{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,3]],"date-time":"2026-06-03T01:06:23Z","timestamp":1780448783052,"version":"3.54.1"},"reference-count":50,"publisher":"MDPI AG","issue":"22","license":[{"start":{"date-parts":[[2023,11,18]],"date-time":"2023-11-18T00:00:00Z","timestamp":1700265600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Basic Frontier Science Research Program of the Chinese Academy of Sciences","award":["ZDBS-LY-DQC028"],"award-info":[{"award-number":["ZDBS-LY-DQC028"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Remote Sensing"],"abstract":"<jats:p>This study presents a methodology for constructing a quasi-geoid model with millimeter-level accuracy over the Shangyu area in China, following the guidelines of the International Association of Geodesy Joint Working Group 2.2.2, known as \u201cThe 1 cm geoid experiment\u201d. Our approach combines two steps to ensure exceptional accuracy. First, we employ Molodensky\u2019s theory to model the gravity field, accounting for non-level surfaces and considering complex terrain effects. Through an exhaustive analysis of these influential factors, we implement a comprehensive suite of applicable formulae within Molodensky\u2019s series solution, enabling a thorough assessment of their impacts on height anomalies within the gravimetric quasi-geoid model. Second, we utilize a hybrid method that involves a multi-surface function using the least-squares method and a robust estimation technique. This approach enables the interpolation of quasi-geoid heights by incorporating ellipsoidal and leveling normal heights, as well as gravimetric quasi-geoid data. Through a numerical example, we demonstrate the efficiency of our solution concept, achieving an accuracy of 0.79 cm compared to independent global navigation satellite system (GNSS)\/leveling measurements. By developing this methodology, our study contributes to the advancement of geodesy research and provides a valuable methodology for creating highly precise quasi-geoid models in geodetic applications.<\/jats:p>","DOI":"10.3390\/rs15225414","type":"journal-article","created":{"date-parts":[[2023,11,18]],"date-time":"2023-11-18T07:33:18Z","timestamp":1700292798000},"page":"5414","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["High-Accuracy Quasi-Geoid Determination Using Molodensky\u2019s Series Solutions and Integrated Gravity\/GNSS\/Leveling Data"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8592-9844","authenticated-orcid":false,"given":"Dongmei","family":"Guo","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":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8684-7194","authenticated-orcid":false,"given":"Xiaodong","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":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhixin","family":"Xue","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":[{"vocabulary":"crossref","role":"author"}]},{"given":"Huiyou","family":"He","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":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lelin","family":"Xing","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":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7765-7965","authenticated-orcid":false,"given":"Xian","family":"Ma","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":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiaowei","family":"Niu","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":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,11,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1007\/BF02521540","article-title":"The deviations of the sea surface from the geoid and their effect on geoid computation","volume":"51","author":"Brennecke","year":"1977","journal-title":"Bull. Geod."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Sans\u00f2, F., and Sideris, M.G. (2013). Geoid Determination: Theory and Methods, Lecture Notes in Earth System Sciences, Springer.","DOI":"10.1007\/978-3-540-74700-0"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"229037","DOI":"10.1016\/j.tecto.2021.229037","article-title":"The Indian Ocean Geoid Low at a plume-slab","volume":"817","author":"Steinbergerk","year":"2021","journal-title":"Tectonophysics"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Mertikas, S. (2010). Gravity, Geoid and Earth Observation, Springer. International Association of Geodesy, Symposia.","DOI":"10.1007\/978-3-642-10634-7"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1007\/s00190-005-0449-y","article-title":"Calibration of geoid error models via a combined adjustment of ellipsoidal, orthometric and gravimetric geoid height data","volume":"79","author":"Fotopoulos","year":"2005","journal-title":"J. Geod."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1007\/s00190-021-01507-7","article-title":"Collocation and FFT-based geoid estimation within the Colorado 1 cm geoid experiment","volume":"95","author":"Grigoriadis","year":"2021","journal-title":"J. Geod."},{"key":"ref_7","unstructured":"Sans\u00f2, F., and Rummel, R. (1977). Geodetic Boundary Value Problems in View of the One Centimeter Geoid, Springer."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Marti, U. (2014). Gravity, Geoid and Height Systems, Springer. International Association of Geodesy, Symposia.","DOI":"10.1007\/978-3-319-10837-7"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"571","DOI":"10.1007\/s001900050125","article-title":"A method to validate gravimetric-geoid computation software based on Stokes\u2019s integral formula","volume":"71","author":"Featherstone","year":"1997","journal-title":"J. Geod."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1007\/s001900050284","article-title":"Topographic effects by the Stokes-Helmert method of geoid and quasi-geoid determinations","volume":"74","year":"2000","journal-title":"J. Geod."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1023\/A:1022156303247","article-title":"Geoid determination through ellipsoidal Stokes boundary-value problem","volume":"4","author":"Ardestani","year":"2000","journal-title":"Stud. Geophys. Et Geod."},{"key":"ref_12","unstructured":"Molodensky, M.S., Eremeev, V.F., and Yurkina, M.I. (1962). Methods for the Study of the External Gravitational Field and Figure of the Earth, Israeli Program for Scientific Translations."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1007\/BF02526971","article-title":"On the solution of Molodensky\u2019s boundary value problem","volume":"72","author":"Brovar","year":"1964","journal-title":"Bull. Geod."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1007\/s00190-001-0238-1","article-title":"Molodensky potential telluroid based on a minimum-distance map. Case study: The quasi-geoid of East Germany in the World Geodetic Datum 2000","volume":"76","author":"Ardalan","year":"2002","journal-title":"J. Geod."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"428","DOI":"10.1007\/s00190-002-0266-5","article-title":"A new hybrid geoid model for Japan, GSIGE02000","volume":"76","author":"Kuroishi","year":"2002","journal-title":"J. Geod."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1007\/s11200-006-0010-3","article-title":"Improved determination of heights using a conversion surface by combining gravimetric quasi-geoid\/geoid and GPS-levelling height differences","volume":"50","author":"Nahavandchi","year":"2006","journal-title":"Stud. Geophys. Geod."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Kenyon, S., Pacino, M., and Marti, U. (2012). Geodesy for Planet Earth, Springer. International Association of Geodesy, Symposia.","DOI":"10.1007\/978-3-642-20338-1"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1007\/s00190-020-01412-5","article-title":"New gravimetric-only and hybrid geoid models of Taiwan for height modernisation, cross-island datum connection and airborne LiDAR mapping","volume":"94","author":"Hwang","year":"2020","journal-title":"J. Geod."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"268","DOI":"10.1007\/s001900050243","article-title":"Robust estimation of geodetic datum transformation","volume":"73","author":"Yang","year":"1999","journal-title":"J. Geod."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Awange, J.L., and Pal\u00e1ncz, B. (2016). Geospatial Algebraic Computations, Springer.","DOI":"10.1007\/978-3-319-25465-4"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"218","DOI":"10.1007\/3-540-26932-0_38","article-title":"Evaluation of SRTM3 and GTOPO30 terrain data in Germany","volume":"Volume 129","author":"Denker","year":"2005","journal-title":"International Association of Geodesy Symposia"},{"key":"ref_22","unstructured":"Amante, C., and Eakins, B.W. (2000). ETOPO1, 1 Arc-Minute Global Relief Model: Procedures, Data Sources and Analysis, NOAA Technical Memorandum NESDIS."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1093\/gji\/ggz264","article-title":"The tree canopy effect in gravity forward modelling","volume":"219","author":"Yang","year":"2019","journal-title":"Geophys. J. Int."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1007\/s00190-020-01409-0","article-title":"Superconducting gravimeter and seismometer shedding light on FG5\u2019s offsets, trends and noise: What observations at Onsala Space Observatory can tell us","volume":"94","author":"Scherneck","year":"2020","journal-title":"J. Geod."},{"key":"ref_25","unstructured":"(2019). Specifications for the Gravimetry Control (Standard No. GB\/T 20256-2019)."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1007\/s00190-021-01587-5","article-title":"Correction to: Performance assessment of the relative gravimeter Scintrex CG-6","volume":"96","author":"Francis","year":"2022","journal-title":"J. Geod."},{"key":"ref_27","unstructured":"(2019). Specifications for the Dense Gravity Measurement (Standard No. GB\/T 17944-2018)."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1016\/0031-9201(81)90046-7","article-title":"Preliminary reference earth model","volume":"25","author":"Dziewonski","year":"1981","journal-title":"Phys. Earth Planet. Inter."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"488","DOI":"10.1111\/j.1365-246X.1992.tb00112.x","article-title":"Atmospheric pressure and gravity","volume":"190","author":"Merriam","year":"1992","journal-title":"Geophys. J. Int."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1111\/j.1365-246X.1982.tb04991.x","article-title":"Modelling the effect of atmospheric pressure variations on gravity","volume":"71","author":"Spratt","year":"1982","journal-title":"Geophys. J. R. Astron. Soc."},{"key":"ref_31","first-page":"511","article-title":"Piece-wise linear dynamic adjustment for gravity network","volume":"45","author":"Wei","year":"2016","journal-title":"Acta Geod. Cartogr. Sin."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"95","DOI":"10.24895\/JIG.2019.25-2.991","article-title":"pyGABEUR-ITB: A free software for adjustment of relative gravimeter data","volume":"25","author":"Wijaya","year":"2019","journal-title":"Geomatika"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1005","DOI":"10.1016\/S0098-3004(02)00005-5","article-title":"Adjustment of relative gravity measurements using weighted and datum-free constraints","volume":"28","author":"Hwang","year":"2002","journal-title":"Comput. Geosci."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1016\/0098-3004(84)90026-8","article-title":"A Fortran IV program for a least-squares gravity base-station network adjustment","volume":"10","author":"Lagios","year":"1983","journal-title":"Comput. Geosci."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1665","DOI":"10.1016\/j.cageo.2008.02.018","article-title":"FA2BOUG-A FORTRAN 90 code to compute Bouguer gravity anomalies from gridded free-air anomalies: Application to the Atlantic-Mediterranean transition zone","volume":"34","author":"Fullea","year":"2008","journal-title":"Comput. Geosci."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Shepard, D. (1968, January 27\u201329). A Two-Dimensional Interpolation Function for Irregularly Spaced Data. Proceedings of the 23rd National Conference ACM, New York, NY, USA.","DOI":"10.1145\/800186.810616"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Pavlis, N.K., Holmes, S.A., Kenyon, S.C., and Factor, J.K. (2008, January 13\u201318). An earth gravitational model degree 2160: EGM2008. Proceedings of the 2008 General Assembly of the European Geosciences Union, Vienna, Austria.","DOI":"10.1190\/1.3063757"},{"key":"ref_38","unstructured":"S\u00e1nchez, L., \u00c5gren, J., Huang, J., Wang, Y.M., and Forsberg, R. (2018, January 17\u201321). Basic Agreements for the Computation of Station Potential Values as IHRS Coordinates, Geoid Undulations and Height Anomalies within the Colorado 1 cm Geoid Experiment. Proceedings of the International Symposium on Gravity, Geoid and Height Systems 2018 (GGHS2018), Copenhagen, Denmark."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"819","DOI":"10.1007\/s00190-011-0467-x","article-title":"First GOCE gravity field models derived by three different approaches","volume":"85","author":"Pail","year":"2011","journal-title":"J. Geod."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1007\/s11200-008-0006-2","article-title":"Comparison of two different solutions to Molodensky\u2019s G1 term","volume":"52","year":"2008","journal-title":"Stud. Geophys. Geod."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1007\/BF02607322","article-title":"Some problems regarding the way of solving Molodensky\u2019s integral equation for the earth considered as a plane","volume":"9","author":"Tiron","year":"1965","journal-title":"Stud. Geophys. Et Geod."},{"key":"ref_42","first-page":"70","article-title":"Series solutions of Molodensky\u2019s problem","volume":"A","author":"Moritz","year":"1971","journal-title":"Publ. Deut. Geod. Komm."},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Guo, J.Y. (2023). Physical Geodesy. Springer Textbooks in Earth Sciences, Geography and Environment, Springer.","DOI":"10.1007\/978-3-031-23320-3"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1007\/BF02519354","article-title":"Solving Molodensky\u2019s series by fast Fourier transform techniques","volume":"60","author":"Sideris","year":"1986","journal-title":"Bull. Geod."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"536","DOI":"10.1007\/s11200-013-0249-4","article-title":"On the effect of planar approximation in the Geodetic Boundary Value Problem","volume":"58","author":"Heck","year":"2014","journal-title":"Stud. Geophys. Geod."},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Heiskanen, W.A., and Moritz, H. (1967). Physical Geodesy, Freeman.","DOI":"10.1007\/BF02525647"},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Bian, F., Xie, Y., Cui, X., and Zeng, Y. (2013). Geo-Informatics in Resource Management and Sustainable Ecosystem GRMSE, Springer. Communications in Computer and Information, Science.","DOI":"10.1007\/978-3-642-41908-9"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"731","DOI":"10.1007\/s00190-010-0406-2","article-title":"The combination of GNSS-levelling data and gravimetric (quasi-) geoid heights in the presence of noise","volume":"84","author":"Klees","year":"2010","journal-title":"J. Geod."},{"key":"ref_49","first-page":"315","article-title":"A refinement of inverse distance weighted interpolation","volume":"2","author":"Watson","year":"1985","journal-title":"Geo-Processing"},{"key":"ref_50","unstructured":"Kubik, K. (1982, January 7\u201311). An error theory for the Danish method. Proceedings of the Symp Math Models, Accuracy Aspects and Quality Control, ISPRS Commission III Symposium, Helsinki, Finland."}],"container-title":["Remote Sensing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2072-4292\/15\/22\/5414\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T21:25:25Z","timestamp":1760131525000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2072-4292\/15\/22\/5414"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,11,18]]},"references-count":50,"journal-issue":{"issue":"22","published-online":{"date-parts":[[2023,11]]}},"alternative-id":["rs15225414"],"URL":"https:\/\/doi.org\/10.3390\/rs15225414","relation":{},"ISSN":["2072-4292"],"issn-type":[{"value":"2072-4292","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,11,18]]}}}