{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,25]],"date-time":"2025-12-25T07:26:43Z","timestamp":1766647603797,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2024,4,5]],"date-time":"2024-04-05T00:00:00Z","timestamp":1712275200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Key Research and Development Program of China","award":["2020YFA0713501","42174040","42388102","HNGTCH-2023-05","SKLGIE2021-Z-2-1"],"award-info":[{"award-number":["2020YFA0713501","42174040","42388102","HNGTCH-2023-05","SKLGIE2021-Z-2-1"]}]},{"name":"National Natural Science Foundation of China","award":["2020YFA0713501","42174040","42388102","HNGTCH-2023-05","SKLGIE2021-Z-2-1"],"award-info":[{"award-number":["2020YFA0713501","42174040","42388102","HNGTCH-2023-05","SKLGIE2021-Z-2-1"]}]},{"name":"Department of Natural Resources of Hunan Province","award":["2020YFA0713501","42174040","42388102","HNGTCH-2023-05","SKLGIE2021-Z-2-1"],"award-info":[{"award-number":["2020YFA0713501","42174040","42388102","HNGTCH-2023-05","SKLGIE2021-Z-2-1"]}]},{"name":"State Key Laboratory of Geo-Information Engineering","award":["2020YFA0713501","42174040","42388102","HNGTCH-2023-05","SKLGIE2021-Z-2-1"],"award-info":[{"award-number":["2020YFA0713501","42174040","42388102","HNGTCH-2023-05","SKLGIE2021-Z-2-1"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>In the precise point positioning\/real-time kinematic (PPP-RTK) technique, high-precision ionospheric delay correction information is an important prerequisite for rapid PPP convergence. The commonly used ionospheric modeling approaches in the PPP-RTKs only take the trend term of the ionospheric total electron content (TEC) variations into account. As a result, the residual ionospheric delay still affects the positioning solutions. In this study, we propose a two-step regional ionospheric modeling approach that involves combining a polynomial fitting model (PFM) and a Kriging interpolation (KI) model. In the first step, a polynomial fitting method is used to model the trend term of the ionospheric TEC variations. In the second step, a KI method is used to compensate for the residual term of the ionospheric TEC variations. Datasets collected from continuously operating reference stations (CORSs) in Hunan Province, China, are used to validate the PFM\/KI method by comparing with a single PFM method and a combined PFM and inverse distance weighting interpolation (IDWI) method. The experimental results show that the two-step PFM\/KI modeled ionospheric delay achieves an average root mean square (RMS) error of 1.8 cm, which is improved by about 48% and 23% when compared with the PFM and PFM\/IDWI methods, respectively. Regarding the positioning performance, the PPP-RTK with the PFM\/KI method takes an average of 1.8 min or 4.0 min to converge to a positioning accuracy of 1.3 cm or 2.5 cm in the horizontal and vertical directions, respectively. The convergence times are decreased by about 18% and 14% in the horizontal direction and 9% and 5% in the vertical direction over the PFM and the PFM\/IDWI methods, respectively.<\/jats:p>","DOI":"10.3390\/s24072307","type":"journal-article","created":{"date-parts":[[2024,4,5]],"date-time":"2024-04-05T08:27:52Z","timestamp":1712305672000},"page":"2307","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A Two-Step Regional Ionospheric Modeling Approach for PPP-RTK"],"prefix":"10.3390","volume":"24","author":[{"given":"Zhenyu","family":"Xu","sequence":"first","affiliation":[{"name":"School of Geosciences and Info-Physics, Central South University, Changsha 410083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Changsheng","family":"Cai","sequence":"additional","affiliation":[{"name":"School of Geosciences and Info-Physics, Central South University, Changsha 410083, China"},{"name":"State Key Laboratory of Geo-Information Engineering, Xi\u2019an 710054, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lin","family":"Pan","sequence":"additional","affiliation":[{"name":"School of Geosciences and Info-Physics, Central South University, Changsha 410083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1606-117X","authenticated-orcid":false,"given":"Wujiao","family":"Dai","sequence":"additional","affiliation":[{"name":"School of Geosciences and Info-Physics, Central South University, Changsha 410083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bei","family":"He","sequence":"additional","affiliation":[{"name":"School of Geosciences and Info-Physics, Central South University, Changsha 410083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,4,5]]},"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":"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_4","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1016\/j.measurement.2019.05.094","article-title":"Comparative analysis of MGEX products for post-processing multi-GNSS PPP","volume":"145","author":"Bahadur","year":"2019","journal-title":"Measurement"},{"key":"ref_5","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_6","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 ION GNSS 18th International Technical Meeting of the Satellite Division, Long Beach, CA, USA."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1007\/s00190-014-0771-3","article-title":"Review and principles of PPP-RTK methods","volume":"89","author":"Teunissen","year":"2015","journal-title":"J. Geod."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1186\/s43020-022-00089-9","article-title":"Review of PPP\u2013RTK: Achievements, challenges, and opportunities","volume":"3","author":"Li","year":"2022","journal-title":"Satell. Navig."},{"key":"ref_9","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_10","doi-asserted-by":"crossref","first-page":"447","DOI":"10.1007\/s00190-015-0789-1","article-title":"Ionospheric effects in uncalibrated phase delay estimation and ambiguity-fixed PPP based on raw observable model","volume":"89","author":"Gu","year":"2015","journal-title":"J. Geod."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Boisits, J., Glaner, M., and Weber, R. (2020). Regiomontan: A regional high precision ionosphere delay model and its application in Precise Point Positioning. Sensors, 20.","DOI":"10.3390\/s20102845"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Shepard, D. (1968, January 27\u201329). A Two-Dimensional Interpolation Function for Irregularly-Spaced Data. Proceedings of the 1968 23rd ACM National Conference, Association for Computing Machinery, New York, NY, USA.","DOI":"10.1145\/800186.810616"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1246","DOI":"10.2113\/gsecongeo.58.8.1246","article-title":"Principles of geostatistics","volume":"58","author":"Matheron","year":"1963","journal-title":"Econ. Geol."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1002\/j.2161-4296.2003.tb00335.x","article-title":"Comparison of interpolation algorithms in network-based GPS techniques","volume":"50","author":"Dai","year":"2003","journal-title":"Navigation"},{"key":"ref_15","unstructured":"Schaer, S. (1999). Mapping and Predicting the Earth\u2019s Ionosphere Using the Global Positioning System, Institut f\u00fcr Geod\u00e4sie und Photogrammetrie, Eidg. Technische Hochschule."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"415","DOI":"10.1023\/B:ASTR.0000032540.35594.64","article-title":"A new ionosphere monitoring technology based on GPS","volume":"290","author":"Brunini","year":"2004","journal-title":"Astrophys. Space Sci."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1031","DOI":"10.1049\/iet-rsn.2018.5504","article-title":"Least-squares collocation modelling of regional ionospheric TEC for accelerating real-time single-frequency PPP convergence","volume":"13","author":"Gong","year":"2019","journal-title":"IET Radar Sonar Navig."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"107067","DOI":"10.1016\/j.measurement.2019.107067","article-title":"A comprehensive assessment of interpolation methods for regional augmented PPP using reference networks with different scales and terrains","volume":"150","author":"Wang","year":"2020","journal-title":"Measurement"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"134","DOI":"10.1007\/s10291-023-01476-8","article-title":"A new large-area hierarchical PPP-RTK service strategy","volume":"27","author":"Cui","year":"2023","journal-title":"GPS Solut."},{"key":"ref_20","unstructured":"Stein, M.L. (2012). Interpolation of Spatial Data: Some Theory for Kriging, Springer Science & Business Media."},{"key":"ref_21","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_22","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_23","doi-asserted-by":"crossref","first-page":"4350","DOI":"10.1016\/j.asr.2023.08.056","article-title":"High-rate GNSS multi-frequency uncombined PPP-AR for dynamic deformation monitoring","volume":"72","author":"Pan","year":"2023","journal-title":"Adv. Space Res."},{"key":"ref_24","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_25","unstructured":"Teunissen, P., Joosten, P., and Tiberius, C. (1999, January 25\u201327). Geometry-Free Ambiguity Success Rates in Case of Partial Fixing. Proceedings of the 1999 National Technical Meeting of the Institute of Navigation, Catamaran Resort Hotel, San Diego, CA, USA."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1007\/s10291-018-0699-9","article-title":"GAMP: An open-source software of multi-GNSS precise point positioning using undifferenced and uncombined observations","volume":"22","author":"Zhou","year":"2018","journal-title":"GPS Solut."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1007\/s10291-010-0174-8","article-title":"Spherical cap harmonic model for mapping and predicting regional TEC","volume":"15","author":"Liu","year":"2011","journal-title":"GPS Solut."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1598","DOI":"10.1016\/j.jastp.2005.07.017","article-title":"Improvement of global ionospheric VTEC maps by using kriging interpolation technique","volume":"67","author":"Juan","year":"2005","journal-title":"J. Atmos. Sol.-Terr. Phys."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/24\/7\/2307\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:23:41Z","timestamp":1760106221000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/24\/7\/2307"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,4,5]]},"references-count":28,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2024,4]]}},"alternative-id":["s24072307"],"URL":"https:\/\/doi.org\/10.3390\/s24072307","relation":{},"ISSN":["1424-8220"],"issn-type":[{"type":"electronic","value":"1424-8220"}],"subject":[],"published":{"date-parts":[[2024,4,5]]}}}