{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,3]],"date-time":"2026-03-03T17:07:46Z","timestamp":1772557666391,"version":"3.50.1"},"reference-count":45,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2023,1,7]],"date-time":"2023-01-07T00:00:00Z","timestamp":1673049600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Technology Innovation Center for Land Engineering and Human Settlements, Shaanxi Land Engineering Construction Group Co., Ltd., and Xi\u2019an Jiaotong University","award":["2021WHZ0080"],"award-info":[{"award-number":["2021WHZ0080"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Remote Sensing"],"abstract":"<jats:p>The high-precision regional geoid model provides important fundamental geospatial information for developing and applying many disciplines. Deterministic and stochastic modifications are applied to Stokes\u2019s and Hotine\u2019s formulas of geoid modeling to reduce errors. Based on the Experimental Geopotential Model 2019 (XGM2019), this paper used Stokes\u2019s and Hotine\u2019s formulas to analyze the variation of global root mean square error (RMSE) with modification parameters for two deterministic (Wong and Gore; and Van\u00ed\u010dek and Kleusberg) and three stochastic modifications (biased, unbiased, and optimum). Taking the quasigeoid refinement of Jilin Province as an example, the global RMSE, approximate geoid undulation, and additive corrections were calculated. The parameter analysis and the global RMSE calculation showed that the variation of the modification limits and the terrestrial gravity data error variance had a centimeter-level effect on the global RMSE. In contrast, the impact of the integration radius was relatively small. The stochastic modifications were better than the deterministic ones in calculating the global RMSE. The global RMSE of Hotine\u2019s formula was smaller than that of Stokes\u2019s, and its unbiased and optimum modifications reached the minimum value of 12.9 mm. The validation of XGM2019 and the refined quasigeoid based on the high accuracy GPS\/leveling points showed that the standard deviation (STD) of XGM2019 was 5.8 cm in Jilin Province, and the refined optimal quasigeoid model was 2.9 cm. Stokes\u2019s and Hotine\u2019s formulas provided the same accuracy in the study area. In the western plain area, the accuracy of the deterministic modifications was 2.0 cm, which was about 0.4 cm higher than that of the stochastic modifications. In the eastern mountainous area, the stochastic modifications were better than the deterministic ones, and the accuracy was about 3.2 cm. Stokes\u2019s and Hotine\u2019s formulas based on deterministic and stochastic modifications significantly improve the accuracy of the XGM2019. The deterministic and stochastic modifications show millimeter-level differences in plain and mountainous areas.<\/jats:p>","DOI":"10.3390\/rs15020376","type":"journal-article","created":{"date-parts":[[2023,1,9]],"date-time":"2023-01-09T04:47:08Z","timestamp":1673239628000},"page":"376","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Performance Comparison of Deterministic and Stochastic Modifications in Stokes\u2019s and Hotine\u2019s Formulas: The Case of Jilin Province, China"],"prefix":"10.3390","volume":"15","author":[{"given":"Qiong","family":"Wu","sequence":"first","affiliation":[{"name":"College of Geo-Exploration Science and Technology, Jilin University, 938 West Democratic Street, Changchun 130026, China"}]},{"given":"Guoyu","family":"Zhang","sequence":"additional","affiliation":[{"name":"College of Geo-Exploration Science and Technology, Jilin University, 938 West Democratic Street, Changchun 130026, China"}]},{"given":"Bin","family":"Wang","sequence":"additional","affiliation":[{"name":"Geodetic Data Processing Centre, National Administration of Surveying, Mapping and Geoinformation, Xi\u2019an 710054, China"}]},{"given":"Linshan","family":"Zhong","sequence":"additional","affiliation":[{"name":"College of Geo-Exploration Science and Technology, Jilin University, 938 West Democratic Street, Changchun 130026, China"}]},{"given":"Feng","family":"Xiao","sequence":"additional","affiliation":[{"name":"College of Geo-Exploration Science and Technology, Jilin University, 938 West Democratic Street, Changchun 130026, China"}]}],"member":"1968","published-online":{"date-parts":[[2023,1,7]]},"reference":[{"key":"ref_1","unstructured":"Wang, C. 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