{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:02:22Z","timestamp":1760148142257,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2023,4,4]],"date-time":"2023-04-04T00:00:00Z","timestamp":1680566400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Nature Science Foundation of China","doi-asserted-by":"publisher","award":["42070430","2022A1515011039"],"award-info":[{"award-number":["42070430","2022A1515011039"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Natural Science Foundation of Guangdong Province of China","award":["42070430","2022A1515011039"],"award-info":[{"award-number":["42070430","2022A1515011039"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Remote Sensing"],"abstract":"<jats:p>Ill-posedness of GNSS-based ionospheric tomography affects the stability and the accuracy of the inversion results. Truncated singular value decomposition (TSVD) is a common algorithm of ionospheric tomography reconstruction. However, the TSVD method usually has low inversion accuracy and reconstruction efficiency. To resolve the above problem, a truncated mapping singular value decomposition (TMSVD) algorithm is presented to improve the reconstructed accuracy and computational efficiency. To authenticate the effectiveness and the advantages of the TMSVD algorithm, a numerical test scheme is devised. Finally, ionospheric temporal\u2013spatial variations of the selected reconstructed region are studied using the GNSS observations under different geomagnetic conditions. The reconstructed results of TMSVD can accurately reflect semiannual anomalies, diurnal variations, and geomagnetic storm effects. In contrast with the ionosonde data, it is found that the reconstructed profiles of the TMSVD method are more consistent with than those of the IRI 2016. The study suggests that TMSVD is an efficient algorithm for the tomographic reconstruction of ionospheric electron density (IED).<\/jats:p>","DOI":"10.3390\/rs15071930","type":"journal-article","created":{"date-parts":[[2023,4,4]],"date-time":"2023-04-04T04:09:50Z","timestamp":1680581390000},"page":"1930","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["A New Algorithm for Ill-Posed Problem of GNSS-Based Ionospheric Tomography"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1529-9749","authenticated-orcid":false,"given":"Debao","family":"Wen","sequence":"first","affiliation":[{"name":"School of Geography and Remote Sensing, Guangzhou University, Guangzhou 510006, China"}]},{"given":"Kangyou","family":"Xie","sequence":"additional","affiliation":[{"name":"School of Geography and Remote Sensing, Guangzhou University, Guangzhou 510006, China"}]},{"given":"Yinghao","family":"Tang","sequence":"additional","affiliation":[{"name":"School of Geography and Remote Sensing, Guangzhou University, Guangzhou 510006, China"}]},{"given":"Dengkui","family":"Mei","sequence":"additional","affiliation":[{"name":"School of Geodesy and Geomatics, Wuhan University, Wuhan 430072, China"}]},{"given":"Xi","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Geography and Remote Sensing, Guangzhou University, Guangzhou 510006, China"}]},{"given":"Hanqing","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Geography and Remote Sensing, Guangzhou University, Guangzhou 510006, China"}]}],"member":"1968","published-online":{"date-parts":[[2023,4,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"5801408","DOI":"10.1109\/TGRS.2021.3113667","article-title":"Three-step tomographic algorithm for ionospheric electron density reconstruction","volume":"60","author":"Wen","year":"2022","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1794","DOI":"10.1049\/el:19931194","article-title":"Ionospheric mapping by computer contouring techniques","volume":"29","author":"Samardjiev","year":"1993","journal-title":"Electron. Lett."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2241","DOI":"10.1016\/j.asr.2003.05.063","article-title":"Operational communication systems and relationships to the ionosphere and space weather","volume":"36","author":"Goodman","year":"2005","journal-title":"Adv. Space Res."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"105628","DOI":"10.1016\/j.jastp.2021.105628","article-title":"The three-dimensional ionospheric electron density imaging in Japan using the approximate Kalman filter algorithm","volume":"219","author":"Song","year":"2021","journal-title":"J. Atmos. Sol.-Terr. Phy."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2215","DOI":"10.1016\/j.asr.2013.09.030","article-title":"Application of hybrid regularization method for tomographic reconstruction of millatitude ionospheric electron density","volume":"12","author":"Yao","year":"2013","journal-title":"Adv. Space Res."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2529","DOI":"10.1016\/j.asr.2020.03.002","article-title":"Ionospheric TEC disturbances over China during the strong geomagnetic storm in September 2017","volume":"65","author":"Wen","year":"2020","journal-title":"Adv. Space Res."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2969","DOI":"10.1016\/j.asr.2017.03.031","article-title":"Variable pixel size ionospheric tomography","volume":"59","author":"Zheng","year":"2017","journal-title":"Adv. Space Res."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Wen, D., Mei, D., and Du, Y. (2020). Adaptive smoothness constraint ionospheric tomography algorithm. Sensors, 20.","DOI":"10.3390\/s20082404"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1186\/s40623-016-0412-6","article-title":"Medium-scale traveling ionospheric disturbances by three-dimensional ionospheric GPS tomography","volume":"68","author":"Chen","year":"2016","journal-title":"Earth Planets Space"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1529","DOI":"10.1007\/s00190-019-01266-6","article-title":"Reconstruction of 2D\/3D ionospheric disturbances in high-latitude and Arctic regions during a geomagnetic storm using GNSS carrier TEC: A case study of the 2015 great storm","volume":"93","author":"Kong","year":"2019","journal-title":"J. Geodesy"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Wen, D., Mei, D., and Du, Y. (2020). Imaging the Three-Dimensional Ionospheric Structure with a Blob Basis Functional Ionospheric Tomography Model. Sensors, 20.","DOI":"10.3390\/s20082182"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1953","DOI":"10.1029\/97RS00837","article-title":"Possibilities of the near-space environment radio tomography","volume":"32","author":"Kunitsyn","year":"1997","journal-title":"Radio Sci."},{"key":"ref_13","first-page":"412","article-title":"A constrained ionospheric tomography algorithm with smoothing method","volume":"38","author":"Li","year":"2013","journal-title":"Geomat. Inf. Sci. Wuhan Univ."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"727","DOI":"10.1186\/BF03352821","article-title":"Constrained simultaneous algebraic reconstruction technique (C-SART)\u2014A new and simple algorithm applied to ionospheric tomography","volume":"60","author":"Hobiger","year":"2008","journal-title":"Earth Planets Space"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"96","DOI":"10.1007\/s10291-019-0893-4","article-title":"A new parameterized approach for ionospheric tomography","volume":"23","author":"Chen","year":"2019","journal-title":"GPS Solut."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"807","DOI":"10.1007\/s10291-015-0490-0","article-title":"Ionospheric tomography using GNSS: Multiplicative algebraic reconstruction technique applied to the area of Brazil","volume":"20","author":"Prol","year":"2016","journal-title":"GPS Solut."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1029\/2011RS004760","article-title":"Numerical validations of neural-network based ionospheric tomography for disturbed ionospheric conditions and sparse data","volume":"46","author":"Hirooka","year":"2011","journal-title":"Radio Sci."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"84","DOI":"10.3103\/S0027134909010196","article-title":"The ionosphere over Alaska during a storm period in October 2003: Radio tomography and data obtained with GAIM\/IFM ionospheric models","volume":"64","author":"Andreeva","year":"2009","journal-title":"Mosc. Univ. Phys. Bull."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2455","DOI":"10.1109\/TGRS.2010.2040829","article-title":"Inversion of ionospheric electron density based on a constrained simultaneous iteration reconstruction technique","volume":"48","author":"Liu","year":"2010","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1007\/s10291-010-0161-0","article-title":"Tomographic reconstruction of ionospheric electron density based on constrained algebraic reconstruction technique","volume":"14","author":"Wen","year":"2010","journal-title":"GPS Solut."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"4696","DOI":"10.1109\/TGRS.2013.2283736","article-title":"An improved iterative algorithm for 3-D ionospheric tomography reconstruction","volume":"52","author":"Yao","year":"2014","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1051","DOI":"10.1029\/2018RS006588","article-title":"An improved iterative algorithm for ionospheric tomography by using the automatic search technology of relaxation factor","volume":"53","author":"Zheng","year":"2018","journal-title":"Radio Sci."},{"key":"ref_23","first-page":"345","article-title":"A computerized ionospheric tomography algorithm based on TSVD regularization","volume":"29","author":"Ou","year":"2014","journal-title":"Chin. J. Radio Sci."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1002\/2015RS005869","article-title":"Ionospheric thermospheric UV tomography: 1. Image space reconstruction algorithms","volume":"52","author":"Dymond","year":"2017","journal-title":"Radio Sci."},{"key":"ref_25","first-page":"1547","article-title":"Ionospheric electron density reconstruction using TSVD methods over Iran","volume":"79","author":"Razin","year":"2010","journal-title":"Curr. Sci."},{"key":"ref_26","first-page":"1254","article-title":"Comparison of several ionospheric tomography algorithms","volume":"13","author":"Raymund","year":"1994","journal-title":"Ann. Geophys."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"851","DOI":"10.1016\/j.asr.2006.12.041","article-title":"International reference ionosphere as a potential regularization profile for computerized ionospheric tomography","volume":"39","author":"Bhuyan","year":"2007","journal-title":"Adv. Space Res."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1142","DOI":"10.1137\/0913066","article-title":"The modified truncated SVD method for regularization in general form","volume":"13","author":"Hansen","year":"1992","journal-title":"SIAM J. Sci. Stat. Comput."},{"key":"ref_29","first-page":"1117","article-title":"Tomogrpahic reconstruction of the ionosphere using generalized singular value decomposition","volume":"83","author":"Bhuyan","year":"2002","journal-title":"J. Geog. Infor. Techn."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1029\/RS023i003p00299","article-title":"Ionospheric imagingusing computerized tomography","volume":"23","author":"Austen","year":"1988","journal-title":"Radio Sci."}],"container-title":["Remote Sensing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2072-4292\/15\/7\/1930\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:09:37Z","timestamp":1760123377000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2072-4292\/15\/7\/1930"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4,4]]},"references-count":30,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2023,4]]}},"alternative-id":["rs15071930"],"URL":"https:\/\/doi.org\/10.3390\/rs15071930","relation":{},"ISSN":["2072-4292"],"issn-type":[{"type":"electronic","value":"2072-4292"}],"subject":[],"published":{"date-parts":[[2023,4,4]]}}}