{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T11:55:52Z","timestamp":1778673352154,"version":"3.51.4"},"reference-count":32,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2023,6,5]],"date-time":"2023-06-05T00:00:00Z","timestamp":1685923200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Department of Science and Technology of Guangdong Province","award":["2019A1515110418"],"award-info":[{"award-number":["2019A1515110418"]}]},{"name":"Department of Science and Technology of Guangdong Province","award":["2021A1515110280"],"award-info":[{"award-number":["2021A1515110280"]}]},{"name":"Department of Science and Technology of Guangdong Province","award":["2020GXJK399"],"award-info":[{"award-number":["2020GXJK399"]}]},{"name":"Department of Science and Technology of Guangdong Province","award":["2019A1515110418"],"award-info":[{"award-number":["2019A1515110418"]}]},{"name":"Department of Science and Technology of Guangdong Province","award":["2021A1515110280"],"award-info":[{"award-number":["2021A1515110280"]}]},{"name":"Department of Science and Technology of Guangdong Province","award":["2020GXJK399"],"award-info":[{"award-number":["2020GXJK399"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>The onboard atomic frequency standard (AFS) is a crucial element of Global Navigation Satellite System (GNSS) satellites. However, it is widely accepted that periodic variations can influence the onboard AFS. The presence of non-stationary random processes in AFS signals can lead to inaccurate separation of the periodic and stochastic components of satellite AFS clock data when using least squares and Fourier transform methods. In this paper, we characterize the periodic variations of AFS using Allan and Hadamard variances and demonstrate that the Allan and Hadamard variances of the periodics are independent of the variances of the stochastic component. The proposed model is tested against simulated and real clock data, revealing that our approach provides more precise characterization of periodic variations compared to the least squares method. Additionally, we observe that overfitting periodic variations can improve the precision of GPS clock bias prediction, as indicated by a comparison of fitting and prediction errors of satellite clock bias.<\/jats:p>","DOI":"10.3390\/s23115356","type":"journal-article","created":{"date-parts":[[2023,6,6]],"date-time":"2023-06-06T02:08:15Z","timestamp":1686017295000},"page":"5356","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Characterizing Periodic Variations of Atomic Frequency Standards via Their Frequency Stability Estimates"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6139-0298","authenticated-orcid":false,"given":"Weiwei","family":"Cheng","sequence":"first","affiliation":[{"name":"School of Transportation, Civil Engineering and Architecture, Foshan University, Foshan 528000, China"},{"name":"GNSS Center, Wuhan University, Wuhan 430079, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1271-7968","authenticated-orcid":false,"given":"Guigen","family":"Nie","sequence":"additional","affiliation":[{"name":"GNSS Center, Wuhan University, Wuhan 430079, China"},{"name":"Collaborative Innovation Center for Geospatial Information Technology, Wuhan University, Wuhan 430079, China"},{"name":"Hubei Luojia Laboratory, Wuhan 430079, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jian","family":"Zhu","sequence":"additional","affiliation":[{"name":"School of Transportation, Civil Engineering and Architecture, Foshan University, Foshan 528000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,6,5]]},"reference":[{"key":"ref_1","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_2","doi-asserted-by":"crossref","first-page":"2465","DOI":"10.1007\/s00190-019-01310-5","article-title":"GPS satellite inter-frequency clock bias estimation using triple-frequency raw observations","volume":"93","author":"Fan","year":"2019","journal-title":"J. Geod."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Cernigliaro, A., Valloreia, S., Galleani, L., and Tavella, P. (2013, January 25\u201327). GNSS space clocks: Performance analysis. Proceedings of the International Conference on Localization and GNSS (ICL-GNSS), IEEE, Torino, Italy.","DOI":"10.1109\/ICL-GNSS.2013.6577251"},{"key":"ref_4","unstructured":"Brown, K.R. (1991, January 11\u201313). The Theory of the GPS Composite Clock. Proceedings of the 4th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GPS 1991), Albuquerque, NM, USA."},{"key":"ref_5","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_6","doi-asserted-by":"crossref","first-page":"1253","DOI":"10.1109\/TUFFC.2005.1509784","article-title":"Enhancements to GPS Operations and Clock Evaluations Using a \u201cTotal\u201d Hadamard Deviation","volume":"52","author":"Howe","year":"2005","journal-title":"IEEE Trans. Ultrason. Ferroelectr. Freq. Control"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"170","DOI":"10.1007\/s10291-004-0102-x","article-title":"Improved relativistic transformations in GPS","volume":"8","author":"Kouba","year":"2004","journal-title":"GPS Solut."},{"key":"ref_8","first-page":"367","article-title":"Extended orbit modeling 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":"Manuscripta Geod."},{"key":"ref_9","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":"2011","journal-title":"GPS Solut."},{"key":"ref_10","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_11","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1088\/0026-1394\/29\/6\/004","article-title":"Real-time Restitution of GPS Time Through a Kalman Estimation","volume":"29","author":"Thomas","year":"1993","journal-title":"Metrologia"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Bogdanov, P.P., Druzhin, A.V., Tiuliakov, A.E., and Feoktistov, A.Y. (2014, January 16\u201323). GLONASS time and UTC(SU). Proceedings of the 2014 XXXIth URSI General Assembly and Scientific Symposium (URSI GASS), Beijing, China.","DOI":"10.1109\/URSIGASS.2014.6928996"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"S47","DOI":"10.1088\/0026-1394\/45\/6\/S08","article-title":"The Compass and its time reference system","volume":"45","author":"Dong","year":"2008","journal-title":"Metrologia"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Bedrick, S., Bauch, A., Moudrak, A., and Sch\u00e4fe, W. (2004, January 7\u20139). Design of the precise time facility for GALILEO. Proceedins of the 36th Annual Precise Time and Time Interval (PTTI) Meeting, Washington, DC, USA.","DOI":"10.1049\/cp:20040913"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"042001","DOI":"10.1088\/1681-7575\/ab1e68","article-title":"The Coordinated Universal Time (UTC)","volume":"56","author":"Panfilo","year":"2019","journal-title":"Metrologia"},{"key":"ref_16","unstructured":"Senior, K., Koppang, P., and Matsakis, D. (2001, January 8). Developing an IGS time scale. Proceedings of the 2001 IEEE International Frequncy Control Symposium and PDA Exhibition, Seattle, WA, USA."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"115004","DOI":"10.1088\/1361-6501\/aa8373","article-title":"JY1 time scale: A new Kalman-filter time scale designed at NIST","volume":"28","author":"Yao","year":"2017","journal-title":"Meas. Sci. Technol."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1088\/0026-1394\/40\/4\/501","article-title":"IGS\/BIPM pilot project: GPS carrier phase for time\/frequency transfer and timescale formation","volume":"40","author":"Ray","year":"2003","journal-title":"Metrologia"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"019514","DOI":"10.1360\/SSPMA-2020-0225","article-title":"On-orbit evaluation of influence of magnetic-induced frequency shift on space borne atomic clock","volume":"51","author":"Lin","year":"2021","journal-title":"Sci. Sin.-Phys. Mech. Astron."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"013409","DOI":"10.1103\/PhysRevA.100.013409","article-title":"Measurements of 25Mg+ and 27Al+ magnetic constants for improved ion clock accuracy","volume":"100","author":"Brewer","year":"2019","journal-title":"Phys. Rev. A"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"019512","DOI":"10.1360\/SSPMA-2020-0245","article-title":"Characteristics of the space-borne rubidium atomic clocks for the BeiDou III navigation satellite system","volume":"51","author":"Shidong","year":"2021","journal-title":"Sci. Sin. Phys. Mech. Astron."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Baranov, A., Ermak, S., Lozov, R., and Semenov, V. (2020). Comparison of Gravitational and Light Frequency Shifts in Rubidium Atomic Clock. Universe, 7.","DOI":"10.3390\/universe7010003"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"381","DOI":"10.11728\/cjss2019.03.381","article-title":"Method of temperature control and its validation for atomic clock cabin on navigation satellite","volume":"39","author":"Lin","year":"2019","journal-title":"Chin. J. Space Sci."},{"key":"ref_24","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_25","doi-asserted-by":"crossref","unstructured":"Kobayashi, H., Mark, B.L., and Turin, W. (2011). Probability, Random Processes, and Statistical Analysis, Cambridge University Press.","DOI":"10.1017\/CBO9780511977770"},{"key":"ref_26","first-page":"B03405","article-title":"Spatiotemporal filtering using principal component analysis and Karhunen-Loeve expansion approaches for regional GPS network analysis","volume":"111","author":"Dong","year":"2021","journal-title":"J. Geophys. Res."},{"key":"ref_27","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_28","doi-asserted-by":"crossref","first-page":"804","DOI":"10.1109\/TUFFC.2019.2898109","article-title":"Predicting Long-Term Frequency Stability: Stochastic Oscillator Noise Analysis","volume":"66","author":"Cheng","year":"2019","journal-title":"IEEE Trans. Ultrason. Ferroelect. Freq. Control"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Cheng, W., and Nie, G. (2018). Predicting Long-Term Stability of Precise Oscillators under Influence of Frequency Drift. Sensors, 18.","DOI":"10.3390\/s18020502"},{"key":"ref_30","unstructured":"Montenbruck, O., Steigenberger, P., Khachikyan, R., Weber, G., Langley, R.B., Mervart, L., and Hugentobler, U. (2013, January 4\u20136). IGS-MGEX: Preparing the Ground for Multi-Constellation GNSS Science. Proceedings of the 4th International Colloquium on Scientific and Fundamental Aspects of the Galileo System, Prague, Czech Republic."},{"key":"ref_31","unstructured":"Villiger, A., and Dach, R. (2021, December 27). International GNSS Service Technical Report 2020 (IGS Annual Report), IGS Central Bureau and University of Bern. Available online: https:\/\/files.igs.org\/pub\/resource\/pubs\/2020_techreport.pdf."},{"key":"ref_32","unstructured":"Yao, J., and Levine, J. (2012, January 26\u201329). GPS carrier-phase time transfer boundary discontinuity investigation. Proceedings of the 44th Annual Precise Time and Time Interval Systems and Applications (PTTI) Meeting, Reston, VA, USA."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/23\/11\/5356\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:48:46Z","timestamp":1760125726000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/23\/11\/5356"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,6,5]]},"references-count":32,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2023,6]]}},"alternative-id":["s23115356"],"URL":"https:\/\/doi.org\/10.3390\/s23115356","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,6,5]]}}}