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

Abstract

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.

Appendices

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.