in Fig. 6 (right). The signature of the inverse of a fully populated VCM is clearly seen when no noise matrix is taken in consideration, i.e. small oscillations around the 0–value. The amplitude of the variations increases as \( \alpha \) decreases. If a noise matrix is added only the first oscillation remains, the other ones being replaced by a ramp, thus damping the effect of the correlations. This was highlighted by studying the whitening of the residuals and this is the reason why the results given with or without noise matrix are similar up to a given value of \( \alpha \) and \( \nu \) where the equivalence between the smoothed and the original curve is getting too high.
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Kermarrec, G., Schön, S. On modelling GPS phase correlations: a parametric model. Acta Geod Geophys 53, 139–156 (2018). https://doi.org/10.1007/s40328-017-0209-5
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DOI: https://doi.org/10.1007/s40328-017-0209-5