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

Content Provider	Springer Nature Link
Author	Kermarrec, Gaël Schön, Steffen
Copyright Year	2016
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.
Starting Page	793
Ending Page	805
Page Count	13
File Format	PDF
ISSN	09497714
Journal	Journal of Geodesy
Volume Number	90
Issue Number	9
e-ISSN	14321394
Language	English
Publisher	Springer Berlin Heidelberg
Publisher Date	2016-05-11
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	GPS Correlations Weighted least squares Equivalent kernel Mátern covariance family Geophysics/Geodesy Earth Sciences
Content Type	Text
Resource Type	Article
Subject	Geophysics Geochemistry and Petrology Computers in Earth Sciences

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