{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,31]],"date-time":"2025-12-31T12:15:43Z","timestamp":1767183343667,"version":"build-2065373602"},"reference-count":38,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2019,5,30]],"date-time":"2019-05-30T00:00:00Z","timestamp":1559174400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Multi- system and multi-frequency are two key factors that determine the performance of precise point positioning. Both multi-frequency and multi-system lead to new biases, which are not solved systematically. This paper concentrates on mathematical models of biases, influences of these biases, and positioning performance analysis of different observation models. The biases comprise the inter-frequency clock bias in multi-frequency and the inter-system clock bias in multi-system. The former is the residual differential code biases (DCBs) from receiver clock and satellite clock and usually occurs at the third frequency, the latter is the deviation of the receiver clock errors in different systems. Unified mathematical models of the biases are presented by analyzing the general formula of observation equations. The influences of these biases are validated by experiments with corresponding observation models. Subsequently, the experiments, which are based on the data at five globally distributed stations in Multi-Global Navigation Satellite System (GNSS) Experiment (MGEX) on day of year 100, 2018, assess positioning performance of different observation models with combination of frequencies (dual-frequency or triple- frequency) and systems (BeiDou Navigation Satellite System (BDS) or Global Positioning System (GPS)). The results show that the performances of triple-frequency models are almost as the same level as the dual-frequency models. They provide scientific support for the triple-frequency ambiguity-fixed solution which has a better convergence characteristic than dual-frequency ambiguity-fixed solution. Furthermore, the biases are expressed as an unified form that gives an important and valuable reference for future research on multi-frequency and multi-system precise point positioning.<\/jats:p>","DOI":"10.3390\/s19112469","type":"journal-article","created":{"date-parts":[[2019,5,30]],"date-time":"2019-05-30T11:07:44Z","timestamp":1559214464000},"page":"2469","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["The Unified Form of Code Biases and Positioning Performance Analysis in Global Positioning System (GPS)\/BeiDou Navigation Satellite System (BDS) Precise Point Positioning Using Real Triple-Frequency Data"],"prefix":"10.3390","volume":"19","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1852-4414","authenticated-orcid":false,"given":"Peng","family":"Liu","sequence":"first","affiliation":[{"name":"School of Electronic and Information Engineering, Beihang University, No. 37 Xueyuan Road, Beijing 100191, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5670-5064","authenticated-orcid":false,"given":"Honglei","family":"Qin","sequence":"additional","affiliation":[{"name":"School of Electronic and Information Engineering, Beihang University, No. 37 Xueyuan Road, Beijing 100191, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1036-6113","authenticated-orcid":false,"given":"Li","family":"Cong","sequence":"additional","affiliation":[{"name":"School of Electronic and Information Engineering, Beihang University, No. 37 Xueyuan Road, Beijing 100191, China"}]}],"member":"1968","published-online":{"date-parts":[[2019,5,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"5005","DOI":"10.1029\/96JB03860","article-title":"Precise Point Positioning for the efficient and robust analysis of GPS data from large networks","volume":"102","author":"Zumberge","year":"1997","journal-title":"J. Geophys. Res. Solid Earth"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1007\/PL00012883","article-title":"Precise Point Positioning using IGS orbit and clock products","volume":"5","author":"Kouba","year":"2001","journal-title":"GPS Solut."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"427","DOI":"10.1007\/s00190-015-0788-2","article-title":"Timing group delay and differential code bias corrections for BeiDou positioning","volume":"89","author":"Guo","year":"2015","journal-title":"J. Geod."},{"key":"ref_4","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_5","first-page":"42","article-title":"The WAAS L5 Signal: An Assessment of Its Behavior and Potential End Use","volume":"20","author":"Rho","year":"2009","journal-title":"GPS World"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1007\/s10291-011-0232-x","article-title":"Apparent clock variations of the Block IIF-1 (SVN62) GPS satellite","volume":"16","author":"Montenbruck","year":"2012","journal-title":"GPS Solut."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1007\/s10291-012-0272-x","article-title":"Initial assessment of the COMPASS\/BeiDou-2 regional navigation satellite system","volume":"17","author":"Montenbruck","year":"2013","journal-title":"GPS Solut."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1225","DOI":"10.1007\/s00190-017-1020-3","article-title":"Initial assessment of the COMPASS\/BeiDou-3: New-generation navigation signals","volume":"91","author":"Zhang","year":"2017","journal-title":"J. Geod."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"480","DOI":"10.1017\/S0373463314000824","article-title":"Precise point positioning using triple-frequency GPS measurements","volume":"68","author":"Elsobeiey","year":"2015","journal-title":"J. Navig."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1223","DOI":"10.1007\/s00190-016-0920-y","article-title":"Modeling and assessment of triple-frequency BDS precise point positioning","volume":"90","author":"Guo","year":"2016","journal-title":"J. Geod."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"751","DOI":"10.1007\/s10291-015-0486-9","article-title":"Improved method for estimating the inter-frequency satellite clock bias of triple-frequency GPS","volume":"20","author":"Li","year":"2016","journal-title":"GPS Solut."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"811","DOI":"10.1007\/s10291-016-0571-8","article-title":"Characteristics of inter-frequency clock bias for Block IIF satellites and its effect on triple-frequency GPS precise point positioning","volume":"21","author":"Pan","year":"2017","journal-title":"GPS Solut."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2048","DOI":"10.1016\/j.asr.2017.02.002","article-title":"Handling the satellite inter-frequency biases in triple-frequency observations","volume":"59","author":"Zhao","year":"2017","journal-title":"Adv. Space Res."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"897","DOI":"10.1007\/s10291-016-0578-1","article-title":"Impact of GPS differential code bias in dual-and triple-frequency positioning and satellite clock estimation","volume":"21","author":"Li","year":"2017","journal-title":"GPS Solut."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Pan, L., Li, X., Zhang, X., Li, X., Lu, C., Zhao, Q., and Liu, J. (2017). Considering inter-frequency clock bias for BDS triple-frequency precise point positioning. Remote. Sens., 9.","DOI":"10.3390\/rs9070734"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"76","DOI":"10.1007\/s10291-018-0741-y","article-title":"GPS inter-frequency clock bias modeling and prediction for real-time precise point positioning","volume":"22","author":"Pan","year":"2018","journal-title":"GPS Solut."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1007\/s10291-017-0680-z","article-title":"Analysis of estimated satellite clock biases and their effects on precise point positioning","volume":"22","author":"Ye","year":"2018","journal-title":"GPS Solut."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1133","DOI":"10.1007\/s00190-017-1106-y","article-title":"GPS satellite clock determination in case of inter-frequency clock biases for triple-frequency precise point positioning","volume":"92","author":"Guo","year":"2018","journal-title":"J. Geod."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"035102","DOI":"10.1088\/0957-0233\/27\/3\/035102","article-title":"On biases in precise point positioning with multi-constellation and multi-frequency GNSS data","volume":"27","author":"Deo","year":"2016","journal-title":"Meas. Sci. Technol."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Aggrey, J., and Bisnath, S. (February, January 30). Analysis of multi-GNSS PPP initialization using dual-and triple-frequency data. Proceedings of the 2017 International Technical Meeting of The Institute of Navigation, Monterey, CA, USA.","DOI":"10.33012\/2017.14909"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Teunissen, P., and Kleusberg, A. (1998). GPS for Geodesy, Springer.","DOI":"10.1007\/978-3-642-72011-6"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Leick, A., Rapoport, L., and Tatarnikov, D. (2015). GPS Satellite Surveying, John Wiley & Sons.","DOI":"10.1002\/9781119018612"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Xu, G., and Xu, Y. (2016). GPS: Theory, Algorithms and Applications, Springer.","DOI":"10.1007\/978-3-662-50367-6"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1","DOI":"10.5081\/jgps.12.1.1","article-title":"The BeiDou navigation message","volume":"12","author":"Montenbruck","year":"2013","journal-title":"J. Glob. Position Syst."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"475","DOI":"10.1007\/s00190-013-0622-7","article-title":"Initial results of precise orbit and clock determination for COMPASS navigation satellite system","volume":"87","author":"Zhao","year":"2013","journal-title":"J. Geod."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"4692","DOI":"10.1038\/srep04692","article-title":"Precise orbit determination of BeiDou constellation based on BETS and MGEX network","volume":"4","author":"Lou","year":"2014","journal-title":"Sci. Rep."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1007\/s12040-007-0039-x","article-title":"GPS satellite and receiver instrumental biases estimation using least squares method for accurate ionosphere modelling","volume":"116","author":"Rao","year":"2007","journal-title":"J. Earth Syst. Sci."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"927","DOI":"10.3390\/s140100927","article-title":"Precise point positioning with the BeiDou navigation satellite system","volume":"14","author":"Li","year":"2014","journal-title":"Sensors"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1002\/navi.64","article-title":"Differential Code Bias Estimation using Multi-GNSS Observations and Global Ionosphere Maps","volume":"61","author":"Montenbruck","year":"2014","journal-title":"Navig. J. Inst. Navig."},{"key":"ref_30","unstructured":"Schaer, S., and Steigenberger, P. (2006, January 8\u201312). Determination and use of GPS differential code bias values. Proceedings of the IGS Workshop, Darmstadt, Germany."},{"key":"ref_31","unstructured":"Schaer, S. (2012). Overview of GNSS Biases, Workshop on GNSS Biases."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Zhao, Q., Wang, G., Liu, Z., Hu, Z., Dai, Z., and Liu, J. (2016). Analysis of BeiDou satellite measurements with code multipath and geometry-free ionosphere-free combinations. Sensors, 16.","DOI":"10.3390\/s16010123"},{"key":"ref_33","unstructured":"Guo, F. (2016). Theory and Methodology of Quality Control and Quality Analysis for GPS Precise Point Positioning, Wuhan University Press. (In Chinese)."},{"key":"ref_34","unstructured":"Witchayangkoon, B. (2000). Elements of GPS Precise Point Positioning. [Ph.D. Thesis, University of Maine]."},{"key":"ref_35","unstructured":"Shen, X. (2002). Improving Ambiguity Convergence in Carrier Phase-based Precise Point Positioning. [Ph.D. Thesis, University of Calgary]. Available online: https:\/\/www.ucalgary.ca\/engo_webdocs\/YG\/02.20170.XiaobingShen.pdf."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"38618","DOI":"10.1109\/ACCESS.2019.2906413","article-title":"The Feasible Combining Observation Models and Equivalence in Dual-Frequency Precise Point Positioning","volume":"7","author":"Liu","year":"2019","journal-title":"IEEE Access"},{"key":"ref_37","unstructured":"Rizos, C., Montenbruck, O., Weber, R., Weber, G., and Hugentobler, U. (2013, January 23\u201325). The IGS MGEX experiment as a milestone for a comprehensive multi-GNSS service. Proceedings of the ION 2013 Pacific PNT Meeting, Honolulu, HI, USA."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1007\/s10291-016-0545-x","article-title":"Uncovering common misconceptions in GNSS Precise Point Positioning and its future prospect","volume":"21","author":"Choy","year":"2017","journal-title":"GPS Solut."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/19\/11\/2469\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:54:39Z","timestamp":1760187279000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/19\/11\/2469"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,5,30]]},"references-count":38,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2019,6]]}},"alternative-id":["s19112469"],"URL":"https:\/\/doi.org\/10.3390\/s19112469","relation":{},"ISSN":["1424-8220"],"issn-type":[{"type":"electronic","value":"1424-8220"}],"subject":[],"published":{"date-parts":[[2019,5,30]]}}}