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Application of optimal stopping theory in batch partial ambiguity resolution

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As the number of satellites increases, the risk of fixing wrong integer ambiguities may reduce the accuracy and efficiency of ambiguity resolution (AR); Thus, partial ambiguity resolution (PAR) is proposed for solving full AR (FAR) based on fixing the subset of ambiguities. Batch PAR with integer least-squares (ILS) is helpful, but selecting an optimal subset must compare all possible combinations, which is time-consuming. In this study, we introduced the classic optimal stopping theory (OST) to dynamically identify ambiguity subsets in batch PAR, which aims to maintain the accuracy and reduce the process time. The process of selecting the ambiguity subset was divided into two stages: observation and decision. All the options in the observation stage were discarded, and the choice was only made in the decision stage. The classic OST has demonstrated that there is a maximum 37% probability of selecting an optimal option when the range of observations is 37% based on six assumptions. To better explain the principle of OST in subset selection and provide a reference method in batch PAR, we chose the ambiguity dilution of precision (ADOP) PAR as the subject and put forward the OST-assisted ADOP PAR (OAPAR) to obtain the fixed solutions. The static relative positioning experiment began with reproducing a 37% observation range in OST (OST-37%) using the global positioning system (GPS) L1 static observation. The 37% observation range has a maximum probability of 41% in selecting the subset with the minimum ADOP and has less consumed time compared to other larger observation ranges; Then, FAR and minimum ADOP PAR (MAPAR) models are used to verify the accuracy and efficiency of OAPAR. The static results of different system combinations show that, compared to MAPAR, OAPAR could achieve the same accuracy while saving consumed time, especially when the number of ambiguities is higher. Finally, PAR based on success rate (SRPAR) is used to analyze the advantages and disadvantages of OAPAR. Although the probability of correct fixing of OAPAR is slightly smaller than SRPAR, the convergence time of OAPAR is better than SRPAR. Meanwhile, with the number of ambiguities increase, the RMSEs of OAPAR gradually perform smaller than SRPAR. Furthermore, the classic OST proposes a solution to the extreme value problem in global navigation satellite system (GNSS) positioning and navigation.

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Data availability

The observation data used in this manuscript are available from the corresponding author upon request.



Ambiguity dilution of precision


Ambiguity resolution


Central processing unit


Day of year




Full ambiguity resolution


Global navigation satellite system


Global positioning system


Integer least-squares


Kalman filter


Least-square ambguity decorrelation adjustment




Minimum ambiguity dilution of precision partial ambiguity resolution




OST-assisted ambiguity dilution of precision partial ambiguity resolution


Optimal stopping theory


Partial ambiguity resolution


Root-mean-square error


Real-time kinematic


Signal-to-noise ratio


Standard deviation


Success rate


Partial ambiguity resolution based on success rate




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This work was sponsored by the National Natural Science Foundation of China (No. 41771475), Social Development Project of Science and Technology Innovation Action Plan of Shanghai (No. 20dz1207107), the Fund of Director of Key Laboratory of Geographic Information Science (Ministry of Education), East China Normal University (Grant No. KLGIS2020C05) and the authors gratefully acknowledge the financial support from China Scholarship Council. I would also like to express my thanks to Mr. Guibie for his explanation about optimal stopping theory.

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Chenglong Zhang did the experiment and wrote the manuscript. Danan Dong and Nobuaki Kubo provide the idea, suggestions and modified the manuscript. Wen Chen provided the funding support. Jianping Wu gave suggestions on the writing of this manuscript and the support of experiment.

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Correspondence to Wen Chen.

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Zhang, C., Chen, W., Dong, D. et al. Application of optimal stopping theory in batch partial ambiguity resolution. GPS Solut 27, 121 (2023).

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