基于科学阶段SWOT/KaRIn测高数据反演高精度的垂直重力异常梯度模型

doi:10.11947/j.AGCS.2025.20240520

测绘学报 ›› 2025, Vol. 54 ›› Issue (9): 1583-1595.doi: 10.11947/j.AGCS.2025.20240520

基于科学阶段SWOT/KaRIn测高数据反演高精度的垂直重力异常梯度模型

押少帅¹(), 刘新¹(), 周瑞宸²^,³, 李真¹, 边少锋⁴, 郭金运¹

^1.山东科技大学测绘与空间信息学院，山东　青岛　266590
^2.中国科学院精密测量科学与技术创新研究院精密大地测量与定位全国重点实验室，湖北　武汉　430077
^3.中国科学院大学，北京　100049
^4.中国地质大学（武汉）地质探测与评估教育部重点实验室，湖北　武汉　430074

收稿日期:2025-01-06 修回日期:2025-08-15 出版日期:2025-10-10 发布日期:2025-10-10
通讯作者: 刘新 E-mail:13526663057@163.com;xinliu1969@126.com
作者简介:押少帅（1992—），男，博士，研究方向为卫星测高数据处理及海洋重力梯度反演。E-mail：13526663057@163.com
基金资助:
国家自然科学基金(42430101)

High accuracy vertical gradient of gravity anomaly model determined from SWOT/KaRIn altimetry data during scientific phase

Shaoshuai YA¹(), Xin LIU¹(), Ruichen ZHOU²^,³, Zhen LI¹, Shaofeng BIAN⁴, Jinyun GUO¹

^1.College of Geodesy and Geomatics, Shandong University of Science and Technology, Qingdao 266590, China
^2.State Key Lab of Precision Geodesy, Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences, Wuhan 430077, China
^3.University of Chinese Academy of Sciences, Beijing 100049, China
^4.Key Laboratory of Geological Survey and Evaluation of Ministry Education, China University of Geosciences, Wuhan 430074, China

Received:2025-01-06 Revised:2025-08-15 Online:2025-10-10 Published:2025-10-10
Contact: Xin LIU E-mail:13526663057@163.com;xinliu1969@126.com
About author:YA Shaoshuai (1992—), male, PhD, majors in altimeter data processing and marine gravity anomalies recovery. E-mail: 13526663057@163.com
Supported by:
The National Natural Science Foundation of China(42430101)

摘要/Abstract

摘要：

卫星测高技术是反演海洋垂直重力异常梯度的重要手段之一。针对常规的一维测高数据采样间隔大、跨轨数据稀疏、精度低的问题，SWOT（surface water and ocean topography）测高卫星可以提供二维宽刈幅海洋信息，获取更高空间分辨率和精度的海面高。本文基于1～20周期的SWOT海面高数据反演垂直重力异常梯度模型（SWOT_VGGA），首先，采用移去-恢复法的处理策略，将XGM2019e_2159作为参考场模型；然后，联合沿轨、跨轨和对角线3个方向的测高数据，并利用最小二乘配置法解算垂线偏差；最后，根据垂线偏差和参考场反演出SWOT_VGGA模型。本文选择菲律宾海为试验区域，将SIO V32.1版本的垂直重力异常梯度（SIO_curv_32.1）作为参考模型，结果表明SWOT_VGGA与SIO_curv_32.1模型的一致性为8.25 E，验证了一年SWOT测高数据反演垂直重力异常梯度的可靠性。此外，对SWOT_VGGA模型在不同水深、离岸距离和海底坡角的一致性进行了统计分析。结果表明，多周期SWOT_VGGA模型的一致性要优于单周期，而且1～10和11～20周期的SWOT_VGGA模型之间的一致性为1.81 E。因此，SWOT卫星在不同周期的数据质量比较稳定，可以用来反演高精度的海洋重力信息。

关键词: SWOT测高卫星, 海面高, 垂线偏差, 垂直重力异常梯度, 移去-恢复法

Abstract:

Satellite altimetry is a crucial technique for determining the marine vertical gradient of gravity anomaly. One-dimensional altimetry data suffer from large sampling intervals, sparse across-track data, and low precision. In comparison, the surface water and ocean topography (SWOT) altimetry satellite provides two-dimensional wide-range ocean information, achieving higher spatial resolution and accuracy in sea surface height measurements. Therefore, this paper presents a vertical gradient of gravity anomaly (SWOT_VGGA) model based on SWOT sea surface height data from cycles 1 to 20. The remove-restore technique was employed as the processing strategy, selecting XGM2019e_2159 as the reference gravity field model. The deflection of vertical was computed by combining along-track, cross-track, and diagonal-track altimetry data through least squares collocation. Based on the deflection of the vertical and the reference gravity field, the SWOT_VGGA model was obtained. This study focused on the Philippine Sea, with the vertical gravity gradient anomaly from SIO V32.1 employed as the reference. The results show an 8.25 E consistency between the SWOT_VGGA and SIO_curv_32.1 models, confirming the reliability of SWOT-derived vertical gradient of gravity anomaly from one year of SWOT altimetry data. Additionally, the model consistency was assessed across varying water depths, distances, and seafloor slope angles. Furthermore, the multi-cycle model exhibited better consistency than the single-cycle model. Specifically, the discrepancy between its 1~10 and 11~20 cycle subsets was only 1.81 E. This demonstrates that the data quality of the SWOT satellite exhibits stability across different cycles, making it suitable for inverting high-precision marine gravity information.

Key words: SWOT altimetry satellite, sea surface height, deflection of the vertical, vertical gradient of gravity anomaly, remove-restore method

中图分类号:

P223

押少帅, 刘新, 周瑞宸, 李真, 边少锋, 郭金运. 基于科学阶段SWOT/KaRIn测高数据反演高精度的垂直重力异常梯度模型[J]. 测绘学报, 2025, 54(9): 1583-1595.

Shaoshuai YA, Xin LIU, Ruichen ZHOU, Zhen LI, Shaofeng BIAN, Jinyun GUO. High accuracy vertical gradient of gravity anomaly model determined from SWOT/KaRIn altimetry data during scientific phase[J]. Acta Geodaetica et Cartographica Sinica, 2025, 54(9): 1583-1595.

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参考文献 48

[1]	ODALOVIC O, MEDVED K, NAOD S. Modeling of vertical gravity gradient by normal gravity field and digital terrain models[J]. Journal of Geodesy, 2022, 96(10): 74.
[2]	胡敏章, 李建成, 邢乐林et al.. 由垂直重力梯度异常反演全球海底地形模型[J]. 测绘学报, 2014, 43(6): 558-565, 574. DOI: . doi: 10.13485/j.cnki.11-2089.2014.0090
	HU Minzhang, LI Jiancheng, XING Lelinet al.. Global bathymetry model predicted from vertical gravity gradient anomalies[J]. Acta Geodaetica et Cartographical Sinica, 2014, 43(6): 558-565, 574. DOI: . doi: 10.13485/j.cnki.11-2089.2014.0090
[3]	ZHENG Wei, HSU H T, ZHONG Min, et al. A contrastive study on the influences of radial and three-dimensional satellite gravity gradiometry on the accuracy of the Earth's gravitational field recovery[J]. Chinese Physics B, 2012, 21(10): 109101.
[4]	WAN Xiaoyun, RAN Jiangjun. An alternative method to improve gravity field models by incorporating GOCE gradient data[J]. Earth Sciences Research Journal, 2018, 22(3): 187-193.
[5]	郭金运, 吴渴知, 金鑫, 等. 高精度三维跟踪抛物运动的重力垂直梯度测量方法[J]. 武汉大学学报(信息科学版), 2025, 50(3): 462-468.
	GUO Jinyun, WU Kezhi, JIN Xin, et al. A method to determine vertical gravity gradient by high-precision 3D tracking parabolic motion[J]. Geomatics and Information Science of Wuhan University, 2025, 50(3): 462-468.
[6]	赵永奇, 李建成, 徐新禹, 等. 利用GOCE和GRACE卫星观测数据确定静态重力场模型[J]. 地球物理学报, 2023, 66(6): 2322-2336.
	ZHAO Yongqi, LI Jiancheng, XU Xinyu, et al. Determination of static gravity field model by using satellite data of GOCE and GRACE[J]. Chinese Journal of Geophysics, 2023, 66(6): 2322-2336.
[7]	STRAY B, LAMB A, KAUSHIK A, et al. Quantum sensing for gravity cartography[J]. Nature, 2022, 602(7898): 590-594.
[8]	黄佳喜, 边少锋, 纪兵et al.. 航空重力梯度地形改正[J]. 测绘学报, 2024, 53(8): 1540-1551. DOI: . doi: 10.11947/j.AGCS.2024.20230342
	HUANG Jiaxi, BIAN Shaofeng, JI Binget al.. Terrain corrections for airborne gravity gradiometry[J]. Acta Geodaetica et Cartographica Sinica, 2024, 53(8): 1540-1551. DOI: . doi: 10.11947/j.AGCS.2024.20230342
[9]	HSIAO Y S, HWANG C, WU Mengling, et al. Improved geoid modeling using observed and modeled gravity gradients in Taiwan[J]. Journal of Surveying Engineering, 2017, 143(2): 04016027.
[10]	ZHOU Ruichen, LIU Xin, LI Zhen, et al. On performance of vertical gravity gradient determined from CryoSat-2 altimeter data over Arabian SeaFree[J]. Geophysical Journal International, 2023, 234(2): 1519-1529.
[11]	吴怿昊, 何秀凤, 罗志才, 等. 基于航空重力梯度数据重建高分辨率局部重力场的径向基函数方法[J]. 地球物理学报, 2024, 67(5): 1913-1926.
	WU Yihao, HE Xiufeng, LUO Zhicai, et al. The approach for high-resolution local gravity field recovery from airborne gravity gradient data based on radial basis functions[J]. Chinese Journal of Geophysics, 2024, 67(5): 1913-1926.
[12]	EVSTIFEEV M I. The state of the art in the development of onboard gravity gradiometers[J]. Gyroscopy and Navigation, 2017, 8(1): 68-79.
[13]	DRANSFIELD M H, CHRISTENSEN A N. Performance of airborne gravity gradiometers[J]. The Leading Edge, 2013, 32(8): 908-922.
[14]	LEE J. Falcon gravity gradiometer technology[J]. Exploration Geophysics, 2001, 32(3-4): 247-250.
[15]	HINKS D, MCINTOSH S, LANE R. A comparison of the Falcon and Air-FTG airborne gravity gradiometer systems at the Kokong Test Block, Botswana[C]//Proceedings of 2004 Airborne Gravity. Sydney: Geoscience Australia Record, 2004: 125-134.
[16]	钟波, 刘滔, 李贤炮, 等. 卫星重力梯度精化局部重力场的径向基函数法[J]. 华中科技大学学报(自然科学版), 2022, 50(9): 141-148.
	ZHONG Bo, LIU Tao, LI Xianpao, et al. Radial basis function method for refining regional gravity field from satellite gravity gradient[J]. Journal of Huazhong University of Science and Technology (Natural Science Edition), 2022, 50(9): 141-148.
[17]	郭金运, 张鸿飞, 李真, 等. 多源船载重力异常数据的联合再处理：以墨西哥湾为例[J]. 武汉大学学报(信息科学版), 2025, 50(7): 1277-1290.
	GUO Jinyun, ZHANG Hongfei, LI Zhen, et al. Joint reprocessing of multi-sources shipborne gravity anomalies: a case study of gulf of Mexico[J]. Geomatics and Information Science of Wuhan University, 2025, 50(7): 1277-1290.
[18]	李真, 郭金运, 孙中苗, 等. 基于ICESat-2多波束激光测高数据的全球海洋重力异常反演分析[J]. 测绘学报, 2024, 53(2): 252-262. DOI: . doi: 10.11947/j.AGCS.2024.20230207
	LI Zhen, GUO Jinyun, SUN Zhongmiao, et al. Global marine gravity anomalies recovered from multi-beam laser altimeter data of ICESat-2[J]. Acta Geodaetica et Cartographical Sinica. 2024, 53(2): 252-262. DOI: . doi: 10.11947/j.AGCS.2024.20230207
[19]	LIU Xin, SONG Menghao, LI Chao, et al. Inversion method of deflection of the vertical based on SWOT wide-swath altimeter data[J]. Geodesy and Geodynamics, 2024, 15(4): 419-428.
[20]	王虎彪, 王勇, 陆洋, 等. 联合多种测高资料确定西太平洋海域2′×2′重力梯度[J]. 地球物理学进展, 2009, 24(3): 852-858.
	WANG Hubiao, WANG Yong, LU Yang, et al. Determination of gravity gradients in the Western Pacific Ocean by combinating multi-altimeter data[J]. Progress in Geophysics, 2009, 24(3): 852-858.
[21]	BAO Lifeng, XU Houze, LI Zhicai. Towards a 1 mGal accuracy and 1 Min resolution altimetry gravity field[J]. Journal of Geodesy, 2013, 87(10): 961-969.
[22]	BOUMAN J, BOSCH W, SEBERA J. Assessment of systematic errors in the computation of gravity gradients from satellite altimeter data[J]. Marine Geodesy, 2011, 34(2): 85-107.
[23]	ANNAN R F, WAN Xiaoyun, HAO Ruijie, et al. Global marine gravity gradient tensor inverted from altimetry-derived deflections of the vertical: CUGB2023GRAD[J]. Earth System Science Data, 2024, 16(3): 1167-1176.
[24]	ZHOU Shuai, LIU Xin, GUO Jinyun, et al. Bathymetry of the gulf of Mexico predicted with multilayer perceptron from multisource marine geodetic data[J]. IEEE Transactions on Geoscience and Remote Sensing, 2023, 61: 1-11.
[25]	AKDOĞAN Y, YILDIZ H, AHI G, et al. Evaluation of global gravity models from absolute gravity and vertical gravity gradient measurements in Turkey[J]. Measurement Science and Technology. 2019, 30(11): 115009.
[26]	WU Lin, WANG Hubiao, CHAI Hua, et al. Research on the relative positions-constrained pattern matching method for underwater gravity-aided inertial navigation[J]. Journal of Navigation, 2015, 68(5): 937-950.
[27]	ORUÇ B. Edge detection and depth estimation using a tilt angle map from gravity gradient data of the Kozaklı-Central Anatolia region, Turkey[J]. Pure and Applied Geophysics, 2011, 168(10): 1769-1780.
[28]	FU L L, PAVELSKY T, CRETAUX J F, et al. The surface water and ocean topography mission: a breakthrough in radar remote sensing of the ocean and land surface water[J]. Geophysical Research Letters, 2024, 51(4): e2023GL107652.
[29]	YU Y, SANDWELL D T, DIBARBOURE G, et al. Accuracy and resolution of SWOT altimetry: foundation seamounts[J]. Earth and Space Science, 2024, 11(6): e2024EA003581.
[30]	YU Daocheng, WENG Zequn, HWANG C, et al. Seamount detection using SWOT-derived vertical gravity gradient: advancements and challenges open access[J]. Geophysical Journal International, 2024, 237(3): 1780-1793.
[31]	JIN Taoyong, ZHOU Mao, ZHANG Huan, et al. Analysis of vertical deflections determined from one cycle of simulated SWOT wide-swath altimeter data[J]. Journal of Geodesy, 2022, 96(4): 30.
[32]	CASULLA M A A, MIZUNAGA H, TANAKA T, et al. Imaging crustal features and Moho depths through enhancements and inversion of gravity data from the Philippine island arc system[J]. Progress in Earth and Planetary Science, 2022, 9(1): 16.
[33]	ZHU C, GUO J, YA S, et al. Moving geoid gradient method for high-precision and high-resolution gravity recovery from SWOT wide-Swath data[J]. IEEE Transactions on Geoscience and Remote Sensing. 2025, 63: 4205613.
[34]	SCHAEFFER P, PUJOL M I, VEILLARD P, et al. The CNES CLS 2022 mean sea surface: short wavelength improvements from CryoSat-2 and SARAL/AltiKa high-sampled altimeter data[J]. Remote Sensing, 2023, 15(11): 2910.
[35]	JOUSSET S, MULET S, WILKIN J, et al. New global mean dynamic topography CNES-CLS-22 combining drifters, hydrological profiles and High Frequency radar data[C]//Proceedings of 2022 Ocean Surface Topography Science Team Meeting. Venice: IEEE, 2022.
[36]	ZINGERLE P, PAIL R, SCHEINERT M, et al. Evaluation of terrestrial and airborne gravity data over Antarctica-a generic approach[J]. Journal of Geodetic Science, 2019, 9(1): 29-40.
[37]	ZINGERLE P, PAIL R, GRUBER T, et al. The combined global gravity field model XGM2019e[J]. Journal of Geodesy, 2020, 94(7): 66-72.
[38]	ZAKI A, ABDALLAH M, EL-ASHQUER M, et al. Marine gravity modelling from SARAL/AltiKA data using the least square collocation for the Red Sea[J]. The Egyptian Journal of Remote Sensing and Space Sciences, 2023, 26(3): 607-617.
[39]	ANDERSEN O B, ZHANG Shengjun, SANDWELL D T, et al. The unique role of the Jason geodetic missions for high resolution gravity field and mean sea surface modelling[J]. Remote Sensing, 2021, 13(4): 646-651.
[40]	ZHU Chengcheng, GUO Jinyun, GAO Jinyao, et al. Marine gravity determined from multi-satellite GM/ERM altimeter data over the South China Sea: SCSGA V1.0[J]. Journal of Geodesy, 2020, 94(5): 50-55.
[41]	TSCHERNING C, RAPP R. Closed covariance expressions for gravity anomalies, geoid undulations, and deflections of the vertical implied by anomaly degree variance models[R]. Ohio State University: Division of Geodetic Science, 1974.
[42]	HWANG C, PARSONS B. Gravity anomalies derived from Seasat, Geosat, ERS-1 and TOPEX/POSEIDON altimetry and ship gravity: a case study over the Reykjanes Ridge[J]. Geophysical Journal International, 1995, 122(2): 551-568.
[43]	HOFMANN-WELLENHOF B, LEGAT K, WIESER M. Physical fundamentals[M]. Vienna: Springer, 2003: 59-76.
[44]	HEISKANEN W A, MORITZ H. Physical geodesy[J]. Bulletin Géodésique (1946—1975), 1967, 86(1): 491-492.
[45]	BOUMAN J. Relation between geoidal undulation, deflection of the vertical and vertical gravity gradient revisited[J]. Journal of Geodesy, 2012, 86(4): 287-304.
[46]	陈旭, 孔祥雪, 周润生, 等. 一种基于SWOT宽刈幅模拟数据的沿轨垂线偏差求解方法[J]. 海洋学报, 2023, 45(11): 175-184.
	CHEN Xu, KONG Xiangxue, ZHOU Runsheng, et al. A method for solving along-track vertical deflection based on SWOT wide-Swath simulated data[J]. Haiyang Xuebao, 2023, 45(11): 175-184.
[47]	WELCH P. The use of fast Fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms[J]. IEEE Transactions on Audio and Electroacoustics, 1967, 15(2): 70-73.
[48]	ZHOU Ruichen, LIU Xin, GUO Jinyun, et al. Inverting vertical gravity anomaly gradients using multidirectional data from a mean sea surface model: the case of the Arabian Sea[J]. Earth, Planets and Space, 2024, 76(1): 170-177.

周期	数量	最大值	最小值	均值	标准差	均方根
1～20	1 442 392	0.061	－0.061	0.000	0.016	0.016
5	1 278 929	0.203	－0.203	0.002	0.059	0.059
10	1 332 836	0.183	－0.183	0.003	0.053	0.053
13	1 303 739	0.202	－0.202	－0.001	0.059	0.059
20	1 234 200	0.194	－0.194	0.000	0.058	0.058

不同分量	最大值	最小值	均值	标准差	均方根
北分量	80.86	－62.76	0.02	1.93	1.93
东分量	132.62	－67.95	0.05	2.53	2.53

不同水深/m	最大值/E	最小值/E	均值/E	标准差/E	均方根/E
（0，1000]	358.06	－373.53	－0.24	16.22	16.22
（1000，2000]	193.31	－391.04	0.30	12.94	12.95
（2000，3000]	144.86	－75.15	－0.13	9.05	9.05
（3000，4000]	80.89	－56.94	0.04	7.26	7.26
（4000，5000]	70.61	－41.99	0.08	5.90	5.90
（5000，∞）	42.52	－43.61	0.19	4.92	4.93

离岸距离/km	最大值/E	最小值/E	均值/E	标准差/E	均方根/E
（0，50]	358.06	－391.04	0.44	18.93	18.93
（50，100]	154.91	－228.64	0.03	8.19	8.19
（100，150]	65.81	－79.51	0.05	6.94	6.95
（150，200]	62.85	－55.56	0.05	6.48	6.48
（200，250]	56.06	－44.07	0.03	6.22	6.22
（250，300]	43.61	－71.04	0.02	6.16	6.16
（300，∞）	94.65	－76.67	0.02	5.74	5.74

坡角/（°）	最大值/E	最小值/E	均值/E	标准差/E	均方根/E
[0，2]	358.06	－274.77	0.15	7.00	7.00
（2，4]	273.40	－228.53	0.05	8.00	8.00
（4，6]	278.89	－316.45	－0.03	9.11	9.11
（6，8]	222.55	－253.63	－0.10	9.77	9.77
（8，10]	220.55	－248.45	－0.01	10.55	10.55
（10，∞）	211.07	－391.04	－0.10	11.97	11.97

基于科学阶段SWOT/KaRIn测高数据反演高精度的垂直重力异常梯度模型

High accuracy vertical gradient of gravity anomaly model determined from SWOT/KaRIn altimetry data during scientific phase

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图/表 22

参考文献 48

相关文章 15

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不同区域	最大值	最小值	均值	标准差	均方根
区域1	20.10	－20.74	0.04	4.77	4.77
区域2	48.76	－95.41	－0.05	6.82	6.82
区域3	122.25	－124.55	0.13	8.64	8.64

窗口长度/（°）	最大值/E	最小值/E	均值/E	标准差/E	均方根/E
0.1	29.59	－26.56	0.02	5.08	5.08
0.2	29.19	－27.45	0.02	4.95	4.95
0.3	29.39	－30.14	0.02	4.98	4.98
0.4	29.28	－29.56	0.02	4.98	4.98

不同周期	最大值	最小值	均值	标准差	均方根
1～10	357.84	－393.59	0.07	8.26	8.26
11～20	358.31	－392.88	0.07	8.27	8.27
4	357.19	－391.27	0.08	8.43	8.43
15	357.88	－398.70	0.04	9.71	9.71

不同模型	最大值	最小值	均值	标准差	均方根
SWOT_VGGA与curv_SWOT_02	247.74	－253.08	0.07	7.83	7.83
SIO_curv_32.1与curv_SWOT_02	249.46	－169.95	0.06	6.54	6.54

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