{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,7]],"date-time":"2026-01-07T07:51:08Z","timestamp":1767772268035,"version":"build-2065373602"},"reference-count":38,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2021,4,6]],"date-time":"2021-04-06T00:00:00Z","timestamp":1617667200000},"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":["42030109","42074012"],"award-info":[{"award-number":["42030109","42074012"]}],"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":["2016YFC0803102"],"award-info":[{"award-number":["2016YFC0803102"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100019033","name":"Key Research and Development Program of Liaoning Province","doi-asserted-by":"publisher","award":["2020JH2\/10100044"],"award-info":[{"award-number":["2020JH2\/10100044"]}],"id":[{"id":"10.13039\/501100019033","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Concerning the triple-frequency ambiguity resolution, in principle there are three different realizations. The first one is to fix all the ambiguities of the original frequencies together. However, it is also believed that fixing the combined integer ambiguities with longer wavelength, such as extra-wide-lane (EWL), wide-lane (WL), should be advantageous. Also, it is demonstrated that fixing sequentially EWL, WL and one type of original ambiguities provides better results, as the previously fixed ambiguities increase parameters\u2019 precision for later fixings. In this paper, we undertake a comparative study of the three fixing approaches by means of experimental validation. In order to realize the three fixing approaches from the same information in terms of adjustment, we developed a processing strategy to provide fully consistent normal equations. We first generate the normal equation with the original undifferentiated carrier phase ambiguities, then map it into that with the combined and double-differenced ambiguities required by the individual approach for fixing. Four baselines of 258 m, 22 km, 47 km and 53 km are selected and processed in both static and kinematic mode using the three ambiguity-fixing approaches. Indicators including time of first fixed solution (TFFS), the correct fixing rate, positioning accuracy and RATIO are used to evaluate and investigate results. We also made a preliminary theoretical explanation of the results by looking into the decorrelation procedure of the ambiguity searching algorithm and the intermediate results. As conclusions, integrated searching of original ambiguities or combined ambiguities has almost the same fixing performance, whereas the sequential fixing of EWL, WL and B1 ambiguities overperforms the integrated searching. By the way, the third-frequency data can shorten the TFFS significantly but can hardly improve the positioning.<\/jats:p>","DOI":"10.3390\/s21072565","type":"journal-article","created":{"date-parts":[[2021,4,6]],"date-time":"2021-04-06T10:34:12Z","timestamp":1617705252000},"page":"2565","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["A Comparative Study of BDS Triple-Frequency Ambiguity Fixing Approaches for RTK Positioning"],"prefix":"10.3390","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8605-6536","authenticated-orcid":false,"given":"Huizhong","family":"Zhu","sequence":"first","affiliation":[{"name":"School of Geomatics, Liaoning Technical University (LNTU), Fuxin 123000, China"}]},{"given":"Yangyang","family":"Lu","sequence":"additional","affiliation":[{"name":"School of Geomatics, Liaoning Technical University (LNTU), Fuxin 123000, China"}]},{"given":"Longjiang","family":"Tang","sequence":"additional","affiliation":[{"name":"School of Geomatics, Liaoning Technical University (LNTU), Fuxin 123000, China"},{"name":"German Research Centre for Geosciences (GFZ), 14473 Potsdam, Germany"}]},{"given":"Jun","family":"Li","sequence":"additional","affiliation":[{"name":"School of Geomatics, Liaoning Technical University (LNTU), Fuxin 123000, China"}]},{"given":"Aigong","family":"Xu","sequence":"additional","affiliation":[{"name":"School of Geomatics, Liaoning Technical University (LNTU), Fuxin 123000, China"}]},{"given":"Maorong","family":"Ge","sequence":"additional","affiliation":[{"name":"German Research Centre for Geosciences (GFZ), 14473 Potsdam, Germany"}]}],"member":"1968","published-online":{"date-parts":[[2021,4,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1007\/BF00863419","article-title":"The least-squares ambiguity decorrelation adjustment: A method for fast GPS integer ambiguity estimation","volume":"70","author":"Temiissen","year":"1995","journal-title":"J. Geod."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"190","DOI":"10.1016\/j.asr.2010.01.017","article-title":"Effects of physical correlations on long-distance GPS positioning and zenith tropospheric delay estimates","volume":"46","author":"Jin","year":"2010","journal-title":"Adv. Space Res."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Xu, Y., Wu, C., Li, L., Yan, L., Liu, M., and Wang, S. (2018). GPS\/BDS medium\/long-range RTK constrained with tropospheric delay parameters from NWP model. Remote Sens., 10.","DOI":"10.3390\/rs10071113"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Shu, B., Liu, H., Xu, L., Qian, C., Gong, X., and An, X. (2018). Performance analysis of BDS medium-long baseline RTK positioning using an empirical troposphere model. Sensors, 18.","DOI":"10.3390\/s18041199"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"524","DOI":"10.1007\/s00190-005-0003-y","article-title":"Analysis of long-range network RTK during a severe ionospheric storm","volume":"79","author":"Wielgosz","year":"2005","journal-title":"J. Geod."},{"key":"ref_6","unstructured":"Dai, L., Eslinger, D., and Sharpe, T. (2007, January 28\u201330). Innovative algorithms to improve long range RTK reliability and availability. Proceedings of the ION National Technical Meeting, San Diego, CA, USA."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1007\/s10291-009-0131-6","article-title":"Three carrier ambiguity resolution: Distance-independent performance demonstrated using semi-generated triple frequency GPS signals","volume":"14","author":"Li","year":"2010","journal-title":"GPS Solut."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1007\/s00190-013-0611-x","article-title":"A method for improving uncalibrated phase delay estimation and ambiguity-fixing in real-time precise point positioning","volume":"87","author":"Li","year":"2013","journal-title":"J. Geod."},{"key":"ref_9","first-page":"325","article-title":"Rapid static positioning based on the fast ambiguity resolution approach FARA\u2019: Theory and first results","volume":"15","author":"Frei","year":"1990","journal-title":"Manuscr. Geod."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"731","DOI":"10.1007\/s11430-010-0032-0","article-title":"Wide area real time kinematic decimeter positioning with multiple carrier GNSS signals","volume":"53","author":"Feng","year":"2010","journal-title":"Sci. China Earth Sci."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s43020-020-00014-y","article-title":"PPP models and performances from single-to quad-frequency BDS observations","volume":"1","author":"Jin","year":"2020","journal-title":"Satell. Navig."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00190-019-01330-1","article-title":"Speeding up PPP ambiguity resolution using triple-frequency GPS\/BeiDou\/Galileo\/QZSS data","volume":"94","author":"Geng","year":"2020","journal-title":"J. Geod."},{"key":"ref_13","unstructured":"Forssell, B., Martin-Neira, M., and Harrisz, R.A. (1997, January 16\u201319). Carrier phase ambiguity resolution in GNSS-2. Proceedings of the 10th International Technical Meeting of the Satellite Division of the Institute of Navigation (ION GPS 1997), Kansas, MO, USA."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"587","DOI":"10.1007\/s001900050269","article-title":"An optimality property of the integer least-squares estimator","volume":"73","author":"Teunissen","year":"1999","journal-title":"J. Geod."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1002\/j.2161-4296.1999.tb02392.x","article-title":"Analysis of Three-Carrier Ambiguity Resolution Technique for Precise Relative Positioning in GNSS-2","volume":"46","author":"Vollath","year":"1999","journal-title":"Navigation"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/PL00012810","article-title":"Civilian GPS: The benefits of three frequencies","volume":"3","author":"Hatch","year":"2000","journal-title":"GPS Solut."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"408","DOI":"10.1007\/s001900100192","article-title":"Random simulation and GPS decorrelation","volume":"75","author":"Xu","year":"2001","journal-title":"J. Geod."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"3254","DOI":"10.1109\/TAES.2015.140643","article-title":"Real-time kinematic positioning over long baselines using triple-frequency BeiDou signals","volume":"51","author":"Li","year":"2015","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1031","DOI":"10.1007\/s00190-017-1006-1","article-title":"ERTK: Extra-wide-lane RTK of triple-frequency GNSS signals","volume":"91","author":"Li","year":"2017","journal-title":"J. Geod."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1080\/14498596.2019.1642250","article-title":"Combined BeiDou-2 and BeiDou-3 instantaneous RTK positioning: Stochastic modeling and positioning performance assessment","volume":"65","author":"Miao","year":"2020","journal-title":"J. Spat. Sci."},{"key":"ref_21","unstructured":"Teunissen, J.G., Joosten, P., and Tiberius, C. (2002, January 24\u201327). A comparison of TCAR, CIR and LAMBDA GNSS ambiguity resolution. Proceedings of the ION-GPS-2002, Institute of Navigation, Portland, OR, USA."},{"key":"ref_22","unstructured":"Zhang, W., Cannon, M.E., Julien, O., and Alves, P. (2003, January 9\u201312). Investigation of combined GPS\/GALILEO cascading ambiguity resolution schemes. Proceedings of ION GPS\/GNSS, Portland, OR, USA."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s43020-019-0006-0","article-title":"Basic performance and future developments of BeiDou global navigation satellite system","volume":"1","author":"Yang","year":"2020","journal-title":"Satell. Navig."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1007\/s10291-007-0070-z","article-title":"Simplified equivalent representation of GPS observation equations","volume":"12","author":"Shen","year":"2008","journal-title":"GPS Solut."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1088\/0957-0233\/22\/4\/045101","article-title":"Troposphere modeling for precise GPS rapid static positioning in mountainous areas","volume":"22","author":"Wielgosz","year":"2011","journal-title":"Meas. Sci. Technol."},{"key":"ref_26","first-page":"113","article-title":"The evaluation of precision about Hopfield, Saastamoinen and EGNOS tropospheric delay correction model","volume":"49","author":"Qu","year":"2008","journal-title":"Acta Astron. Sin."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1069","DOI":"10.1002\/grl.50288","article-title":"GPT2: Empirical slant delay model for radio space geodetic techniques","volume":"40","author":"Lagler","year":"2013","journal-title":"Geophys. Res. Lett."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"B\u00f6hm, J., Niell, A., Tregoning, P., and Schuh, H. (2006). Global Mapping Function (GMF): A new empirical mapping function based on numerical weather model data. Geophys. Res. Lett., 33.","DOI":"10.1029\/2005GL025546"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"B\u00f6hm, J., and Schuh, H. (2004). Vienna mapping functions in VLBI analyses. Geophys. Res. Lett., 31.","DOI":"10.1029\/2003GL018984"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"10187","DOI":"10.1029\/JB094iB08p10187","article-title":"Carrier phase ambiguity resolution for the Global Positioning System applied to geodetic baselines up to 2000 km","volume":"94","author":"Blewitt","year":"1989","journal-title":"J. Geophys. Res. Solid Earth"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"3949","DOI":"10.1029\/JB094iB04p03949","article-title":"Global Positioning System network analysis with phase ambiguity resolution applied to crustal deformation studies in California","volume":"94","author":"Dong","year":"1989","journal-title":"J. Geophys. Res. Solid Earth"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"545","DOI":"10.1007\/s00190-018-1180-9","article-title":"Refining ionospheric delay modeling for undifferenced and uncombined GNSS data processing","volume":"93","author":"Zhao","year":"2019","journal-title":"J. Geod."},{"key":"ref_33","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_34","doi-asserted-by":"crossref","first-page":"1359","DOI":"10.1007\/s00190-016-0928-3","article-title":"Improved ambiguity resolution for URTK with dynamic atmosphere constraints","volume":"90","author":"Tang","year":"2016","journal-title":"J. Geod."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"639","DOI":"10.1007\/s001900050203","article-title":"Grid point search algorithm for fast integer ambiguity resolution","volume":"72","author":"Iz","year":"1998","journal-title":"J. Geod."},{"key":"ref_36","unstructured":"Dai, L.L., Eslinger, D.J., Sharpe, R.T., and Hatch, R.R. (2011). NavCorn Technology Inc. Partial Search Carrier-Phase Integer Ambiguity Resolution. (7,961,143), U.S. Patent."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"603","DOI":"10.1007\/BF02899825","article-title":"PANDA software and its preliminary result of positioning and orbit determination","volume":"8","author":"Liu","year":"2003","journal-title":"Wuhan Univ. J. Nat. Sci."},{"key":"ref_38","unstructured":"Odijk, D. (2000, January 19\u201322). Weighting Ionospheric Corrections to Improve Fast GPS Positioning Over Medium Distances. Proceedings of the 13th International Technical Meeting of the Satellite Division of the Institute of Navigation (ION GPS 2000), Salt Lake City, UT, USA."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/7\/2565\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T13:34:06Z","timestamp":1760362446000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/7\/2565"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,4,6]]},"references-count":38,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2021,4]]}},"alternative-id":["s21072565"],"URL":"https:\/\/doi.org\/10.3390\/s21072565","relation":{},"ISSN":["1424-8220"],"issn-type":[{"type":"electronic","value":"1424-8220"}],"subject":[],"published":{"date-parts":[[2021,4,6]]}}}