Tighter GDOP Bounds and their Use in Satellite Subset Selection

Peter F. Swaszek, Richard J. Hartnett, Kelly C. Seals, Rebecca M.A. Swaszek

Abstract: GNSS receivers convert the measured pseudoranges from the visible and viable GNSS satellites into an estimate of the position and clock offset of the receiver. The accuracy of the resulting solution can be described statistically by its error covariance matrix; a common scalar reduction of this covariance that highlights the impact of satellite geometry is the GDOP (Geometric Dilution of Precision). In some instances, a GNSS receiver cannot process all of the visible satellites. For example, the issue might be that the receiver has limited computational power (perhaps a hardware or energy limit). Alternatively, the receiver might be using corrections from a ground-based augmentation system and the bandwidth of the correction channel is insufficient to provide information for all of the visible satellites. In such a case the question arises: If only m of the n visible satellites can be processed, which ones should they be? Since the GDOP is nonlinear and non-separable in the satellites’ locations in the sky, finding the best subset has combinatorial complexity. For example, if the receiver limits its attention to the 15 or so visible GPS satellites then a brute force comparison of all subsets to find the optimal subset is possible (whatever the value of m, if n = 15 then there are at most 6, 500 potential cases to check, well within modern computational capability). The advent of other GNSS constellations exacerbates this problem. For example, desiring to select 12 of 35 visible satellites (such as frequently occurs with GPS, GLONASS, and Galileo) brute force comparison is expensive. This question of selecting a subset of the possible satellites is not new to the navigation literature; multiple authors have described sub-optimal methods, usually greedy algorithms, for choosing the satellite subset. In prior works these authors developed a lower bound to GDOP for GNSS constellations purely as a function of the number of satellites employed. This simple bound shows the merit of the high and low (in elevation) satellites to the GDOP performance. This paper presents two new bounds that employ partial information on the satellites positions, notably information about their elevations. Not only does this provide tighter bounds, the results also suggest how to choose satellites for the subset selection problem.
Published in: Proceedings of the 32nd International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2019)
September 16 - 20, 2019
Hyatt Regency Miami
Miami, Florida
Pages: 637 - 649
Cite this article: Swaszek, Peter F., Hartnett, Richard J., Seals, Kelly C., Swaszek, Rebecca M.A., "Tighter GDOP Bounds and their Use in Satellite Subset Selection," Proceedings of the 32nd International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2019), Miami, Florida, September 2019, pp. 637-649.
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