{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,17]],"date-time":"2026-04-17T15:52:08Z","timestamp":1776441128693,"version":"3.51.2"},"reference-count":44,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2023,6,16]],"date-time":"2023-06-16T00:00:00Z","timestamp":1686873600000},"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":["12073034"],"award-info":[{"award-number":["12073034"]}],"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":["XAB2021YN25"],"award-info":[{"award-number":["XAB2021YN25"]}],"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":["E167SC14"],"award-info":[{"award-number":["E167SC14"]}],"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":["2022KW-29"],"award-info":[{"award-number":["2022KW-29"]}],"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":["DP 190102444"],"award-info":[{"award-number":["DP 190102444"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"CAS \u201cLight of West China\u201d Program","award":["12073034"],"award-info":[{"award-number":["12073034"]}]},{"name":"CAS \u201cLight of West China\u201d Program","award":["XAB2021YN25"],"award-info":[{"award-number":["XAB2021YN25"]}]},{"name":"CAS \u201cLight of West China\u201d Program","award":["E167SC14"],"award-info":[{"award-number":["E167SC14"]}]},{"name":"CAS \u201cLight of West China\u201d Program","award":["2022KW-29"],"award-info":[{"award-number":["2022KW-29"]}]},{"name":"CAS \u201cLight of West China\u201d Program","award":["DP 190102444"],"award-info":[{"award-number":["DP 190102444"]}]},{"name":"National Time Service Center, Chinese Academy of Sciences (CAS)","award":["12073034"],"award-info":[{"award-number":["12073034"]}]},{"name":"National Time Service Center, Chinese Academy of Sciences (CAS)","award":["XAB2021YN25"],"award-info":[{"award-number":["XAB2021YN25"]}]},{"name":"National Time Service Center, Chinese Academy of Sciences (CAS)","award":["E167SC14"],"award-info":[{"award-number":["E167SC14"]}]},{"name":"National Time Service Center, Chinese Academy of Sciences (CAS)","award":["2022KW-29"],"award-info":[{"award-number":["2022KW-29"]}]},{"name":"National Time Service Center, Chinese Academy of Sciences (CAS)","award":["DP 190102444"],"award-info":[{"award-number":["DP 190102444"]}]},{"name":"Shaanxi Province Key R&amp;D Program Project","award":["12073034"],"award-info":[{"award-number":["12073034"]}]},{"name":"Shaanxi Province Key R&amp;D Program Project","award":["XAB2021YN25"],"award-info":[{"award-number":["XAB2021YN25"]}]},{"name":"Shaanxi Province Key R&amp;D Program Project","award":["E167SC14"],"award-info":[{"award-number":["E167SC14"]}]},{"name":"Shaanxi Province Key R&amp;D Program Project","award":["2022KW-29"],"award-info":[{"award-number":["2022KW-29"]}]},{"name":"Shaanxi Province Key R&amp;D Program Project","award":["DP 190102444"],"award-info":[{"award-number":["DP 190102444"]}]},{"DOI":"10.13039\/501100000923","name":"Australian Research Council","doi-asserted-by":"publisher","award":["12073034"],"award-info":[{"award-number":["12073034"]}],"id":[{"id":"10.13039\/501100000923","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000923","name":"Australian Research Council","doi-asserted-by":"publisher","award":["XAB2021YN25"],"award-info":[{"award-number":["XAB2021YN25"]}],"id":[{"id":"10.13039\/501100000923","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000923","name":"Australian Research Council","doi-asserted-by":"publisher","award":["E167SC14"],"award-info":[{"award-number":["E167SC14"]}],"id":[{"id":"10.13039\/501100000923","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000923","name":"Australian Research Council","doi-asserted-by":"publisher","award":["2022KW-29"],"award-info":[{"award-number":["2022KW-29"]}],"id":[{"id":"10.13039\/501100000923","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000923","name":"Australian Research Council","doi-asserted-by":"publisher","award":["DP 190102444"],"award-info":[{"award-number":["DP 190102444"]}],"id":[{"id":"10.13039\/501100000923","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Remote Sensing"],"abstract":"<jats:p>High-accuracy Low Earth Orbit (LEO) satellite clock and orbital products are preconditions to realize LEO augmentation for high-accuracy GNSS-based positioning on the ground. There is a high correlation between the orbit and clock parameters in the kinematic Precise Orbit Determination (POD) process. While future LEO satellites are planned to be equipped with better clocks, the benefits of modeling high-stability LEO satellite clocks are not yet thoroughly investigated, particularly when mid- to long-term systematic effects induced by the complex LEO relativistic effects and the external environment remain in the clocks. Through clock modeling, this study attempts to reduce not only the short-term noise of radial kinematic orbits, but also mis-modeled effects caused by, e.g., real-time GNSS orbital and clock errors. To explore the benefits of clock modeling, the clocks need to be first detrended by the mid- to long-term systematic effects. While over-detrending limits the orbital improvements, weak detrending would also hamper strong clock modeling and easily lead to performance degradations. A balance between the strengths of the detrending and the model thus needs to be investigated for different clock types. In this study, the Piece-Wise Linear (PWL) model of different time lengths and a 2.5-state filter with different strengths (h values) are tested using real data from GRACE FO-1 with an Ultra-Stable Oscillator (USO) on board. Using the CNES real-time GPS products, it was found that when detrending the clocks with a smoothing window of 300 to 500 s, one could generally expect an improvement larger than 10% in the estimation of radial orbits when applying a PWL model with a length from 300 to 1200 s. Improvements of this size can also be expected when using the 2.5-state model with h\u22121 (for Flicker Frequency Noise) from 10\u221228 to 10\u221230.<\/jats:p>","DOI":"10.3390\/rs15123149","type":"journal-article","created":{"date-parts":[[2023,6,16]],"date-time":"2023-06-16T08:56:01Z","timestamp":1686905761000},"page":"3149","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["LEO Satellite Clock Modeling and Its Benefits for LEO Kinematic POD"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5688-6937","authenticated-orcid":false,"given":"Kan","family":"Wang","sequence":"first","affiliation":[{"name":"National Time Service Center, Chinese Academy of Sciences, Xi\u2019an 710600, China"},{"name":"University of Chinese Academy of Sciences, Beijing 100049, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7060-4123","authenticated-orcid":false,"given":"Ahmed","family":"El-Mowafy","sequence":"additional","affiliation":[{"name":"School of Earth and Planetary Sciences, Curtin University, Perth, WA 6845, Australia"}]},{"given":"Xuhai","family":"Yang","sequence":"additional","affiliation":[{"name":"National Time Service Center, Chinese Academy of Sciences, Xi\u2019an 710600, China"},{"name":"University of Chinese Academy of Sciences, Beijing 100049, China"}]}],"member":"1968","published-online":{"date-parts":[[2023,6,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1002\/navi.234","article-title":"Broadband LEO constellations for navigation","volume":"65","author":"Reid","year":"2018","journal-title":"Navig. J. Inst. Navig."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"4753","DOI":"10.1016\/j.asr.2021.03.008","article-title":"Precise orbit and Earth parameter determination supported by LEO satellites, inter-satellite links and synchronized clocks of a future GNSS","volume":"68","author":"Michalak","year":"2021","journal-title":"Adv. Space Res."},{"key":"ref_3","unstructured":"Lawrence, D., Cobb, H.S., Gutt, G., O\u2019Connor, M., Reid, T.G., and Walter, T. (2023, June 14). Navigation from LEO. Available online: https:\/\/www.gpsworld.com\/innovation-navigation-from-leo\/."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Li, K., Zhou, X., Wang, W., Gao, Y., Zhao, G., Tao, E., and Xu, K. (2018). Centimeter-level orbit determination for TG02 spacelab using onboard GNSS data. Sensors, 18.","DOI":"10.3390\/s18082671"},{"key":"ref_5","unstructured":"Faragher, R., and Ziebart, M. (2020). OneWeb LEO PNT: Progress or Risky Gamble. Inside GNSS, 28, Available online: https:\/\/insidegnss.com\/oneweb-leo-pnt-progress-or-risky-gamble\/."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"687","DOI":"10.1007\/s10291-015-0480-2","article-title":"Benefits of receiver clock modeling in code-based GNSS navigation","volume":"20","author":"Krawinkel","year":"2016","journal-title":"GPS Solut."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1901","DOI":"10.1016\/j.asr.2014.07.016","article-title":"Enhancing the kinematic precise orbit determination of low earth orbiters using GPS receiver clock modelling","volume":"54","author":"Yang","year":"2014","journal-title":"Adv. Space Res."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"B27","DOI":"10.1115\/1.1451162","article-title":"Satellite orbits: Models, methods, and applications","volume":"55","author":"Montenbruck","year":"2002","journal-title":"Appl. Mech. Rev."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1515\/jag-2018-0008","article-title":"Research on the impact factors of GRACE precise orbit determination by dynamic method","volume":"12","author":"Guo","year":"2018","journal-title":"J. Appl. Geod."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"788","DOI":"10.1016\/j.asr.2020.10.012","article-title":"Dynamic GPS-based LEO orbit determination with 1 cm precision using the Bernese GNSS Software","volume":"67","author":"Mao","year":"2021","journal-title":"Adv. Space Res."},{"key":"ref_11","unstructured":"Case, K., Kruizinga, G., and Wu, S. (2010). GRACE Level 1B Data Product User Handbook, Available online: https:\/\/earth.esa.int\/eogateway\/documents\/20142\/37627\/GRACE-L1B-Handbook-v1.3.pdf."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"931","DOI":"10.2514\/1.A34326","article-title":"GRACE-FO: The gravity recovery and climate experiment follow-on mission","volume":"56","author":"Kornfeld","year":"2019","journal-title":"J. Spacecr. Rocket."},{"key":"ref_13","unstructured":"Fletcher, K. (2012). Sentinel-3: ESA\u2019s Global Land and Ocean Mission for GMES Operational Services, ESA Communications. Available online: https:\/\/sentinel.esa.int\/documents\/247904\/351187\/S3_SP-1322_3.pdf."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Zhou, X., Jiang, W., Chen, H., Li, Z., and Liu, X. (2019). Improving the GRACE kinematic precise orbit determination through modified clock estimating. Sensors, 19.","DOI":"10.3390\/s19194347"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"511","DOI":"10.1007\/s10291-012-0297-1","article-title":"Improved GRACE kinematic orbit determination using GPS receiver clock modeling","volume":"17","author":"Weinbach","year":"2013","journal-title":"GPS Solut."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1007\/s10291-020-01078-8","article-title":"POD of small LEO satellites based on precise real-time MADOCA and SBAS-aided PPP corrections","volume":"25","author":"Wang","year":"2021","journal-title":"GPS Solut."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"484","DOI":"10.1088\/0026-1394\/44\/6\/007","article-title":"An assessment of relativistic effects for low Earth orbiters: The GRACE satellites","volume":"44","author":"Larson","year":"2007","journal-title":"Metrologia"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1080\/10095020.2021.1917310","article-title":"LEO satellite clock analysis and prediction for positioning applications","volume":"25","author":"Wang","year":"2022","journal-title":"Geo Spat. Inf. Sci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1007\/s00190-021-01563-z","article-title":"Sentinel-6A precise orbit determination using a combined GPS\/Galileo receiver","volume":"95","author":"Montenbruck","year":"2021","journal-title":"J. Geod."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1007\/s10291-022-01377-2","article-title":"Characteristics analysis and prediction of Low Earth Orbit (LEO) satellite clock corrections by using least-squares harmonic estimation","volume":"27","author":"Ge","year":"2022","journal-title":"GPS Solut."},{"key":"ref_21","unstructured":"Wen, H.Y., Kruizinga, G., Paik, M., Landerer, F., Bertiger, W., Sakumura, C., Bandikova, T., and Mccullough, C. (2023, June 14). Gravity Recovery and Climate Experiment Follow-on (GRACE-FO) Level-1 Data Product User Handbook, Available online: https:\/\/podaac-tools.jpl.nasa.gov\/drive\/files\/allData\/gracefo\/docs\/GRACE-FO_L1_Handbook.pdf."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1007\/s00190-008-0281-2","article-title":"Global Multi-GNSS Processing at CODE","volume":"83","author":"Dach","year":"2009","journal-title":"J. Geod."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"3314","DOI":"10.1016\/j.asr.2021.06.030","article-title":"Characteristics analysis of the GNSS satellite clock","volume":"68","author":"Ye","year":"2021","journal-title":"Adv. Space Res."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Kudrys, J., Prochniewicz, D., Zhang, F., Jakubiak, M., and Maciuk, K. (2021). Identification of BDS Satellite Clock Periodic Signals Based on Lomb-Scargle Power Spectrum and Continuous Wavelet Transform. Energies, 14.","DOI":"10.3390\/en14217155"},{"key":"ref_25","unstructured":"GMV (2023, June 14). Sentinels POD Product Handbook. Copernicus Sentinel-1, -2 and -3 Precise Orbit Determination Service (SENTINELSPOD). Available online: https:\/\/sentinel.esa.int\/documents\/247904\/3372484\/Sentinels-POD-Product-Handbook.pdf."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1007\/s10291-020-0962-8","article-title":"LEO\u2013BDS\u2013GPS integrated precise orbit modeling using FengYun-3D, FengYun-3C onboard and ground observations","volume":"24","author":"Li","year":"2020","journal-title":"GPS Solut."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Wang, L., Xu, B., Fu, W., Chen, R., Li, T., Han, Y., and Zhou, H. (2020). Centimeter-level precise orbit determination for the Luojia-1A satellite using BeiDou observations. Remote Sens., 12.","DOI":"10.3390\/rs12122063"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Yang, H., Yang, X., Zhang, Z., Sun, B., and Qin, W. (2020). Evaluation of the Effect of High-order Ionopsheric Delay on GPS Precise Point Positioning Time Transfer. Remote Sens., 12.","DOI":"10.3390\/rs12132129"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1007\/s10291-021-01200-4","article-title":"Integrity monitoring for precise orbit determination of LEO satellites","volume":"26","author":"Wang","year":"2021","journal-title":"GPS Solut."},{"key":"ref_30","unstructured":"Dach, R., Lutz, S., Walser, P., and Fridez, P. (2015). Bernese GNSS Software Version 5.2, University of Bern, Bern Open Publishing."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"749","DOI":"10.1007\/s00190-018-1195-2","article-title":"LEO constellation-augmented multi-GNSS for rapid PPP convergence","volume":"93","author":"Li","year":"2019","journal-title":"J. Geod."},{"key":"ref_32","unstructured":"Pavlis, N., Holmes, S., Kenyon, S., and Factor, J. (2008, January 13\u201318). An Earth Gravitational Model to Degree 2160: EGM2008. Proceedings of the European Geosciences Union General Assembly, Vienna, Austria. Available online: http:\/\/earth-info.nga.mil\/GandG\/wgs84\/gravitymod\/egm2008\/index.html."},{"key":"ref_33","unstructured":"Standish, E. (2023, June 14). JPL Planetary and Lunar Ephemerides, DE405\/LE405 (Memo IOM 312. F-98-048; Pasadena: JPL), Available online: http:\/\/ssd.jpl.nasa.gov\/iau-comm4\/de405iom\/de405iom.pdf."},{"key":"ref_34","first-page":"2010","article-title":"IERS conventions","volume":"36","author":"Petit","year":"2010","journal-title":"IERS Tech. Note"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"394","DOI":"10.1007\/s10236-006-0086-x","article-title":"Modelling the global ocean tides: Modern insights from FES2004","volume":"56","author":"Lyard","year":"2006","journal-title":"Ocean. Dyn."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"427","DOI":"10.1007\/s00190-013-0616-5","article-title":"Stochastic modeling of high-stability ground clocks in GPS analysis","volume":"87","author":"Wang","year":"2013","journal-title":"J. Geod."},{"key":"ref_37","unstructured":"Wang, K. (2016). Advanced Modeling and Algorithms for High-Precision GNSS Analysis, ETH."},{"key":"ref_38","unstructured":"Van Dierendonck, A.J., and McGraw, J. (1984, January 27\u201329). Relationship between Allan variances and Kalman filter parameters. Proceedings of the 16th Annual Precise Time and Time Interval Systems and Applications Meeting, Greenbelt, MD, USA."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1109\/MCS.2010.938485","article-title":"Kalman filtering with Newton\u2019s method [lecture notes]","volume":"30","author":"Humpherys","year":"2010","journal-title":"IEEE Control Syst. Mag."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1007\/s00190-015-0854-9","article-title":"On the estimability of parameters in undifferenced, uncombined GNSS network and PPP-RTK user models by means of S-system theory","volume":"90","author":"Odijk","year":"2016","journal-title":"J. Geod."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"647","DOI":"10.1109\/T-UFFC.1987.26997","article-title":"Time and Frequency (Time-Domain) Characterization, Estimation, and Prediction of Precision Clocks and Oscillators","volume":"34","author":"Allan","year":"1987","journal-title":"IEEE Trans. Ultrason. Ferroelectr. Freq. Control"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1007\/s10291-017-0678-6","article-title":"Quality assessment of multi-GNSS orbits and clocks for real-time precise point positioning","volume":"22","author":"Kazmierski","year":"2018","journal-title":"GPS Solut."},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Riley, W.J., and Howe, D.A. (2008). Handbook of Frequency Stability Analysis.","DOI":"10.6028\/NIST.SP.1065"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"44","DOI":"10.1007\/s10291-021-01089-z","article-title":"Sub-nanosecond one-way real-time time service system based on UTC","volume":"25","author":"Wu","year":"2021","journal-title":"GPS Solut."}],"container-title":["Remote Sensing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2072-4292\/15\/12\/3149\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:56:29Z","timestamp":1760126189000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2072-4292\/15\/12\/3149"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,6,16]]},"references-count":44,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2023,6]]}},"alternative-id":["rs15123149"],"URL":"https:\/\/doi.org\/10.3390\/rs15123149","relation":{},"ISSN":["2072-4292"],"issn-type":[{"value":"2072-4292","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,6,16]]}}}