A new efficient fusion positioning method for single-epoch multi-GNSS based on the theoretical analysis of the relationship between ADOP and PDOP

The global navigation satellite system (GNSS) can provide single-epoch differential positioning services for geological disasters with a sudden and instantaneous nature. It needs fast and precise monitoring, which lies in the rapidly and correctly fixing ambiguities of GNSS. Compared to a single-frequency single system (SF-SS), multiple GNSSs (multi-GNSS) can achieve a high success rate (SR), but the positioning becomes time- and power-consuming due to its large number of visible satellites. Satellite selection and partial ambiguity resolution (PAR) can improve the positioning efficiency of multi-GNSS, but they cannot achieve precise and high-SR rapid positioning. How to effectively utilize multi-GNSS observations to achieve fast, precise, and high-SR single-epoch positioning becomes crucial. Hence, the following theory and method are developed. The roles of code and carrier observations in precise and high-SR positioning are theoretically analyzed. Then, the relationships between position dilution of precision and ambiguity dilution of precision (ADOP) are established by adopting the Schur-Horn Theorem, Majorization Theorem, and Weyl Theorem. Based on the above analyses, a PAR method of ADOP-based BeiDou navigation satellite system (BDS)/Galileo system (Galileo) augmenting global positioning system (GPS) (A-GPS/BDS/Galileo) is proposed. The single-epoch relative positioning results of SR, positioning accuracy, time consumption, and the R-ratio test-based fixed reliability demonstrate that A-GPS/BDS/Galileo outperforms the traditional SF-SS and single/dual-frequency multi-GNSS methods: it can achieve fast and precise positioning with an empirical SR of 100.0%; its R-ratio test-based accept, successfully fixed, failure, detection, and false alarm rates can be up to 98.5%, 100.0%, 0.0%, 0.01%, and 1.5%, respectively.

GNSS data used in this study from Hong Kong base station, China, can be accessed at: https://www.geodetic.gov.hk/en/rinex/downv.aspx.

The authors are very grateful for the comments and remarks of the reviewers who helped to improve the manuscript. This work was supported by the Natural Science Foundation of Jiangsu Province (No. BK20221146 and No. BK20191342), the Project funded by the China Postdoctoral Science Foundation (No. 2021M703496), and the Open Research Fund of Jiangsu Key Laboratory of Resources and Environmental Information Engineering, CUMT (No. JS202109).

Authors and Affiliations

1. Jiangsu Key Laboratory of Resources and Environmental Information Engineering, China University of Mining and Technology, Xuzhou, 221116, China

   Xin Liu, Qianxin Wang, Shubi Zhang & Shuhui Wu

2. School of Environment and Spatial Informatics, China University of Mining and Technology, Xuzhou, 221116, China

   Xin Liu, Qianxin Wang, Shubi Zhang & Shuhui Wu

Corresponding authors

Correspondence to Xin Liu or Qianxin Wang.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix A: Schur-Horn Theorem (Horn et al. 1998; Schur 1923)

If \(\varvec{C} = \left( {c_{ij} } \right)_{n \times n} \in \varvec{M}_{n}\) is a Hermite matrix, \(c_{11}^{ \uparrow } + \cdots + c_{mm}^{ \uparrow } \ge \gamma_{1}^{ \uparrow } + \cdots + \gamma_{m}^{ \uparrow }\) holds for every \(m = 1, \cdots ,n\) with equality for m = n, where \({\upgamma }_{{\text{i}}}\) (\(i = 1, \cdots ,n\)) is the eigenvalue of C, ‘\(\uparrow\)’ denotes the increasingly ordered symbol, and \(\varvec{M}_{n}\) denotes the space of \(n \times n\) complex matrices.

Appendix B: Majorization Theorem (Marshall and Olkin 1979)

Let \(\varvec{x} = \left[ {\begin{array}{*{20}c} {x_{1}^{ \uparrow } } & \cdots & {x_{n}^{ \uparrow } } \\ \end{array} } \right]\) and \(\varvec{y} = \left[ {\begin{array}{*{20}c} {y_{1}^{ \uparrow } } & \cdots & {y_{n}^{ \uparrow } } \\ \end{array} } \right]\) \(\in {\varvec{R}}^{{\text{n}}}\), where \({\varvec{R}}^{{\text{n}}}\) denotes the space of n-order real vectors. If \(\mathop \sum \nolimits_{i = 1}^{m} x_{i} \ge \mathop \sum \nolimits_{i = 1}^{m} y_{i} (1 \le m \le n - 1)\) and \(\mathop \sum \nolimits_{i = 1}^{n} x_{i} = \mathop \sum \nolimits_{i = 1}^{n} y_{i}\) hold, it is said that x is majorized by y, which is denoted by \({\varvec{x}} \prec {\varvec{y}}\). Specially, \(\underbrace {{\left[ {\begin{array}{*{20}c} {\overline{x}} & \cdots & {\overline{x}} \\ \end{array} } \right]}}_{n} \prec {\varvec{x}}\) with \(\overline{x} = \left( {\mathop \sum \nolimits_{i = 1}^{n} x_{i} } \right)/n\) always holds, which can be easily proved by mathematical induction.

Appendix C: Weyl Theorem (Lancaster and Tismenetsky 1985)

Let U and V \(\in {\varvec{M}}_{{n}}\) are Hermite matrices. Then \(\eta_{i} ({\varvec{U}} + {\varvec{V}}) \ge \left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\eta_{i} ({\varvec{U}}) + \eta_{1} ({\varvec{V}})} \\ {\eta_{i - 1} ({\varvec{U}}) + \eta_{2} ({\varvec{V}})} \\ \end{array} } \\ {\begin{array}{*{20}c} \cdots \\ {\eta_{1} ({\varvec{U}}) + \eta_{i} ({\varvec{V}})} \\ \end{array} } \\ \end{array} } \right.\) and \(\eta_{i} ({\varvec{U}} + {\varvec{V}}) \le \left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\eta_{i} ({\varvec{U}}) + \eta_{n} ({\varvec{V}})} \\ {\eta_{i + 1} ({\varvec{U}}) + \eta_{n - 1} ({\varvec{V}})} \\ \end{array} } \\ {\begin{array}{*{20}c} \cdots \\ {\eta_{n} ({\varvec{U}}) + \eta_{i} ({\varvec{V}})} \\ \end{array} } \\ \end{array} } \right.\) hold, where \({\varvec{\eta}} (*)= \left[ {\begin{array}{*{20}c} {{{\eta}_{1}^{\uparrow}}(*)} & {\cdots} & {{{\eta}_{n}^{\uparrow}} (*)} \\ \end{array} } \right]\) is the eigenvalue vector of ‘*’ with ‘*’ be U, V, or U + V.

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions