{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,15]],"date-time":"2025-11-15T04:01:46Z","timestamp":1763179306555,"version":"build-2065373602"},"reference-count":36,"publisher":"MDPI AG","issue":"16","license":[{"start":{"date-parts":[[2021,8,13]],"date-time":"2021-08-13T00:00:00Z","timestamp":1628812800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["41774017"],"award-info":[{"award-number":["41774017"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012166","name":"National Key Research and Development Program of China","doi-asserted-by":"publisher","award":["2016YFB0501900"],"award-info":[{"award-number":["2016YFB0501900"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Remote Sensing"],"abstract":"<jats:p>The restart of the receiver will lead to the change in the non-overlapping frequency inter-system biases (ISB), which will make it difficult to apply the tightly combined RTK method of pre-calibrating ISB to the actual scene. Particle swarm optimization (PSO) algorithm can be used to estimate the fractional part of the inter-system phase bias (F-ISPB) in real time, which is not affected by the receiver restart. However, the standard PSO can easily fall into local optimum and cannot accurately estimate the value of F-ISPB. In this contribution, based on the characteristics of F-ISPB, we propose an improved PSO with adaptive search space and elite reservation strategy to estimate the F-ISPB in real time. When the value of F-ISPB is close to the boundary of the search space, the improved PSO will transform the search space so that F-ISPB will be located near the central region of the new search space, which will greatly reduce the situation of the standard PSO easily falling into local optimum. Since F-ISPB is very stable, an elite retention strategy will help us to estimate F-ISPB faster and more accurately. Three sets of short baseline static data were selected for testing. The results show that the inter-system differenced model based on the improved PSO has a higher ambiguity fixed rate and positioning accuracy than the inter-system differenced model based on the standard PSO and the classical intra-system differenced model, and the fewer the number of satellites, the more obvious the effect.<\/jats:p>","DOI":"10.3390\/rs13163214","type":"journal-article","created":{"date-parts":[[2021,8,13]],"date-time":"2021-08-13T09:22:38Z","timestamp":1628846558000},"page":"3214","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Real-Time Estimation of GPS-BDS Inter-System Biases: An Improved Particle Swarm Optimization Algorithm"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2798-7186","authenticated-orcid":false,"given":"Wenhao","family":"Zhao","sequence":"first","affiliation":[{"name":"Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences, Wuhan 430077, China"},{"name":"College of Earth and Planetary Sciences, University of Chinese Academy of Sciences, Beijing 100049, China"}]},{"given":"Genyou","family":"Liu","sequence":"additional","affiliation":[{"name":"Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences, Wuhan 430077, China"}]},{"given":"Shengliang","family":"Wang","sequence":"additional","affiliation":[{"name":"Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences, Wuhan 430077, China"},{"name":"College of Earth and Planetary Sciences, University of Chinese Academy of Sciences, Beijing 100049, China"}]},{"given":"Ming","family":"Gao","sequence":"additional","affiliation":[{"name":"Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences, Wuhan 430077, China"},{"name":"College of Earth and Planetary Sciences, University of Chinese Academy of Sciences, Beijing 100049, China"}]},{"given":"Dong","family":"Lv","sequence":"additional","affiliation":[{"name":"Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences, Wuhan 430077, China"},{"name":"College of Earth and Planetary Sciences, University of Chinese Academy of Sciences, Beijing 100049, China"}]}],"member":"1968","published-online":{"date-parts":[[2021,8,13]]},"reference":[{"key":"ref_1","unstructured":"Montenbruck, O., Hauschild, A., and Steigenberger, P. (2014, January 27\u201329). Differential Code Bias Estimation using Multi-GNSS Observations and Global Ionosphere Maps. Proceedings of the 2014 International Technical Meeting of The Institute of Navigation, San Diego, CA, USA."},{"key":"ref_2","unstructured":"Odolinski, R., Teunissen, P.J.G., Odijk, D., and Inst, N. (2014, January 8\u201312). Combined GPS plus BDS plus Galileo plus QZSS for Long Baseline RTK Positioning. Proceedings of the 27th International Technical Meeting of the Satellite Division of the Institute of Navigation, Tampa, FL, USA."},{"key":"ref_3","unstructured":"Julien, O. (2003, January 22\u201325). A Tightly Coupled GPS\/GALILEO Combination for Improved Ambiguity Resolution. Proceedings of the European Navigation Conference, Graz, Austria."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"686","DOI":"10.1017\/S0373463316000825","article-title":"Accounting for Inter-System Bias in DGNSS Positioning with GPS\/GLONASS\/BDS\/Galileo","volume":"70","author":"Liu","year":"2017","journal-title":"J. Navig."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1161","DOI":"10.1007\/s10291-017-0599-4","article-title":"Double differencing within GNSS constellations","volume":"21","author":"Li","year":"2017","journal-title":"GPS Solut."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1007\/s10291-017-0687-5","article-title":"Combined GPS and BDS for single-frequency continuous RTK positioning through real-time estimation of differential inter-system biases","volume":"22","author":"Gao","year":"2017","journal-title":"GPS Solut."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1007\/s00190-007-0187-4","article-title":"Resolution of GPS carrier-phase ambiguities in Precise Point Positioning (PPP) with daily observations","volume":"82","author":"Ge","year":"2008","journal-title":"J. Geod."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"521","DOI":"10.1007\/s10291-012-0298-0","article-title":"Characterization of between-receiver GPS-Galileo inter-system biases and their effect on mixed ambiguity resolution","volume":"17","author":"Odijk","year":"2012","journal-title":"GPS Solut."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1007\/s10291-014-0392-6","article-title":"The mixed-receiver BeiDou inter-satellite-type bias and its impact on RTK positioning","volume":"19","author":"Nadarajah","year":"2014","journal-title":"GPS Solut."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1007\/s10291-015-0513-x","article-title":"GPS\/BDS short-term ISB modelling and prediction","volume":"21","author":"Jiang","year":"2017","journal-title":"GPS Solut."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1007\/s10291-014-0388-2","article-title":"An analysis of intersystem biases for multi-GNSS positioning","volume":"19","author":"Torre","year":"2014","journal-title":"GPS Solut."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1007\/s10291-015-0450-8","article-title":"Assessing the IRNSS L5-signal in combination with GPS, Galileo, and QZSS L5\/E5a-signals for positioning and navigation","volume":"20","author":"Nadarajah","year":"2016","journal-title":"GPS Solut."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1007\/s10291-016-0536-y","article-title":"GPS, Galileo, QZSS and IRNSS differential ISBs: Estimation and application","volume":"21","author":"Odijk","year":"2017","journal-title":"GPS Solut."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Gao, W., Gao, C., Pan, S., Meng, X., and Xia, Y. (2017). Inter-System Differencing between GPS and BDS for Medium-Baseline RTK Positioning. Remote Sens., 9.","DOI":"10.3390\/rs9090948"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1007\/s10291-017-0670-1","article-title":"Mixed GPS\u2013BeiDou RTK with inter-systems bias estimation aided by CSAC","volume":"22","author":"Kubo","year":"2017","journal-title":"GPS Solut."},{"key":"ref_16","first-page":"1","article-title":"An Algorithm and Results Analysis for GPS+BDS Inter-System Mix Double-Difference RTK","volume":"36","author":"Lou","year":"2016","journal-title":"J. Geod. Geodyn."},{"key":"ref_17","first-page":"175","article-title":"The Stability of GPS\/BDS Inter-system Biases at the Receiver End and its Effect on Ambiguity Resolution","volume":"43","author":"Sui","year":"2018","journal-title":"Geomat. Inf. Sci. Wuhan Univ."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1007\/s00190-011-0502-y","article-title":"Carrier-phase inter-frequency biases of GLONASS receivers","volume":"86","author":"Wanninger","year":"2012","journal-title":"J. Geod."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10291-020-01027-5","article-title":"On the temperature sensitivity of multi-GNSS intra- and inter-system biases and the impact on RTK positioning","volume":"24","author":"Mi","year":"2020","journal-title":"GPS Solut."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1007\/s10291-019-0873-8","article-title":"Multi-GNSS inter-system biases: Estimability analysis and impact on RTK positioning","volume":"23","author":"Mi","year":"2019","journal-title":"GPS Solut."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1007\/s00190-014-0763-3","article-title":"Accounting for Galileo\u2013GPS inter-system biases in precise satellite positioning","volume":"89","author":"Paziewski","year":"2015","journal-title":"J. Geod."},{"key":"ref_22","unstructured":"Eberhart, R.C., and Kennedy, J. (1995, January 4\u20136). A New Optimizer Using Particle Swarm Theory. Proceedings of the Sixth International Symposium on Micro Machine and Human Science, Nagoya, Japan."},{"key":"ref_23","unstructured":"Shi, Y., and Eberhart, R. (1998, January 4\u20139). A modified particle swarm optimizer. Proceedings of the 1998 IEEE International Conference on Evolutionary Computation Proceedings. IEEE World Congress on Computational Intelligence (Cat. No. 98TH83620), Anchorage, AK, USA."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1109\/4235.985692","article-title":"The particle swarm - explosion, stability, and convergence in a multidimensional complex space","volume":"6","author":"Clerc","year":"2002","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1185","DOI":"10.1016\/j.ijepes.2014.09.005","article-title":"Particle swarm optimization with an aging leader and challengers algorithm for optimal power flow problem with FACTS devices","volume":"64","author":"Singh","year":"2015","journal-title":"Int. J. Electr. Power Energy Syst."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"138","DOI":"10.1179\/003962609X390058","article-title":"The GNSS ambiguity ratio-test revisited: A better way of using it","volume":"41","author":"Teunissen","year":"2009","journal-title":"Surv. Rev."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1007\/s10291-012-0299-z","article-title":"The ratio test for future GNSS ambiguity resolution","volume":"17","author":"Verhagen","year":"2012","journal-title":"GPS Solut."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"054006","DOI":"10.1088\/1361-6501\/abdae5","article-title":"Multi-GNSS inter-system model for complex environments based on optimal state estimation","volume":"32","author":"Shang","year":"2021","journal-title":"Meas. Sci. Technol."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"949","DOI":"10.1007\/s10291-016-0584-3","article-title":"Particle filter-based estimation of inter-system phase bias for real-time integer ambiguity resolution","volume":"21","author":"Tian","year":"2017","journal-title":"GPS Solut."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1073","DOI":"10.1007\/s00190-018-01226-6","article-title":"Multi-dimensional particle filter-based estimation of inter-system phase biases for multi-GNSS real-time integer ambiguity resolution","volume":"93","author":"Tian","year":"2019","journal-title":"J. Geod."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"953","DOI":"10.1017\/S0373463320000077","article-title":"Particle Filter-Based Inter-System Positioning Model for Non-Overlapping Frequency Code Division Multiple Access Systems","volume":"73","author":"Shang","year":"2020","journal-title":"J. Navig."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"606","DOI":"10.1007\/s001900050199","article-title":"Success probability of integer GPS ambiguity rounding and bootstrapping","volume":"72","author":"Teunissen","year":"1998","journal-title":"J. Geod."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1016\/j.ins.2020.06.027","article-title":"Heterogeneous comprehensive learning and dynamic multi-swarm particle swarm optimizer with two mutation operators","volume":"540","author":"Wang","year":"2020","journal-title":"Inf. Sci."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1007\/s10291-013-0337-5","article-title":"Reliable single-epoch ambiguity resolution for short baselines using combined GPS\/BeiDou system","volume":"18","author":"Deng","year":"2014","journal-title":"GPS Solut."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"165","DOI":"10.5081\/jgps.10.2.165","article-title":"Impact of the GNSS Time Offsets on Positioning Reliability","volume":"10","author":"Wang","year":"2011","journal-title":"J. Glob. Position. Syst."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"(1995). Teunissen, P The least-squares ambiguity decorrelation adjustment: A method for fast GPS integer ambiguity estimation. J. Geod., 70, 65\u201382.","DOI":"10.1007\/BF00863419"}],"container-title":["Remote Sensing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2072-4292\/13\/16\/3214\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:45:43Z","timestamp":1760165143000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2072-4292\/13\/16\/3214"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,8,13]]},"references-count":36,"journal-issue":{"issue":"16","published-online":{"date-parts":[[2021,8]]}},"alternative-id":["rs13163214"],"URL":"https:\/\/doi.org\/10.3390\/rs13163214","relation":{},"ISSN":["2072-4292"],"issn-type":[{"type":"electronic","value":"2072-4292"}],"subject":[],"published":{"date-parts":[[2021,8,13]]}}}