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.
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.