{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:52:44Z","timestamp":1760233964509,"version":"build-2065373602"},"reference-count":60,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2021,3,15]],"date-time":"2021-03-15T00:00:00Z","timestamp":1615766400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Major Research Plan of China","award":["2016YFB0501702"],"award-info":[{"award-number":["2016YFB0501702"]}]},{"name":"National Nature Science Foundation of China","award":["41774089; 42004073"],"award-info":[{"award-number":["41774089; 42004073"]}]},{"name":"National Nature Science Foundation of China Youth Found","award":["41704013"],"award-info":[{"award-number":["41704013"]}]},{"name":"Funded by State Key Laboratory of Geo-Information Engineering","award":["SKLGIE2019-M-1-2"],"award-info":[{"award-number":["SKLGIE2019-M-1-2"]}]},{"name":"Open Research Fund of Qian Xuesen Laboratory of Space Technology, CAST","award":["GZZKFJJ2020006"],"award-info":[{"award-number":["GZZKFJJ2020006"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Remote Sensing"],"abstract":"<jats:p>When using the dynamic approach to recover the time-variable gravity field, the reference orbit generated by the perturbation model and the non-conservative force observed from the accelerometer should be introduced at first, and then the observation equations of the residual orbit and the residual range rate are established. This introduces a perturbation model error and instrument noise. Thus, there are low-frequency errors in the residual orbit and the residual range rate. Currently, most studies only focus on the low-frequency error of the residual range rate, neglecting the influence of the low-frequency error in the residual orbit. Therefore, under the condition of the perturbation model error and instrument noise including the constant term and 1CPR term, the low-frequency error formulas of the residual orbit and residual range rate are derived according to the characteristics of the solution of the Hill equation. Then, the influence of the low-frequency error on the residuals is analyzed by using the simulation and the real data processing respectively. In the simulation and real data processing, the accuracy of the recovered gravity field can maintain a good consistency for different arc lengths by removing the low-frequency error in the residual orbit. Finally, the time-variable gravity field model UCAS-IGG (University of Chinese Academy of Sciences-Institute of Geodesy and Geophysics) was solved from January 2005 to February 2010 by removing the low-frequency error of the residual orbit and residual range rate. Compared with the official institutions, the UCAS-IGG presents a good consistency in the estimating time-variable gravity field signal. This study demonstrates how the effect of the low-frequency error of the residual orbit should be taken into consideration when the longer arc length is used to recover a time-variable gravity field. Using a long arc length can reduce the variables of the initial state and recover the influence of the small force.<\/jats:p>","DOI":"10.3390\/rs13061118","type":"journal-article","created":{"date-parts":[[2021,3,15]],"date-time":"2021-03-15T22:16:54Z","timestamp":1615846614000},"page":"1118","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Influence of the Low-Frequency Error of the Residual Orbit on Recovering Time-Variable Gravity Field from the Satellite-To-Satellite Tracking Mission"],"prefix":"10.3390","volume":"13","author":[{"given":"Lei","family":"Liang","sequence":"first","affiliation":[{"name":"State Key Laboratory of Geodesy and Earth\u2019s Dynamics, Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences, Wuhan 430077, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jinhai","family":"Yu","sequence":"additional","affiliation":[{"name":"College of Earth Sciences and Planetary Sciences, University of Chinese Academy of Sciences, Beijing 100049, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Changqing","family":"Wang","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Geodesy and Earth\u2019s Dynamics, Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences, Wuhan 430077, China"},{"name":"School of Geospatial Engineering and Science, Sun Yat-sen University, Zhuhai 519082, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Min","family":"Zhong","sequence":"additional","affiliation":[{"name":"School of Geospatial Engineering and Science, Sun Yat-sen University, Zhuhai 519082, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8873-0750","authenticated-orcid":false,"given":"Wei","family":"Feng","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Geodesy and Earth\u2019s Dynamics, Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences, Wuhan 430077, China"},{"name":"School of Geospatial Engineering and Science, Sun Yat-sen University, Zhuhai 519082, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiaoyun","family":"Wan","sequence":"additional","affiliation":[{"name":"School of Land Science and Technology, China University of Geosciences (Beijing), Beijing 100089, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wei","family":"Chen","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Geodesy and Earth\u2019s Dynamics, Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences, Wuhan 430077, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yihao","family":"Yan","sequence":"additional","affiliation":[{"name":"School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,3,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"503","DOI":"10.1126\/science.1099192","article-title":"GRACE measurements of mass variability in the Earth system","volume":"305","author":"Tapley","year":"2004","journal-title":"Science"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"L09607","DOI":"10.1029\/2004GL019920","article-title":"The gravity recovery and climate experiment: Mission overview and early results","volume":"31","author":"Tapley","year":"2004","journal-title":"Geophys. Res. Lett."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"467","DOI":"10.1007\/s00190-005-0480-z","article-title":"GGM02- an improved earth gravity field model from GRACE","volume":"79","author":"Tapley","year":"2005","journal-title":"J. Geod."},{"key":"ref_4","unstructured":"Bettadpur, S. (2021, February 04). Gravity Recovery and Climate Experiment UTCSR Level-2 Processing Standards Document for Level-2 Product Release 005. Available online: https:\/\/www.researchgate.net\/publication\/289630299_UTCSR_Level-2_Processing_Standards_Document_for_Level-2_Product_Release_0005_Center_for_Space_Research_Technical_Report_GRACE."},{"key":"ref_5","first-page":"1","article-title":"GFZ GRACE level-2 processing standards document for level-2 product release 0005","volume":"12","author":"Dahle","year":"2012","journal-title":"Sci. Tech. Rep. Data"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"7547","DOI":"10.1002\/2016JB013007","article-title":"High-resolution CSR GRACE RL05 mascons","volume":"121","author":"Save","year":"2016","journal-title":"J. Geophys. Res. Solid Earth"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2648","DOI":"10.1002\/2014JB011547","article-title":"Improving methods for observing Earth\u2019s time variable mass distribution with GRCE using spherical cap mascons","volume":"120","author":"Watkins","year":"2015","journal-title":"J. Geophys. Res. Solid Earth"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1029\/2018JB015556","article-title":"Impact of different kinematic empirical parameters processing strategies on temporal gravity field model determination","volume":"123","author":"Zhou","year":"2018","journal-title":"J. Geophys. Res. Solid Earth"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"7040","DOI":"10.1029\/2018JB015601","article-title":"Improvements in the monthly gravity field solutions through modeling the colored noise in the GRACE data","volume":"123","author":"Guo","year":"2018","journal-title":"J. Geophys. Res. Solid Earth"},{"key":"ref_10","first-page":"3259","article-title":"Recovered GRACE time-variable gravity field based on dynamic approach with the non-linear corrections","volume":"62","author":"Liang","year":"2019","journal-title":"Chin. J. Geophys."},{"key":"ref_11","unstructured":"Mayer-G\u00fcrr, T. (2006). Gravitationsfeldbestimmung aus der Analyse kurzer Bahnb\u00f6gen am Beispielder Satellitenmissionen CHAMP und GRACE, University of Bonn."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"L17102","DOI":"10.1029\/2009GL039564","article-title":"Deriving daily snapshots of the Earth\u2019s gravity field from GRACE L1B data using Kalman filtering","volume":"36","author":"Kurtenbach","year":"2009","journal-title":"Geophys. Res. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"503","DOI":"10.1007\/s00190-016-0889-6","article-title":"An improved GRACE monthly gravity field solution by modeling the non-conservative acceleration and attitude observation errors","volume":"90","author":"Chen","year":"2016","journal-title":"J. Geod."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"6111","DOI":"10.1029\/2018JB015641","article-title":"Tongji-Grace02s and Tongji-Grace02k: High-precision static GRACE-only global Earth\u2019s gravity field models derived by refined data processing strategies","volume":"123","author":"Chen","year":"2018","journal-title":"J. Geophys. Res. Solid Earth"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"6010","DOI":"10.1029\/2018JB016596","article-title":"An optimized short-arc approach: Methodology and application to develop refined time series of Tongji-Grace2018 GRACE monthly solutions","volume":"124","author":"Chen","year":"2019","journal-title":"J. Geophys. Res. Solid Earth"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1007\/s00190-003-0362-1","article-title":"A technique for modeling the Earth\u2019s gravity field on the basis of satellite accelerations","volume":"78","author":"Ditmar","year":"2004","journal-title":"J. Geod."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"586","DOI":"10.1007\/s00190-005-0008-6","article-title":"\u2018DEOS_CHAMP-01C_70\u2019: A model of the Earth\u2019s gravity field computed from accelerations of the CHAMP satellite","volume":"79","author":"Ditmar","year":"2006","journal-title":"J. Geod."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Liu, X. (2008). Global Gravity Field Recovery from Satellite-to-Satellite Tracking Data with the Acceleration Approach, Delft University of Technology.","DOI":"10.54419\/rmsi6z"},{"key":"ref_19","first-page":"769","article-title":"DEOS Mass Transport model (DMT-1) based on GRACE satellite data: Methodology and validation","volume":"181","author":"Liu","year":"2010","journal-title":"Geophys. J. Int."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"605","DOI":"10.1007\/s00190-010-0401-7","article-title":"The celestial mechanics approach: Theoretical foundations","volume":"84","author":"Beutler","year":"2010","journal-title":"J. Geod."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"661","DOI":"10.1007\/s00190-010-0402-6","article-title":"The celestial mechanics approach: Application to data of the GRACE mission","volume":"84","author":"Beutler","year":"2010","journal-title":"J. Geod."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1023\/A:1008313405488","article-title":"The determination of gravitational potential differences from satellite-to-satellite tracking","volume":"75","author":"Jekeli","year":"1999","journal-title":"Celest. Mech. Dyn. Astron."},{"key":"ref_23","first-page":"B04411","article-title":"Precise estimation of in situ geopotential differences from GRACE low-low satellite-to-satellite tracking and accelerometer data","volume":"111","author":"Han","year":"2006","journal-title":"J. Geophys. Res."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1007\/s00190-010-0438-7","article-title":"GRACE-derived surface water mass anomalies by energy integral approach: Application to continental hydrology","volume":"6","author":"Ramillien","year":"2011","journal-title":"J. Geod."},{"key":"ref_25","first-page":"B11406","article-title":"Regional surface mass anomalies from GRACE KBR measurements: Application of L-curve reqularinzation and a priori hydrological knowledge","volume":"117","author":"Tangdamrongsub","year":"2012","journal-title":"J. Geophys. Res."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1792","DOI":"10.1093\/gji\/ggv248","article-title":"On the formulation of gravitational potential difference between the GRACE satellites based on energy integral in earth fixed frame","volume":"202","author":"Zeng","year":"2015","journal-title":"Geophys. J. Int."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1773","DOI":"10.1093\/gji\/ggv392","article-title":"GRACE time-variable gravity field recovery using an improved energy balance approach","volume":"203","author":"Shang","year":"2015","journal-title":"Geophys. J. Int."},{"key":"ref_28","unstructured":"Watkins, M.M., and Yuan, D.N. (2012). GRACE JPL Level-2 Processing Standards Document for Level-2 Product Release 05, Available online: ftp:\/\/isdcftp.gfz-posdam.de\/grace\/."},{"key":"ref_29","unstructured":"Kim, J. (2000). Simulation Study of a Low-Low Satellite-to-Satellite Tracking Mission, The University of Texas at Austin."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1833","DOI":"10.1016\/j.asr.2010.11.041","article-title":"GRACE gravity field modeling with an investigation on correlation between nuisance parameters and gravity field coefficients","volume":"47","author":"Zhao","year":"2011","journal-title":"Adv. Space Res."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"160","DOI":"10.1007\/s00190-005-0455-0","article-title":"Low-low satellite-to-satellite tracking: A comparison between analytical linear orbit perturbation theory and numerical integration","volume":"79","author":"Visser","year":"2005","journal-title":"J. Geod."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"587","DOI":"10.1016\/j.asr.2009.10.012","article-title":"CNES\/GRGS 10-day gravity field models (release 2) and their evaluation","volume":"45","author":"Bruinsma","year":"2010","journal-title":"Adv. Space Res."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1016\/j.epsl.2012.06.026","article-title":"Monthly gravity field solutions based on GRACE observations generated with the Celestial Mechanics Approach","volume":"345\u2013348","author":"Meyer","year":"2012","journal-title":"Earth Planet Sci. Lett."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"441","DOI":"10.1007\/s00190-011-0531-6","article-title":"Understanding data noise in gravity field recovery on the basis of inter-satellite ranging measurements acquired by the satellite gravimetry mission GRACE","volume":"86","author":"Ditmar","year":"2012","journal-title":"J. Geod."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"843","DOI":"10.1007\/s00190-013-0650-3","article-title":"The static gravity field model DGM-1S from GRACE and GOCE data: Computation, validation and an analysis of GOCE mission\u2019s added value","volume":"87","author":"Ditmar","year":"2013","journal-title":"J. Geod."},{"key":"ref_36","first-page":"756","article-title":"An investigation on GRACE temporal gravity field recovery using the dynamic approach","volume":"58","author":"Wang","year":"2015","journal-title":"Chin. J. Geophys."},{"key":"ref_37","first-page":"1994","article-title":"A new-time-variable gravity field model recovered by dynamic integral approach on the basis of GRACE KBRR data alone","volume":"59","author":"Luo","year":"2015","journal-title":"Chin. J. Geophys."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Colombo, O.L. (1984). The Global Mapping of Gravity with Two Satellite, Netherlands Geodetic Commission.","DOI":"10.54419\/07zzxs"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1111\/j.1365-246X.2007.03611.x","article-title":"GRACE\u2019s spatial aliasing error","volume":"172","author":"Seo","year":"2008","journal-title":"Geophys. J. Int."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Kaula, W.M. (1966). Theory of Satellite Geodesy: Applications of Satellites to Geodesy, Blaisdell.","DOI":"10.1063\/1.3033941"},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Montenbruck, O., and Gill, E. (2000). Satellite Orbits, Springer.","DOI":"10.1007\/978-3-642-58351-3"},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Sans\u03cc, F., and Rummel, R. (1989). Gravity field recovery from satellite tracking data. Theory of Satellite Geodesy and Gravity Field Determination\u2014Lecture Notes in Earth Sciences, Springer.","DOI":"10.1007\/BFb0010546"},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Sans\u03cc, F., and Rummel, R. (1989). Fundamentals of orbit determination. Theory of Satellite Geodesy and Gravity Field Determination\u2014Lecture notes in Earth Sciences, Springer.","DOI":"10.1007\/BFb0010546"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"286","DOI":"10.1002\/cjg2.30046","article-title":"Orbital perturbation differential equations with non-linear corrections for Champ-like satellite","volume":"60","author":"Yu","year":"2017","journal-title":"Chin. J. Geophy."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1007\/s10569-008-9117-x","article-title":"Position and velocity perturbations for the determination of geopotential from space geodetic measurements","volume":"100","author":"Xu","year":"2008","journal-title":"Celest. Mech. Dyn. Astron."},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Heiskanen, W.A., and Moritz, H. (1967). Physical Geodesy, Freeman, W.H. and company.","DOI":"10.1007\/BF02525647"},{"key":"ref_47","unstructured":"Petit, G., and Luzum, B. (2010). IERS Conventions, Bonifatius GmbH."},{"key":"ref_48","unstructured":"Kaplan, M.H. (1976). Modern Spacecraft Dynamics & Control, Wiley."},{"key":"ref_49","unstructured":"McCullough, G.M. (2007). Gravity Field Estimation for Next Generation Satellite Missions, University of Texas at Austin."},{"key":"ref_50","unstructured":"Ries, J., Bettadpur, S., Eanes, R., Kang, Z., Ko, U., McCullough, C., Nagel, P., Pie, N., Poole, S., and Richter, T. (2016). The Combination Global Gravity Model GGM05C, The University of Texas at Austin."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"1818","DOI":"10.1016\/j.asr.2014.07.004","article-title":"Improvement of the GRACE star camera data based on the revision of the combination method","volume":"54","author":"Bandikova","year":"2014","journal-title":"Adv. Space Res."},{"key":"ref_52","unstructured":"Bandikova, T. (2015). The Role of Attitude Determination for Inter-Satellite Ranging. [Ph.D. Thesis, Leibniz University of Hannover]."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"1804","DOI":"10.1002\/2014JB011470","article-title":"Monthly gravity field models derived from GRACE Level 1B data using a modified short-arc approach","volume":"120","author":"Chen","year":"2015","journal-title":"J. Geophys. Res. Solid Earth"},{"key":"ref_54","first-page":"B09402","article-title":"Variations in the Earth\u2019s oblateness during the past 28 years","volume":"109","author":"Cheng","year":"2004","journal-title":"J. Geophys. Res."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"L08402","DOI":"10.1029\/2005GL025285","article-title":"Post-processing removal of correlated errors in GRACE data","volume":"33","author":"Swenson","year":"2006","journal-title":"Geophys. Res. Lett."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"L11501","DOI":"10.1029\/2004GL019779","article-title":"Time-variable gravity from GRACE: First results","volume":"31","author":"Wahr","year":"2004","journal-title":"Geophys. Res. Lett."},{"key":"ref_57","first-page":"37","article-title":"Towards a more accurate temporal gravity model from GRACE observations through the kinematic orbits","volume":"60","author":"Yang","year":"2017","journal-title":"Chin. J. Geophys."},{"key":"ref_58","first-page":"1032","article-title":"Global temporal gravity field recovery using GRACE data","volume":"57","author":"Ran","year":"2014","journal-title":"Chin. J. Geophys."},{"key":"ref_59","first-page":"1308","article-title":"Algorithm characteristics of dynamic approach-based satellite gravimetry and its improvement proposals","volume":"46","author":"Shen","year":"2017","journal-title":"Acta Geod. Cartogr. Sin."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"1055","DOI":"10.1111\/j.1365-246X.2006.03229.x","article-title":"Tide model errors and GRACE gravimetry: Towards a more realistic assessment","volume":"167","author":"Ray","year":"2006","journal-title":"Geophys. J. Int."}],"container-title":["Remote Sensing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2072-4292\/13\/6\/1118\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:36:06Z","timestamp":1760160966000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2072-4292\/13\/6\/1118"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,3,15]]},"references-count":60,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2021,3]]}},"alternative-id":["rs13061118"],"URL":"https:\/\/doi.org\/10.3390\/rs13061118","relation":{},"ISSN":["2072-4292"],"issn-type":[{"type":"electronic","value":"2072-4292"}],"subject":[],"published":{"date-parts":[[2021,3,15]]}}}