Taking correlations in GPS least squares adjustments into account with a diagonal covariance matrix

Abstract

Based on the results of Luati and Proietti (Ann Inst Stat Math 63:673–686, 2011) on an equivalence for a certain class of polynomial regressions between the diagonally weighted least squares (DWLS) and the generalized least squares (GLS) estimator, an alternative way to take correlations into account thanks to a diagonal covariance matrix is presented. The equivalent covariance matrix is much easier to compute than a diagonalization of the covariance matrix via eigenvalue decomposition which also implies a change of the least squares equations. This condensed matrix, for use in the least squares adjustment, can be seen as a diagonal or reduced version of the original matrix, its elements being simply the sums of the rows elements of the weighting matrix. The least squares results obtained with the equivalent diagonal matrices and those given by the fully populated covariance matrix are mathematically strictly equivalent for the mean estimator in terms of estimate and its a priori cofactor matrix. It is shown that this equivalence can be empirically extended to further classes of design matrices such as those used in GPS positioning (single point positioning, precise point positioning or relative positioning with double differences). Applying this new model to simulated time series of correlated observations, a significant reduction of the coordinate differences compared with the solutions computed with the commonly used diagonal elevation-dependent model was reached for the GPS relative positioning with double differences, single point positioning as well as precise point positioning cases. The estimate differences between the equivalent and classical model with fully populated covariance matrix were below the mm for all simulated GPS cases and below the sub-mm for the relative positioning with double differences. These results were confirmed by analyzing real data. Consequently, the equivalent diagonal covariance matrices, compared with the often used elevation-dependent diagonal covariance matrix is appropriate to take correlations in GPS least squares adjustment into account, yielding more accurate cofactor matrices of the unknown.

Appendix: Mátern covariance family

First introduced by Mátern (1960), the covariance function reads:

$$\begin{aligned} C( r)=\phi ( {\alpha r})^\nu K_\nu ( {\alpha r}). \end{aligned}$$

(13)

Other parameterizations are possible as mentioned by Stein (1999), Shkarofsky (1968). Special cases arise when the smoothness factor is taken to:

• 1 / 2 :exponential covariance function

• 1: AR(1) process: autoregressive process of first order also called Markov process of first order. This process is often used in the field of geodesy to analyze gravitational fields (Meier 1981; Grafarend 1976) as well as to fit GPS covariance functions (Jansson and Persson 2013; Wang et al. 2002).

• infinity: squared exponential covariance function. Due to its indefinitely differentiability which is difficult to explain physically, this covariance function (Stein 1999; Handcock and Wallis 1994) should be avoided. Some examples of problems that can arise using this smoothness are shown in Stein (1999).

Because of its flexibility as well as the possibility to estimate the parameters via maximum likelihood (Stein 1999; Handcock and Wallis 1994 for meteorological data field), we adopted this covariance function for all our computation of covariance matrices used in GPS least squares procedure.

Kermarrec and Schön (2014) proposed to compute the elements of the GPS covariance matrices thanks to a close formula: $\sigma _{it}^{i( {t+\tau })} =\frac{\delta }{\sin ( {El_i ( t)})\sin ( {El_i ( {t+\tau })})}( {\alpha \tau })^1K_\nu ( {\alpha \tau })$, where $El_i (t)$ is the elevation of satellite i at t and $El_i (t+\tau )$ at $t+\tau$. Other weightings than elevation-dependent model could be chosen. The ranges $\alpha \in [0.005-0.025],\nu \in \left[ 1/4^{-1} \right]$ were proposed for GPS time series.