{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,17]],"date-time":"2026-04-17T04:59:39Z","timestamp":1776401979576,"version":"3.51.2"},"reference-count":64,"publisher":"MDPI AG","issue":"21","license":[{"start":{"date-parts":[[2020,10,29]],"date-time":"2020-10-29T00:00:00Z","timestamp":1603929600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Remote Sensing"],"abstract":"<jats:p>A new approach is presented to improve the spatial and temporal resolution of the Vertical Total Electron Content (VTEC) estimates for regional positioning applications. The proposed technique utilises a priori information from the Global Ionosphere Maps (GIMs) of the Center for Orbit Determination in Europe (CODE), provided in terms of Spherical Harmonic (SH) coefficients of up to degree and order 15. Then, it updates the VTEC estimates using a new set of base-functions (with better resolution than SHs) while using the measurements of a regional GNSS network. To achieve the highest accuracy possible, our implementation is based on a transformation of the GIM\/CODE VTECs to their equivalent coefficients in terms of (spherical) Slepian functions. These functions are band-limited and reflect the majority of signal energy inside an arbitrarily defined region, yet their orthogonal property is remained. Then, new dual-frequency GNSS measurements are introduced to a Least Squares (LS) updating step that modifies the Slepian VTEC coefficients within the region of interest. Numerical application of this study is demonstrated using a synthetic example and ground-based GPS data in South America. The results are also validated against the VTEC estimations derived from independent GPS stations (that are not used in the modelling), and the VTEC products of international centres. Our results indicate that, by using 62 GPS stations in South America, the ionospheric delay estimation can be considerably improved. For example, using the new VTEC estimates in a Precise Point Positioning (PPP) experiment improved the positioning accuracy compared to the usage of GIM\/CODE and Klobuchar models. The reductions in the root mean squared of errors were \u223c23% and 25% for a day with moderate solar activity while 26% and \u223c35% for a day with high solar activity, respectively.<\/jats:p>","DOI":"10.3390\/rs12213545","type":"journal-article","created":{"date-parts":[[2020,10,29]],"date-time":"2020-10-29T09:44:53Z","timestamp":1603964693000},"page":"3545","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["A Least Squares Solution to Regionalize VTEC Estimates for Positioning Applications"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0534-0632","authenticated-orcid":false,"given":"Saeed","family":"Farzaneh","sequence":"first","affiliation":[{"name":"School of Surveying and Geospatial Engineering, College of Engineering, University of Tehran, Tehran 1439957131, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3055-041X","authenticated-orcid":false,"given":"Ehsan","family":"Forootan","sequence":"additional","affiliation":[{"name":"Geodesy and Earth Observation Group, Department of Planning, Aalborg University, Rendsburggade 14, 9000 Aalborg, Denmark"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,10,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Blaunstein, N., and Plohotniuc, E. (2008). Ionosphere and Applied Aspects of Radio Communication and Radar, CRC Press.","DOI":"10.1201\/9781420055177"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1029\/2002RS002794","article-title":"Global assimilation of ionospheric measurements (GAIM)","volume":"39","author":"Schunk","year":"2004","journal-title":"Radio Sci."},{"key":"ref_3","unstructured":"Notarpietro, R., Dovis, F., Franceschi, G.D., and Aquino, M. (2014). Performance analysis of empirical ionosphere models by comparison with CODE vertical TEC maps. Mitigation of Ionospheric Threats to GNSS, IntechOpen. Chapter 13."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"909","DOI":"10.1007\/s00190-010-0427-x","article-title":"The international reference ionosphere today and in the future","volume":"85","author":"Bilitza","year":"2011","journal-title":"J. Geod."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Angrisano, A., Gaglione, S., Gioia, C., Massaro, M., and Troisi, S. (2013). Benefit of the NeQuick Galileo version in GNSS single-point positioning. Int. J. Navig. Obs., 2013.","DOI":"10.1155\/2013\/302947"},{"key":"ref_6","unstructured":"Schaer, S., Beutler, G., Rothacher, M., and Springer, T.A. (1996, January 19\u201321). Daily global ionosphere maps based on GPS carrier phase data routinely produced by the CODE Analysis Center. Proceedings of the IGS Analysis Center Workshop, Silver Spring, MD, USA."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"662","DOI":"10.1007\/s00190-004-0432-z","article-title":"Efficient spatial and temporal representations of global ionosphere maps over Japan using B-spline wavelets","volume":"78","author":"Mautz","year":"2005","journal-title":"J. Geod."},{"key":"ref_8","first-page":"479","article-title":"Comparison and consistency research of regional ionospheric TEC models based on GPS measurements","volume":"33","author":"Zhu","year":"2008","journal-title":"Geomat. Inf. Sci. Wuhan Univ."},{"key":"ref_9","unstructured":"Wielgosz, P., Grejner-Brzezinska, D., and Kashani, I. (2009). Regional ionosphere mapping with kriging and multiquadric methods. Positioning, 1."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1007\/s10712-017-9446-y","article-title":"Reconstructing regional ionospheric electron density: A combined spherical Slepian function and empirical orthogonal function approach","volume":"39","author":"Farzaneh","year":"2018","journal-title":"Surv. Geophys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"393","DOI":"10.1002\/j.2161-4296.1994.tb01887.x","article-title":"Comparison of real-time ionospheric algorithms for a GPS Wide-Area Augmentation System (WAAS)","volume":"41","author":"Conker","year":"1994","journal-title":"Navigation"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"683","DOI":"10.1016\/S1364-6826(03)00085-3","article-title":"Combining ionosonde with ground GPS data for electron density estimation","volume":"65","author":"Juan","year":"2003","journal-title":"J. Atmos. Sol. Terr. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1965","DOI":"10.1016\/S0273-1177(03)00168-6","article-title":"3Dimensional ionospheric electron density reconstruction based on GPS measurements","volume":"31","author":"Stolle","year":"2003","journal-title":"Adv. Space Res."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"782","DOI":"10.1016\/j.asr.2007.02.050","article-title":"Regional 4-D modelling of the ionospheric electron density","volume":"42","author":"Schmidt","year":"2008","journal-title":"Adv. Space Res."},{"key":"ref_15","unstructured":"Garc\u00eda Fern\u00e1ndez, M. (2004). Contributions to the 3D Ionospheric Sounding with GPS Data, Universitat Polit\u00e8cnica de Catalunya."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"908","DOI":"10.1016\/j.asr.2014.05.005","article-title":"The spatio-spectral localization approach to modelling VTEC over the western part of the USA using GPS observations","volume":"54","author":"Sharifi","year":"2014","journal-title":"Adv. Space Res."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"263","DOI":"10.5194\/angeo-35-263-2017","article-title":"Near real-time estimation of ionosphere vertical total electron content from GNSS satellites using B-splines in a Kalman filter","volume":"35","author":"Erdogan","year":"2017","journal-title":"Ann. Geophys."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1002\/j.2161-4296.1992.tb01874.x","article-title":"Real-time ionospheric monitoring system using GPS","volume":"39","author":"Coster","year":"1992","journal-title":"Navigation"},{"key":"ref_19","unstructured":"Komjathy, A. (1997). Global Ionospheric Total Electron Content Mapping Using the Global Positioning System. [Ph.D. Thesis, University of New Brunswick Fredericton]."},{"key":"ref_20","unstructured":"Schaer, S., Helv\u00e9tique des Sciences Naturelles, and Commission g\u00e9od\u00e9sique (1999). Mapping and Predicting the Earth\u2019s Ionosphere Using the Global Positioning System, Institut f\u00fcr Geod\u00e4sie und Photogrammetrie, Eidg. Technische Hochschule Z\u00fcrich."},{"key":"ref_21","unstructured":"Schaer, S., Beutler, G., Mervart, L., Rothacher, M., and Wild, U. (1996, January 19\u201321). Global and regional ionosphere models using the GPS double difference phase observable. Proceedings of the IGS Workshop Special Topics and New Directions, Silver Spring, MD, USA."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"136","DOI":"10.1093\/gji\/ggs122","article-title":"Spectral and spatial decomposition of lithospheric magnetic field models using spherical Slepian functions","volume":"193","author":"Beggan","year":"2013","journal-title":"Geophys. J. Int."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"504","DOI":"10.1137\/S0036144504445765","article-title":"Spatiospectral concentration on a sphere","volume":"48","author":"Simons","year":"2006","journal-title":"SIAM Rev."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"932","DOI":"10.1016\/j.asr.2006.09.030","article-title":"Wavelet modelling in support of IRI","volume":"39","author":"Schmidt","year":"2007","journal-title":"Adv. Space Res."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"617","DOI":"10.1016\/j.jastp.2010.02.022","article-title":"B-spline modelling of VTEC over Turkey using GPS observations","volume":"72","author":"Nohutcu","year":"2010","journal-title":"J. Atmos. Sol. Terr. Phys."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1673","DOI":"10.1007\/s00190-019-01276-4","article-title":"A 4D tomographic ionospheric model to support PPP-RTK","volume":"93","author":"Terkildsen","year":"2019","journal-title":"J. Geod."},{"key":"ref_27","unstructured":"Zeilhofer, C. (2008). Multi-Dimensional B-Spline Modeling of Spatio-Temporal Ionospheric Signals, DGK."},{"key":"ref_28","unstructured":"Liu, J., Chen, R., Kuusniemi, H., Wang, Z., Zhang, H., and Yang, J. (2009, January 27\u201330). Mapping the regional ionospheric TEC using a spherical cap harmonic model and IGS products in high latitudes and the arctic region. Proceedings of the IAIN 2009 World Congress, Stockholm, Sweden."},{"key":"ref_29","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_30","doi-asserted-by":"crossref","first-page":"601","DOI":"10.1002\/2013JA019501","article-title":"Spherical cap harmonic analysis of the Arctic ionospheric TEC for one solar cycle","volume":"119","author":"Liu","year":"2014","journal-title":"J. Geophys. Res. Space Phys."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Goss, A., Schmidt, M., Erdogan, E., G\u00f6rres, B., and Seitz, F. (2019). High-resolution vertical total electron content maps based on multi-scale B-spline representations. Ann. Geophys., 37.","DOI":"10.5194\/angeo-2019-32"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"477","DOI":"10.1007\/s00190-011-0532-5","article-title":"Separation of global time-variable gravity signals into maximally independent components","volume":"86","author":"Forootan","year":"2012","journal-title":"J. Geod."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1007\/s11200-012-0718-1","article-title":"Separation of deterministic signals using independent component analysis (ICA)","volume":"57","author":"Forootan","year":"2013","journal-title":"Stud. Geophys. Geod."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"435","DOI":"10.1007\/s10712-017-9451-1","article-title":"Developing a complex independent component analysis (CICA) technique to extract non-stationary patterns from geophysical time series","volume":"39","author":"Forootan","year":"2018","journal-title":"Surv. Geophys."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"373","DOI":"10.1002\/j.2161-4296.1997.tb02355.x","article-title":"Performance of the LAMBDA method for fast GPS ambiguity resolution","volume":"44","author":"Teunissen","year":"1997","journal-title":"Navigation"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1007\/s00190-006-0093-1","article-title":"Calibration errors on experimental slant total electron content (TEC) determined with GPS","volume":"81","author":"Ciraolo","year":"2007","journal-title":"J. Geod."},{"key":"ref_37","first-page":"303","article-title":"Satellite Geodesy, 2nd completely revised and extended edition","volume":"10785","author":"Seeber","year":"2003","journal-title":"Walter Gruyter Gmbh Co. KG"},{"key":"ref_38","unstructured":"Hofmann-Wellenhof, B., Lichtenegger, H., and Wasle, E. (2008). GPS, Springer."},{"key":"ref_39","unstructured":"Feltens, J., and Schaer, S. (1998, January 9\u201311). IGS Products for the Ionosphere. Proceedings of the 1998 IGS Analysis Center Workshop Darmstadt, Darmstadt, Germany."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1007\/s00190-008-0266-1","article-title":"The IGS VTEC maps: A reliable source of ionospheric information since 1998","volume":"83","author":"Juan","year":"2009","journal-title":"J. Geod."},{"key":"ref_41","first-page":"373","article-title":"Mitigation of ionospheric effects on GNSS","volume":"52","author":"Warnant","year":"2009","journal-title":"Ann. Geophys."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1237","DOI":"10.1016\/S1364-6826(99)00054-1","article-title":"New approaches in global ionospheric determination using ground GPS data","volume":"61","author":"Juan","year":"1999","journal-title":"J. Atmos. Sol. Terr. Phys."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1081","DOI":"10.1029\/97RS00431","article-title":"Neural network modelling of the ionospheric electron content at global scale using GPS data","volume":"32","author":"Juan","year":"1997","journal-title":"Radio Sci."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1137\/1025078","article-title":"Some comments on Fourier analysis, uncertainty and modelling","volume":"25","author":"Slepian","year":"1983","journal-title":"SIAM Rev."},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Simons, F.J. (2009). Slepian functions and their use in signal estimation and spectral analysis. arXiv.","DOI":"10.1007\/978-3-642-01546-5_30"},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Percival, D.B., and Walden, A.T. (1993). Spectral Analysis for Physical Applications, Cambridge University Press.","DOI":"10.1017\/CBO9780511622762"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"655","DOI":"10.1111\/j.1365-246X.2005.02687.x","article-title":"Localized spectral analysis on the sphere","volume":"162","author":"Wieczorek","year":"2005","journal-title":"Geophys. J. Int."},{"key":"ref_48","first-page":"31","article-title":"The regional estimates of the GPS satellite and receiver differential code biases","volume":"10","author":"Farzaneh","year":"2017","journal-title":"Iran. J. Geophys."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"e25","DOI":"10.1186\/BF03353138","article-title":"Efficient GPS receiver DCB estimation for ionosphere modelling using satellite-receiver geometry changes","volume":"60","author":"Hong","year":"2008","journal-title":"Earth Planets Space"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"541","DOI":"10.1007\/s10291-012-0279-3","article-title":"M_DCB: Matlab code for estimating GNSS satellite and receiver differential code biases","volume":"16","author":"Jin","year":"2012","journal-title":"GPS Solut."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"129","DOI":"10.5047\/eps.2010.12.011","article-title":"Tomographic imaging of the equatorial and low-latitude ionosphere over central-eastern Brazil","volume":"63","author":"Muella","year":"2011","journal-title":"Earth Planets Space"},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Basu, S., Basu, S., Huba, J., Krall, J., McDonald, S., Makela, J.J., Miller, E., Ray, S., and Groves, K. (2009). Day-to-day variability of the equatorial ionization anomaly and scintillations at dusk observed by GUVI and modelling by SAMI3. J. Geophys. Res. Space Phys., 114.","DOI":"10.1029\/2008JA013899"},{"key":"ref_53","doi-asserted-by":"crossref","unstructured":"Stolle, C., Manoj, C., L\u00fchr, H., Maus, S., and Alken, P. (2008). Estimating the daytime equatorial ionization anomaly strength from electric field proxies. J. Geophys. Res. Space Phys., 113.","DOI":"10.1029\/2007JA012781"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"596","DOI":"10.1080\/01621459.1988.10478639","article-title":"Locally-weighted regression: An approach to regression analysis by local fitting","volume":"83","author":"Cleveland","year":"1988","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_55","unstructured":"Sanz Subirana, J., Juan Zornoza, J., and Hern\u00e1ndez-Pajares, M. (2013). GNSS Data Processing, Volume I: Fundamentals and Algorithms, ESA Communications; ESTEC."},{"key":"ref_56","doi-asserted-by":"crossref","unstructured":"Parvazi, K., Farzaneh, S., and Safari, A. (2020). Role of The RLS-VCE-Estimated Stochastic Model for Improvement of Accuracy and Convergence Time in Multi-GNSS Precise Point Positioning. Measurement, 108073.","DOI":"10.1016\/j.measurement.2020.108073"},{"key":"ref_57","unstructured":"Deng, Z., Fritsche, M., Uhlemann, M., Wickert, J., and Schuh, H. (2016, January 8\u201312). Reprocessing of GFZ multi-GNSS product GBM. Proceedings of the IGS Workshop, Sydney, Australia."},{"key":"ref_58","unstructured":"Rebischung, P., and Schmid, R. (2016, January 12\u201316). IGS14\/igs14. atx: A new framework for the IGS products. Proceedings of the AGU Fall Meeting, San Francisco, CA, USA."},{"key":"ref_59","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_60","unstructured":"Kouba, J. (2009). A guide to using International GNSS Service (IGS) products. Nat. Resour., Available online: https:\/\/www.researchgate.net\/profile\/Jan_Kouba\/publication\/228663800_A_guide_to_using_International_GNSS_Service_IGS_products\/links\/54fcc30c0cf270426d102cd3.pdf."},{"key":"ref_61","unstructured":"Wu, J.T., Wu, S.C., Hajj, G.A., Bertiger, W.I., and Lichten, S.M. (1992). Effects of antenna orientation on GPS carrier phase. ASDY, Univelt, Inc."},{"key":"ref_62","unstructured":"Petit, G., and Luzum, B. (2010). IERS Conventions (2010) (No. IERS-TN-36), Bureau International Des Poids et Mesures Sevres."},{"key":"ref_63","doi-asserted-by":"crossref","unstructured":"Klobuchar, J.A. (1987). Ionospheric time-delay algorithm for single-frequency GPS users. IEEE Trans. Aerosp. Electron. Syst., 325\u2013331.","DOI":"10.1109\/TAES.1987.310829"},{"key":"ref_64","doi-asserted-by":"crossref","unstructured":"Su, K., Jin, S., and Hoque, M. (2019). Evaluation of ionospheric delay effects on multi-GNSS positioning performance. Remote Sens., 11.","DOI":"10.3390\/rs11020171"}],"container-title":["Remote Sensing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2072-4292\/12\/21\/3545\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:26:23Z","timestamp":1760178383000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2072-4292\/12\/21\/3545"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,10,29]]},"references-count":64,"journal-issue":{"issue":"21","published-online":{"date-parts":[[2020,11]]}},"alternative-id":["rs12213545"],"URL":"https:\/\/doi.org\/10.3390\/rs12213545","relation":{},"ISSN":["2072-4292"],"issn-type":[{"value":"2072-4292","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,10,29]]}}}