A Generalized GDOP Analysis for PNT that Requires Sequential Filtering

Mark L. Psiaki

Abstract:	A generalization of the GDOP metric for point positioning/timing batch-filter solutions is extended to sequential filters, and an efficient means of computing the generalized GDOP is developed. This work is done in support of navigation systems that cannot yield a full PNT solution based solely on observables from a single point in time. Such cases include systems that use a time series of pseudoranges from fewer than the minimum requirement of 4 satellites, or that use a time series of carrier Doppler shifts from fewer than the minimum requirement of 8 satellites. If one were to compute the standard pseudorangesolution GDOP for such systems, or a newer Doppler-only GDOP, the results would yield infinite GDOP values. It has been demonstrated, however, that a time series of fewer than 4 pseudoranges can provide a full PNT solution when processed using an appropriate recursive filter. Similarly, a time series of fewer than 8 carrier Doppler shifts can provide a full PNT solution when using a recursive filter. Therefore, it would be helpful to have an appropriate generalization of GDOP to such situations in order to characterize PNT accuracy quickly and succinctly. This paper develops such a generalized GDOP. It is based on a non-dimensionalized covariance matrix for a Kalman filter that, in theory, processes data starting infinitely far back in time and ending at the time when GDOP is to be calculated. In practice, the Kalman filter calculations only need to start at a finite prior time in the past because its process noise model causes it to discount the contributions of data from the past, with data far in the past being discounted to the point of irrelevance. One determines an appropriate finite past interval by successively trying longer Kalman filter intervals until the GDOP at the time point of interest converges. A new retrospective Kalman filter backwards recursion is developed to perform the needed calculations. It uses computation time that is linear in the needed number of retrospective samples. This represents a significant improvement over a brute-force method whose computational cost is quadratic in this number of samples.
Published in:	Proceedings of the 38th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2025) September 8 - 12, 2025 Hilton Baltimore Inner Harbor Baltimore, Maryland
Pages:	3505 - 3521
Cite this article:	Psiaki, Mark L., "A Generalized GDOP Analysis for PNT that Requires Sequential Filtering," Proceedings of the 38th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2025), Baltimore, Maryland, September 2025, pp. 3505-3521. https://doi.org/10.33012/2025.20467
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