{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,11]],"date-time":"2026-05-11T23:41:12Z","timestamp":1778542872854,"version":"3.51.4"},"reference-count":32,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2018,11,14]],"date-time":"2018-11-14T00:00:00Z","timestamp":1542153600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["41774005, 41874032"],"award-info":[{"award-number":["41774005, 41874032"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Remote Sensing"],"abstract":"<jats:p>In Global Navigation Satellite System (GNSS) relative positioning applications, multipath errors are non-negligible. Mitigation of the multipath error is an important task for precise positioning and it is possible due to the repeatability, even without any rigorous mathematical model. Empirical modeling is required for this mitigation. In this work, the multipath error modeling using carrier phase measurement residuals is realized by solving a regularization problem. Two Tikhonov regularization schemes, namely with the first and the second order differences, are considered. For each scheme, efficient numerical algorithms are developed to find the solutions, namely the Thomas algorithm and Cholesky rank-one update algorithm for the first and the second differences, respectively. Regularization parameters or Lagrange multipliers are optimized using the bootstrap method. In experiment, data on the first day are processed to construct a multipath model for each satellite (except the reference one), and then the model is used to correct the measurement on the second day, namely following the sidereal filtering approach. The smoothness of the coordinates calculated using the corrected measurements is improved significantly compared to those using the raw measurement. The efficacy of the proposed method is illustrated by the actual calculation.<\/jats:p>","DOI":"10.3390\/rs10111801","type":"journal-article","created":{"date-parts":[[2018,11,14]],"date-time":"2018-11-14T10:58:22Z","timestamp":1542193102000},"page":"1801","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":24,"title":["Tikhonov Regularization Based Modeling and Sidereal Filtering Mitigation of GNSS Multipath Errors"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6392-3265","authenticated-orcid":false,"given":"Guobin","family":"Chang","sequence":"first","affiliation":[{"name":"School of Environmental Science and Spatial Informatics, China University of Mining and Technology, Xuzhou 221116, China"},{"name":"State Key Laboratory of Geo-information Engineering, Xi\u2019an Research Institute of Surveying and Mapping, Xi\u2019an 710054, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chao","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Environmental Science and Spatial Informatics, China University of Mining and Technology, Xuzhou 221116, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5553-948X","authenticated-orcid":false,"given":"Yuanxi","family":"Yang","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Geo-information Engineering, Xi\u2019an Research Institute of Surveying and Mapping, Xi\u2019an 710054, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5818-6264","authenticated-orcid":false,"given":"Tianhe","family":"Xu","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Geo-information Engineering, Xi\u2019an Research Institute of Surveying and Mapping, Xi\u2019an 710054, China"},{"name":"Institute of Space Science, Shandong University, Weihai 264209, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,11,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Chen, Q., Jiang, W., Meng, X., Jiang, P., Wang, K., Xie, Y., and Ye, J. (2018). Vertical deformation monitoring of the suspension bridge tower using GNSS: A case study of the forth road bridge in the UK. Remote Sens., 10.","DOI":"10.3390\/rs10030364"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Li, W., and Shen, Y. (2018). The consideration of formal errors in spatiotemporal filtering using principal component analysis for regional GNSS position time series. Remote Sens., 10.","DOI":"10.3390\/rs10040534"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1179\/1752270614Y.0000000092","article-title":"Mine surface deformation monitoring using modified GPS RTK with surveying rod: Initial results","volume":"47","author":"Liu","year":"2015","journal-title":"Surv. Rev."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Cai, M., Chen, W., Dong, D., Song, L., Wang, M., Wang, Z., Zhou, F., Zheng, Z., and Yu, C. (2016). Reduction of kinematic short baseline multipath effects based on multipath hemispherical map. Sensors, 16.","DOI":"10.3390\/s16101677"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"5100","DOI":"10.1109\/TGRS.2015.2417214","article-title":"Snow depth estimation based on multipath phase combination of GPS triple-frequency signals","volume":"53","author":"Yu","year":"2015","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Wang, N., Xu, T., Gao, F., and Xu, G. (2018). Sea level estimation based on GNSS dual-frequency carrier phase linear combinations and SNR. Remote Sens., 10.","DOI":"10.3390\/rs10030470"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Jin, S., Cardellach, E., and Xie, F. (2014). GNSS Remote Sensing: Theory, Methods and Applications, Springer.","DOI":"10.1007\/978-94-007-7482-7"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1007\/s00190-006-0113-1","article-title":"GPS sidereal filtering: Coordinate- and carrier-phase-level strategies","volume":"81","author":"Ragheb","year":"2007","journal-title":"J. Geod."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1007\/s10291-015-0473-1","article-title":"Effectiveness of observation-domain sidereal filtering for GPS precise point positioning","volume":"20","author":"Atkins","year":"2016","journal-title":"GPS Solut."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"40","DOI":"10.1007\/s10291-018-0708-z","article-title":"A new multipath mitigation method based on adaptive thresholding wavelet denoising and double reference shift strategy","volume":"22","author":"Su","year":"2018","journal-title":"GPS Solut."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1007\/s10291-016-0533-1","article-title":"Wavelet packets based denoising method for measurement domain repeat-time multipath filtering in GPS static high-precision positioning","volume":"21","author":"Lau","year":"2017","journal-title":"GPS Solut."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2882","DOI":"10.1016\/j.asr.2017.01.007","article-title":"Multipath extraction and mitigation for bridge deformation monitoring using a single-difference model","volume":"60","author":"Wang","year":"2017","journal-title":"Adv. Space Res."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"499","DOI":"10.1007\/s10291-015-0458-0","article-title":"A new method for specular and diffuse pseudorange multipath error extraction using wavelet analysis","volume":"20","author":"Pugliano","year":"2016","journal-title":"GPS Solut."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1007\/s00190-005-0474-x","article-title":"Filtering GPS time-series using a Vondrak filter and cross-validation","volume":"79","author":"Zheng","year":"2005","journal-title":"J. Geod."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1007\/s10291-013-0341-9","article-title":"Multipath mitigation via component analysis methods for GPS dynamic deformation monitoring","volume":"18","author":"Dai","year":"2014","journal-title":"GPS Solut."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"152","DOI":"10.1179\/003962609X390067","article-title":"Reducing GPS carrier phase errors with EMD-wavelet for precise static positioning","volume":"41","author":"Wang","year":"2009","journal-title":"Surv. Rev."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Zhang, Q., Yang, W., Zhang, S., and Liu, X. (2018). Characteristics of beidou navigation satellite system multipath and its mitigation method based on kalman filter and rauch-tung-striebel smoother. Sensors, 18.","DOI":"10.3390\/s18010198"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1007\/s00190-015-0870-9","article-title":"Mitigation of multipath effect in GNSS short baseline positioning by the multipath hemispherical map","volume":"90","author":"Dong","year":"2016","journal-title":"J. Geod."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1007\/s10291-014-0367-7","article-title":"Generating statistically robust multipath stacking maps using congruent cells","volume":"19","author":"Fuhrmann","year":"2015","journal-title":"GPS Solut."},{"key":"ref_20","first-page":"76","article-title":"Carrier-phase multipath corrections for GPS-based satellite attitude determination","volume":"48","author":"Reichert","year":"2001","journal-title":"Annu. Navig."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1874","DOI":"10.1016\/S1003-6326(14)63266-0","article-title":"Modeling GPS multipath effect based on spherical cap harmonic analysis","volume":"24","author":"Guo","year":"2014","journal-title":"Trans. Nonferr. Met. Soc. China"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1421","DOI":"10.1126\/science.1084531","article-title":"Using 1-Hz GPS data to measure deformations caused by the Denali fault earthquake","volume":"300","author":"Larson","year":"2003","journal-title":"Science"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1007\/s10291-006-0038-4","article-title":"Finding the repeat times of the GPS constellation","volume":"11","author":"Agnew","year":"2007","journal-title":"GPS Solut."},{"key":"ref_24","first-page":"B05442","article-title":"Improving the precision of high-rate GPS","volume":"112","author":"Larson","year":"2007","journal-title":"J. Geophys. Res."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"L22608","DOI":"10.1029\/2004GL021621","article-title":"Modified sidereal filtering: Implications for high-rate GPS positioning","volume":"31","author":"Choi","year":"2004","journal-title":"Geophys. Res. Lett."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Wang, M., Wang, J., Dong, D., Li, H., Han, L., and Chen, W. (2018). Comparison of three methods for estimating GPS multipath repeat time. Remote Sens., 10.","DOI":"10.3390\/rs10020006"},{"key":"ref_27","unstructured":"Tikhonov, A.N., and Arsenin, V.Y. (1977). Solutions of Ill-Posed Problems, John Wiley & Sons."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Datta, B.N. (2010). Numerical Linear Algebra and Applications, SIAM.","DOI":"10.1137\/1.9780898717655"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Efron, B., and Tibshirani, R.J. (1993). An Introduction to the Bootstrap, CRC Press.","DOI":"10.1007\/978-1-4899-4541-9"},{"key":"ref_30","unstructured":"Takasu, T., and Yasuda, A. (2009). Development of the low-cost RTK-GPS receiver with an open source program package RTKLIB. International Symposium on GPS\/GNSS, Available online: https:\/\/bit.ly\/2QLs5dO."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1007\/s00190-009-0352-z","article-title":"Sidereal filtering based on single differences for mitigating GPS multipath effects on short baselines","volume":"84","author":"Zhong","year":"2009","journal-title":"J. Geod."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"2661","DOI":"10.1029\/2000GL011525","article-title":"Obtaining single path phase delays from GPS double differences","volume":"27","author":"Alber","year":"2000","journal-title":"Geophys. Res. Lett."}],"container-title":["Remote Sensing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2072-4292\/10\/11\/1801\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:29:36Z","timestamp":1760196576000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2072-4292\/10\/11\/1801"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,11,14]]},"references-count":32,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2018,11]]}},"alternative-id":["rs10111801"],"URL":"https:\/\/doi.org\/10.3390\/rs10111801","relation":{},"ISSN":["2072-4292"],"issn-type":[{"value":"2072-4292","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,11,14]]}}}