{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T06:22:14Z","timestamp":1771482134091,"version":"3.50.1"},"reference-count":47,"publisher":"MDPI AG","issue":"20","license":[{"start":{"date-parts":[[2021,10,11]],"date-time":"2021-10-11T00:00:00Z","timestamp":1633910400000},"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":["61773132"],"award-info":[{"award-number":["61773132"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61803115"],"award-info":[{"award-number":["61803115"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100018546","name":"Shandong Province Postdoctoral Innovation Project","doi-asserted-by":"publisher","award":["202003050"],"award-info":[{"award-number":["202003050"]}],"id":[{"id":"10.13039\/501100018546","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"publisher","award":["3072021CFJ0404"],"award-info":[{"award-number":["3072021CFJ0404"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002858","name":"China Postdoctoral Science Foundation","doi-asserted-by":"publisher","award":["2020M681078"],"award-info":[{"award-number":["2020M681078"]}],"id":[{"id":"10.13039\/501100002858","id-type":"DOI","asserted-by":"publisher"}]},{"name":"the Qingdao Postdoctoral Foundation","award":["QDBSHYYYJXM20200101"],"award-info":[{"award-number":["QDBSHYYYJXM20200101"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Remote Sensing"],"abstract":"<jats:p>The periodic noise exists in BeiDou navigation satellite system (BDS) clock offsets. As a commonly used satellite clock prediction model, the spectral analysis model (SAM) typically detects and identifies the periodic terms by the Fast Fourier transform (FFT) according to long-term clock offset series. The FFT makes an aggregate assessment in frequency domain but cannot characterize the periodic noise in a time domain. Due to space environment changes, temperature variations, and various disturbances, the periodic noise is time-varying, and the spectral peaks vary over time, which will affect the prediction accuracy of the SAM. In this paper, we investigate the periodic noise and its variations present in BDS clock offsets, and improve the clock prediction model by considering the periodic variations. The periodic noise and its variations over time are analyzed and quantified by short time Fourier transform (STFT). The results show that both the amplitude and frequency of the main periodic term in BDS clock offsets vary with time. To minimize the impact of periodic variations on clock prediction, a time frequency analysis model (TFAM) based on STFT is constructed, in which the periodic term can be quantified and compensated accurately. The experiment results show that both the fitting and prediction accuracy of TFAM are better than SAM. Compared with SAM, the average improvement of the prediction accuracy using TFAM of the 6 h, 12 h, 18 h and 24 h is in the range of 6.4% to 10% for the GNSS Research Center of Wuhan University (WHU) clock offsets, and 11.1% to 14.4% for the Geo Forschungs Zentrum (GFZ) clock offsets. For the satellites C06, C14, and C32 with marked periodic variations, the prediction accuracy is improved by 26.7%, 16.2%, and 16.3% for WHU clock offsets, and 29.8%, 16.0%, 21.0%, and 9.0% of C06, C14, C28, and C32 for GFZ clock offsets.<\/jats:p>","DOI":"10.3390\/rs13204058","type":"journal-article","created":{"date-parts":[[2021,10,11]],"date-time":"2021-10-11T21:45:32Z","timestamp":1633988732000},"page":"4058","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":18,"title":["BDS Satellite Clock Prediction Considering Periodic Variations"],"prefix":"10.3390","volume":"13","author":[{"given":"Lin","family":"Zhao","sequence":"first","affiliation":[{"name":"College of Intelligent Systems Science and Engineering, Harbin Engineering University, Harbin 150001, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7378-0130","authenticated-orcid":false,"given":"Nan","family":"Li","sequence":"additional","affiliation":[{"name":"College of Intelligent Systems Science and Engineering, Harbin Engineering University, Harbin 150001, China"}]},{"given":"Hui","family":"Li","sequence":"additional","affiliation":[{"name":"College of Intelligent Systems Science and Engineering, Harbin Engineering University, Harbin 150001, China"},{"name":"HEU Qingdao Ship Science and Technology Co., Ltd., Qingdao 266400, China"}]},{"given":"Renlong","family":"Wang","sequence":"additional","affiliation":[{"name":"College of Intelligent Systems Science and Engineering, Harbin Engineering University, Harbin 150001, China"}]},{"given":"Menghao","family":"Li","sequence":"additional","affiliation":[{"name":"College of Intelligent Systems Science and Engineering, Harbin Engineering University, Harbin 150001, China"}]}],"member":"1968","published-online":{"date-parts":[[2021,10,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1007\/s10291-018-0820-0","article-title":"Estimation of GPS LNAV based on IGS products for real-time PPP","volume":"23","author":"Zhou","year":"2019","journal-title":"GPS Solut."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1007\/s10291-014-0369-5","article-title":"IGS RTS precise orbits and clocks verification and quality degradation over time","volume":"19","author":"Hadas","year":"2015","journal-title":"GPS Solut."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1186\/s43020-019-0007-z","article-title":"BDS satellite clock prediction based on a semiparametric adjustment model considering model errors","volume":"1","author":"Yan","year":"2020","journal-title":"Satell. Navig."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1007\/s10291-019-0950-z","article-title":"Fractal behavior of BDS-2 satellite clock offsets and its application to real-time clock offsets prediction","volume":"24","author":"Han","year":"2020","journal-title":"GPS Solut."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"937","DOI":"10.1007\/s10291-016-0583-4","article-title":"Maintaining real-time precise point positioning during outages of orbit and clock corrections","volume":"21","author":"Deo","year":"2017","journal-title":"GPS Solut."},{"key":"ref_6","unstructured":"Sun, J., Yang, C., and Guo, S. (2018). The Long-Term Performance Analysis and Comparison of BDS on-Board Atomic Clock Based on Precise Clock Products of IGS Analysis Centers. China Satellite Navigation Conference (CSNC) 2018 Proceedings, Springer. CSNC 2018, Harbin, China, May 2018."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"109075","DOI":"10.1016\/j.measurement.2021.109075","article-title":"Spaceborne Atomic Clock Performance Review of BDS-3 MEO Satellites","volume":"175","author":"Wang","year":"2021","journal-title":"Measurement"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Lv, Y., Dai, Z., Zhao, Q., Yang, S., Zhou, J., and Liu, J. (2017). Improved Short-Term Clock Prediction Method for Real-Time Positioning. Sensors, 17.","DOI":"10.3390\/s17061308"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"513","DOI":"10.1007\/s40328-019-00270-8","article-title":"An improved model for BDS satellite ultra-rapid clock offset prediction based on BDS-2 and BDS-3 combined estimation","volume":"54","author":"Hu","year":"2019","journal-title":"Acta. Geod. Geophys."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Janicka, J., Tomaszewski, D., Rapinski, J., Jagoda, M., and Rutkowska, M. (2020). The Prediction of Geocentric Corrections during Communication Link Outages in PPP. Sensors, 20.","DOI":"10.3390\/s20030602"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1007\/s10291-017-0681-y","article-title":"An approach to GPS clock prediction for real-time PPP during outages of RTS stream","volume":"22","author":"Nie","year":"2018","journal-title":"GPS Solut."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1007\/s10291-013-0313-0","article-title":"Real-time clock offset prediction with an improved model","volume":"18","author":"Huang","year":"2014","journal-title":"GPS Solut."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1016\/j.chinastron.2009.01.001","article-title":"An Improved Grey Model and Its Application Research on the Prediction of Real-time GPS Satellite Clock Errors","volume":"33","author":"Zheng","year":"2009","journal-title":"Chin. Astron. Astrophys."},{"key":"ref_14","first-page":"116","article-title":"Application of ARIMA (0, 2, q) model to prediction of satellite clock error","volume":"29","author":"Xu","year":"2009","journal-title":"J. Geod. Geodyn."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"105002","DOI":"10.1088\/1361-6501\/abfcec","article-title":"An enhanced prediction model for BDS ultra-rapid clock offset that combines singular spectrum analysis, robust estimation and gray model","volume":"32","author":"Xu","year":"2021","journal-title":"Meas. Sci. Technol."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1681","DOI":"10.1016\/j.asr.2018.10.047","article-title":"Switching and performance variations of on-orbit BDS satellite clocks","volume":"63","author":"Huang","year":"2018","journal-title":"Adv. Space Res."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Lv, Y., Geng, T., Zhao, Q., and Liu, J. (2018). Characteristics of BeiDou-3 Experimental Satellite Clocks. Remote. Sens., 10.","DOI":"10.3390\/rs10111847"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1080\/00396265.2017.1359473","article-title":"Analysis of GPS satellite clock prediction performance with different update intervals and application to real-time PPP","volume":"51","author":"Yang","year":"2017","journal-title":"Surv. Rev."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1007\/s10291-012-0278-4","article-title":"Short-term analysis of GNSS clocks","volume":"17","author":"Hauschild","year":"2013","journal-title":"GPS Solut."},{"key":"ref_20","first-page":"1299","article-title":"Periodic variations of BeiDou Satellite Clock Offsets Derived from Multi-satellite Orbit Determination","volume":"44","author":"Zhou","year":"2015","journal-title":"J. Geod. Geoinf. Sci."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1007\/s00190-015-0862-9","article-title":"Precise orbit determination for quad-constellation satellites at Wuhan University: Strategy, result validation, and comparison","volume":"90","author":"Guo","year":"2016","journal-title":"J. Geod."},{"key":"ref_22","unstructured":"Bhattarai, S. (2015). Satellite Clock Time Offset Prediction in Global Navigation Satellite Systems. [Ph.D. Thesis, University College London]."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"073001","DOI":"10.1088\/0957-0233\/21\/7\/073001","article-title":"Improving prediction accuracy of GPS satellite clocks with periodic variation behavior","volume":"21","author":"Youn","year":"2010","journal-title":"Meas. Sci. Technol."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1007\/s10291-011-0232-x","article-title":"Apparent clock variations of the Block IIF-1 (SVN62) GPS satellite","volume":"16","author":"Montenbruck","year":"2012","journal-title":"GPS Solut."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Xu, B., Wang, L., Fu, W., Chen, R., Li, T., and Zhang, X. (2019). A Practical Adaptive Clock Offset Prediction Model for the Beidou-2 System. Remote. Sens., 11.","DOI":"10.3390\/rs11161850"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Huang, G., Cui, B., Zhang, Q., Fu, W., and Li, P. (2018). An Improved Predicted Model for BDS Ultra-Rapid Satellite Clock Offsets. Remote Sens., 10.","DOI":"10.3390\/rs10010060"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"He, L., Zho, H., Liu, Z., and Wen, Y. (2019). Improving Clock Prediction Algorithm for BDS-2\/3 Satellites Based on LS-SVM Method. Remote Sens., 11.","DOI":"10.3390\/rs11212554"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1016\/j.jog.2013.05.005","article-title":"Singular spectrum analysis for modeling seasonal signals from GPS time series","volume":"72","author":"Chen","year":"2013","journal-title":"J. Geodyn."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1007\/s10291-008-0089-9","article-title":"Characterization of periodic variations in the GPS satellite clocks","volume":"12","author":"Senior","year":"2008","journal-title":"GPS Solut."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"56","DOI":"10.1007\/s10291-021-01094-2","article-title":"Time\u2013frequency analysis of the Galileo satellite clocks: Looking for the J2 relativistic effect and other periodic variations","volume":"25","author":"Formichella","year":"2021","journal-title":"GPS Solut."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"783","DOI":"10.1007\/s10291-015-0488-7","article-title":"Analysis of BDS satellite clocks in orbit","volume":"20","author":"Wang","year":"2016","journal-title":"GPS Solut."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1007\/s10291-018-0757-3","article-title":"On the capabilities of the inaction method for extracting the periodic components from GPS clock data","volume":"22","author":"Wang","year":"2018","journal-title":"GPS Solut."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"2889","DOI":"10.1016\/j.asr.2018.01.020","article-title":"Atomic clock performance and combined clock error prediction for the new generation of BeiDou navigation satellites","volume":"63","author":"Wang","year":"2018","journal-title":"Adv. Space Res."},{"key":"ref_34","first-page":"215","article-title":"A Data Quality Control Method for the Beidou-2 Satellite Clock bias","volume":"37","author":"Tian","year":"2017","journal-title":"J. Geod. Geodyn."},{"key":"ref_35","unstructured":"Wang, C. (2019). Solar radiation pressure modelling for BeiDou navigation satellites. [Ph.D. Thesis, Wuhan University]."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"32967","DOI":"10.1038\/srep32967","article-title":"A new analytical solar radiation pressure model for current BeiDou satellites: IGGBSPM","volume":"6","author":"Tan","year":"2016","journal-title":"Sci. Rep."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"791","DOI":"10.1007\/s00190-018-1199-y","article-title":"Empirically derived model of solar radiation pressure for BeiDou GEO satellites","volume":"93","author":"Wang","year":"2018","journal-title":"J. Geod."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"2633","DOI":"10.1002\/jgrb.50167","article-title":"The development and evaluation of the earth gravitational model 2008 (EGM2008)","volume":"118","author":"Pavlis","year":"2013","journal-title":"J. Geophys. Res. Solid Earth"},{"key":"ref_39","first-page":"394","article-title":"Modelling the global ocean tides: Modern insights from ES2004","volume":"56","author":"Lyard","year":"2004","journal-title":"J. Geophys. Res. Atmos."},{"key":"ref_40","unstructured":"Petit, G., and Luzum, B. (2010). IERS Conventions 2010, IERS Convention Center. Technical Report."},{"key":"ref_41","first-page":"367","article-title":"Extended orbit modelling techniques at the CODE processing center of the international GPS service for geodynamics (IGS): Theory and initial results","volume":"19","author":"Beutler","year":"1994","journal-title":"Eur. Respir. J."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"609","DOI":"10.1007\/s00190-017-1082-2","article-title":"GNSS satellite transmit power and its impact on orbit determination","volume":"92","author":"Steigenberger","year":"2018","journal-title":"J. Geod."},{"key":"ref_43","unstructured":"Rodriguez-Solano, C.J. (2009). Impact of the albedo modeling on GPS orbits. [Ph.D. Thesis, Technische Universit\u00e4t M\u00fcnchen (TUM)]."},{"key":"ref_44","first-page":"639","article-title":"Performance Analysis of BDS Satellite Orbits during Eclipse Periods: Results of Satellite Laser Ranging Validation","volume":"45","author":"Peng","year":"2016","journal-title":"Acta Geod. Cartogr. Sin."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1007\/s10291-019-0857-8","article-title":"Observation of BDS-2 IGSO\/MEOs yaw-attitude behavior during eclipse seasons","volume":"23","author":"Xia","year":"2019","journal-title":"GPS Solut."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1007\/s10291-020-01073-z","article-title":"Enhanced solar radiation pressure model for GPS satellites considering various physical effects","volume":"25","author":"Duan","year":"2021","journal-title":"GPS Solut."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"1571","DOI":"10.1016\/j.asr.2014.06.012","article-title":"The impact of orbital errors on the estimation of satellite clock errors and PPP","volume":"54","author":"Lou","year":"2014","journal-title":"Adv. Space Res."}],"container-title":["Remote Sensing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2072-4292\/13\/20\/4058\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:11:43Z","timestamp":1760166703000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2072-4292\/13\/20\/4058"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,10,11]]},"references-count":47,"journal-issue":{"issue":"20","published-online":{"date-parts":[[2021,10]]}},"alternative-id":["rs13204058"],"URL":"https:\/\/doi.org\/10.3390\/rs13204058","relation":{},"ISSN":["2072-4292"],"issn-type":[{"value":"2072-4292","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,10,11]]}}}