{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T03:25:48Z","timestamp":1774322748656,"version":"3.50.1"},"reference-count":53,"publisher":"MDPI AG","issue":"16","license":[{"start":{"date-parts":[[2021,8,5]],"date-time":"2021-08-05T00:00:00Z","timestamp":1628121600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100010665","name":"H2020 Marie Sk\u0142odowska-Curie Actions","doi-asserted-by":"publisher","award":["722023"],"award-info":[{"award-number":["722023"]}],"id":[{"id":"10.13039\/100010665","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Remote Sensing"],"abstract":"<jats:p>The single-receiver integer ambiguity resolution-enabled variant of precise point positioning (PPP), namely PPP-RTK, has proven to be crucial in reducing the long convergence time of PPP solutions through the recovery of the integerness of the user-ambiguities. The proliferation of global navigation satellite systems (GNSS) supports various improvements in this regard through the availability of more satellites and frequencies. The increased availability of the Galileo E6 signal from GNSS receivers paves the way for speeding up integer ambiguity resolution, as more frequencies provide for a stronger model. In this contribution, the Galileo-based PPP-RTK ambiguity resolution and positioning convergence capabilities are studied and numerically demonstrated as a function of the number and spacing of frequencies, aiming to shed light on which frequencies should be used to obtain optimal performance. Through a formal analysis, we provide insight into the pivotal role of frequency separation in ambiguity resolution. Using real Galileo data on up to five frequencies and our estimated PPP-RTK corrections, representative kinematic user convergence results with partial ambiguity resolution are presented and discussed. Compared to the achieved performance of dual-frequency fixed solutions, it is found that the contribution of multi-frequency observations is significant and largely driven by frequency separation. When using all five available frequencies, it is shown that the kinematic user can achieve a sub-decimeter level convergence in 15.0 min (90% percentile). In our analysis, we also show to what extent the provision of the estimable satellite code biases as standard PPP-RTK corrections accelerates convergence. Finally, we numerically demonstrate that, when integrated with GPS, the kinematic user solution achieves convergence in 3.0 and 5.0 min on average and at 90%, respectively, in the presence of ionospheric delays, thereby indicating the single-receiver user\u2019s fast-convergence capabilities.<\/jats:p>","DOI":"10.3390\/rs13163077","type":"journal-article","created":{"date-parts":[[2021,8,5]],"date-time":"2021-08-05T04:33:29Z","timestamp":1628138009000},"page":"3077","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":33,"title":["A Multi-Frequency Galileo PPP-RTK Convergence Analysis with an Emphasis on the Role of Frequency Spacing"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0829-8644","authenticated-orcid":false,"given":"Dimitrios","family":"Psychas","sequence":"first","affiliation":[{"name":"Department of Geoscience and Remote Sensing, Delft University of Technology, 2600 GA Delft, The Netherlands"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4143-6006","authenticated-orcid":false,"given":"Peter J. G.","family":"Teunissen","sequence":"additional","affiliation":[{"name":"Department of Geoscience and Remote Sensing, Delft University of Technology, 2600 GA Delft, The Netherlands"},{"name":"GNSS Research Centre, Curtin University, Perth, WA 6845, Australia"},{"name":"Department of Infrastructure, University of Melbourne, Melbourne, VIC 3010, Australia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4107-2364","authenticated-orcid":false,"given":"Sandra","family":"Verhagen","sequence":"additional","affiliation":[{"name":"Department of Geoscience and Remote Sensing, Delft University of Technology, 2600 GA Delft, The Netherlands"}]}],"member":"1968","published-online":{"date-parts":[[2021,8,5]]},"reference":[{"key":"ref_1","unstructured":"Leandro, R., Landau, H., Nitschke, M., Glocker, M., Seeger, S., Chen, X., Deking, A., BenTahar, M., Zhang, F., and Ferguson, K. (2011, January 20\u201323). RTX positioning: The next generation of cm-accurate real-time GNSS positioning. Proceedings of the 24th International Technical Meeting of the Satellite Division of the Institute of Navigation (ION GNSS 2011), Portland, OR, USA."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1002\/navi.57","article-title":"Global and regional ionospheric corrections for faster PPP convergence","volume":"61","author":"Banville","year":"2014","journal-title":"Navigation"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1007\/s00190-015-0854-9","article-title":"On the estimability of parameters in undifferenced, uncombined GNSS network and PPP-RTK user models by means of S-system theory","volume":"90","author":"Odijk","year":"2016","journal-title":"J. Geod."},{"key":"ref_4","first-page":"223","article-title":"PPP-RTK: Results of CORS network-based PPP with integer ambiguity resolution","volume":"42","author":"Teunissen","year":"2010","journal-title":"J. Aeronaut. Astronaut. Aviat."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"840","DOI":"10.1016\/j.asr.2013.07.021","article-title":"Predicting atmospheric delays for rapid ambiguity resolution in precise point positioning","volume":"54","author":"Li","year":"2014","journal-title":"Adv. Space Res."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Psychas, D., and Verhagen, S. (2020). Real-Time PPP-RTK Performance Analysis Using Ionospheric Corrections from Multi-Scale Network Configurations. Sensors, 20.","DOI":"10.3390\/s20113012"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Geng, J., Li, X., Zhao, Q., and Li, G. (2019). Inter-system PPP ambiguity resolution between GPS and BeiDou for rapid initialization. J. Geod., 93.","DOI":"10.1007\/s00190-018-1167-6"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Katsigianni, G., Loyer, S., and Perosanz, F. (2019). PPP and PPP-AR Kinematic Post-Processed Performance of GPS-Only, Galileo-Only and Multi-GNSS. Remote Sens., 11.","DOI":"10.3390\/rs11212477"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"810","DOI":"10.1016\/j.asr.2020.04.054","article-title":"Assessing GPS\/Galileo real-time precise point positioning with ambiguity resolution based on phase biases from CNES","volume":"66","author":"Liu","year":"2020","journal-title":"Adv. Space Res."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Nadarajah, N., Khodabandeh, A., Wang, K., Choudhury, M., and Teunissen, P.J.G. (2018). Multi-GNSS PPP-RTK: From Large- to Small-Scale Networks. Sensors, 18.","DOI":"10.3390\/s18041078"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1007\/s10291-020-00992-1","article-title":"GPS+Galileo+BeiDou precise point positioning with triple-frequency ambiguity resolution","volume":"24","author":"Li","year":"2020","journal-title":"GPS Solut."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"579","DOI":"10.1007\/s00190-017-1081-3","article-title":"Multi-GNSS phase delay estimation and PPP ambiguity resolution: GPS, BDS, GLONASS, Galileo","volume":"92","author":"Li","year":"2018","journal-title":"J. Geod."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Ma, H., Zhao, Q., Verhagen, S., Psychas, D., and Liu, X. (2020). Assessing the Performance of Multi-GNSS PPP-RTK in the Local Area. Remote Sens., 12.","DOI":"10.3390\/rs12203343"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1007\/s10291-020-01043-5","article-title":"GLONASS FDMA data for RTK positioning: A five-system analysis","volume":"25","author":"Brack","year":"2021","journal-title":"GPS Solut."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"449","DOI":"10.1007\/s00190-013-0619-2","article-title":"Triple-frequency GPS precise point positioning with rapid ambiguity resolution","volume":"87","author":"Geng","year":"2013","journal-title":"J. Geod."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Xiao, G., Li, P., Gao, Y., and Heck, B. (2019). A Unified Model for Multi-Frequency PPP Ambiguity Resolution and Test Results with Galileo and BeiDou Triple-Frequency Observations. Remote Sens., 11.","DOI":"10.3390\/rs11020116"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Liu, G., Zhang, X., and Li, P. (2019). Improving the Performance of Galileo Uncombined Precise Point Positioning Ambiguity Resolution Using Triple-Frequency Observations. Remote Sens., 11.","DOI":"10.3390\/rs11030341"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1105","DOI":"10.1007\/s00190-019-01229-x","article-title":"Triple-frequency PPP ambiguity resolution with multi-constellation GNSS: BDS and Galileo","volume":"93","author":"Li","year":"2019","journal-title":"J. Geod."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Geng, J., Guo, J., Meng, X., and Kao, G. (2020). Speeding up PPP ambiguity resolution using triple-frequency GPS\/BeiDou\/Galileo\/QZSS data. GPS Solut., 94.","DOI":"10.1007\/s00190-019-01330-1"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Morton, Y.J., van Diggelen, F., Spilker, J.J., Parkinson, B.W., Lo, S., and Gao, G. (2021). GNSS Precise Point Positioning. Position, Navigation, and Timing Technologies in the 21st Century: Integrated Satellite Navigation, Sensor Systems, and Civil Applications, Wiley-IEEE Press. [1st ed.]. Chapter 20.","DOI":"10.1002\/9781119458449"},{"key":"ref_21","unstructured":"GNSS Science Support Centre (2020, December 03). Galileo Signal Plan. Available online: https:\/\/gssc.esa.int\/navipedia\/index.php\/Galileo_Signal_Plan."},{"key":"ref_22","unstructured":"GNSS Science Support Centre (2021, January 02). Galileo High Accuracy Service (HAS). Available online: https:\/\/gssc.esa.int\/navipedia\/index.php\/Galileo_High_Accuracy_Service_(HAS)."},{"key":"ref_23","unstructured":"European GNSS Agency (2020). GNSS User Technology Report. Issue 3, European GNSS Agency. Available online: https:\/\/www.euspa.europa.eu\/simplecount_pdf\/tracker?file=uploads\/technology_report_2020.pdf."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Xin, S., Geng, G., Guo, G., and Meng, X. (2020). On the Choice of the Third-Frequency Galileo Signals in Accelerating PPP Ambiguity Resolution in Case of Receiver Antenna Phase Center Errors. Remote Sens., 12.","DOI":"10.3390\/rs12081315"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"24","DOI":"10.1007\/s10291-019-0930-3","article-title":"Galileo PPP rapid ambiguity resolution with five-frequency observations","volume":"24","author":"Li","year":"2020","journal-title":"GPS Solut."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1007\/s00190-019-01341-y","article-title":"Beyond three frequencies: An extendable model for single-epoch decimeter-level point positioning by exploiting Galileo and BeiDou-3 signals","volume":"94","author":"Geng","year":"2020","journal-title":"J. Geod."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"402","DOI":"10.1007\/s00190-003-0344-3","article-title":"Theory of integer equivariant estimation with application to GNSS","volume":"77","author":"Teunissen","year":"2003","journal-title":"J. Geod."},{"key":"ref_28","unstructured":"Laurichesse, D., and Banville, S. (2020, December 03). Instantaneous Centimeter-Level Multi-Frequency Precise Point Positioning. GPS World, Innovation Column. Available online: https:\/\/www.gpsworld.com\/innovation-instantaneous-centimeter-level-multi-frequency-precise-point-positioning\/."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Guo, J., Geng, J., and Wang, C. (2021). Impact of the third frequency GNSS pseudorange and carrier phase observations on rapid PPP convergences. GPS Solut., 25.","DOI":"10.1007\/s10291-020-01079-7"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Guo, G., and Xin, S. (2019, January 16\u201320). Toward single-epoch 10-centimeter precise point positioning using Galileo E1\/E5a and E6 signals. Proceedings of the 32nd International Technical Meeting of the ION Satellite Division (ION GNSS+ 2019), Miami, FL, USA.","DOI":"10.33012\/2019.16925"},{"key":"ref_31","unstructured":"Hofmann-Wellenhof, B., Lichtenegger, H., and Wasle, E. (2008). GNSS\u2014Global Navigation Satellite Systems: GPS, GLONASS, Galileo, and More, Springer."},{"key":"ref_32","unstructured":"Baarda, W. (1973). S-transformations and criterion matrices. Publications on Geodesy, 18, Netherlands Geodetic Commission."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Grafarend, E., and Sanso, F. (1985). Zero Order Design: Generalized Inverses, Adjustment, the Datum Problem and S-Transformations. Optimization and Design of Geodetic Networks, Springer.","DOI":"10.1007\/978-3-642-70659-2"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"De Jonge, P. (1998). A Processing Strategy for the Application of the GPS in Networks. [Ph.D. Thesis, Delft University of Technology].","DOI":"10.54419\/hcczv9"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1671","DOI":"10.1016\/j.asr.2017.01.011","article-title":"The Multi-GNSS Experiment (MGEX) of the International GNSS Service (IGS)\u2014Achievements, prospects and challenges","volume":"59","author":"Montenbruck","year":"2017","journal-title":"Adv. Space Res."},{"key":"ref_36","unstructured":"Geoscience Australia (2020, December 03). Weekly Station Coordinates from ARGN, SPRGN, AuScope and APREF, Available online: ftp:\/\/ftp.ga.gov.au\/geodesy-outgoing\/gnss\/solutions\/final\/weekly\/."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Teunissen, P.J.G., and Montenbruck, O. (2017). The International GNSS Service. Springer Handbook of Global Navigation Satellite Systems, Springer International Publishing. [1st ed.]. Chapter 33.","DOI":"10.1007\/978-3-319-42928-1"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1007\/BF02521844","article-title":"Contributions to the theory of atmospheric refraction","volume":"105","author":"Saastamoinen","year":"1972","journal-title":"Bull. Geod."},{"key":"ref_39","unstructured":"Ifadis, I. (1986). The Atmospheric Delay of Radio Waves: Modelling the Elevation Dependence on a Global Scale, Chalmers University of Technology. Technical Report No 38L."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1007\/BF00806368","article-title":"On optimal filtering for GPS dual-frequency observations without using orbit information","volume":"65","author":"Euler","year":"1991","journal-title":"Bull. G\u00e9od."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"3219","DOI":"10.1007\/s11434-008-0293-6","article-title":"Assessment of stochastic models for GPS measurements with different types of receivers","volume":"53","author":"Li","year":"2008","journal-title":"Chin. Sci. Bull."},{"key":"ref_42","unstructured":"Teunissen, P.J.G. (1990, January 19\u201321). An integrity and quality control procedure for use in multi sensor integration. Proceedings of the 3rd International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GPS 1990), Colorado Spring, CO, USA."},{"key":"ref_43","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":"Teunissen","year":"1995","journal-title":"J. Geod."},{"key":"ref_44","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_45","unstructured":"Teunissen, P.J.G., Joosten, P., and Tiberius, C.C.J.M. (1999, January 25\u201327). Geometry-free ambiguity success rates in case of partial fixing. Proceedings of the ION NTM 1999, San Diego, CA, USA."},{"key":"ref_46","unstructured":"W\u00fcbbena, G., Schmitz, M., and Bagge, A. (2005, January 13\u201316). PPP-RTK: Precise Point Positioning Using State-Space Representation in RTK Networks. Proceedings of the 18th International Technical Meeting of the Satellite Division of the Institute of Navigation (ION GNSS 2005), Long Beach, CA, USA."},{"key":"ref_47","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":"2013","journal-title":"GPS Solut."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1007\/s10291-016-0523-3","article-title":"Assessment of precise orbit and clock products for Galileo, BeiDou, and QZSS from IGS Multi-GNSS Experiment (MGEX)","volume":"21","author":"Guo","year":"2017","journal-title":"GPS Solut."},{"key":"ref_49","unstructured":"Teunissen, P.J.G., Joosten, P., and Jonkman, N.F. (1999, January 2\u20133). Evaluation of long range GNSS ambiguity resolution. Proceedings of the 8th European CGSIC IISC Meeting, Prague, Czech Republic."},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Li, X., Li, X., Liu, G., Xie, W., Guo, F., Yuan, Y., Zhang, K., and Feng, G. (2020). The phase and code biases of Galileo and BDS-3 BOC signals: Effect on ambiguity resolution and precise positioning. J. Geod., 94.","DOI":"10.1007\/s00190-019-01336-9"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1007\/s10291-018-0700-7","article-title":"Estimation and analysis of multi-GNSS differential code biases using a hardware signal simulator","volume":"22","author":"Ammar","year":"2018","journal-title":"GPS Solut."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"2075","DOI":"10.1016\/j.asr.2020.08.010","article-title":"Precision analysis of partial ambiguity resolution-enabled PPP using multi-GNSS and multi-frequency signals","volume":"66","author":"Psychas","year":"2020","journal-title":"Adv. Space Res."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"S180","DOI":"10.1017\/S0373463311000361","article-title":"A Novel Un-differenced PPP-RTK concept","volume":"64","author":"Zhang","year":"2011","journal-title":"J. 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