Band-limited SRBF quadratic approximation method for local gravity fields in complex regions

doi:10.11947/j.AGCS.2024.20230132

Abstract

Abstract:

Utilizing the SRBF-based two-step cumulative approximation method, this study conducted a detailed numerical analysis and validation for the construction of a local gravity field model in complex areas. First, data from the global gravity field model were combined with airborne gravity data to construct the initial SRBF-based gravity field model for the study area (the first gravity field approximation based on band-limited SRBFs). When compared to the airborne-only gravity field model, the standard deviation of geoid height discrepancies between the initial gravity field model and GPS/leveling data decreased by approximately 0.022 m. This suggests the effectiveness of merging GGM and airborne gravity data for initial modeling. Subsequently, after removing the effects of the reference gravity field model, the initial gravity field model and the residual terrain model from the terrestrial and airborne gravity data, a secondary SRBF-based modeling process was conducted (the second gravity field approximation based on band-limited SRBFs). Compared to the one-step SRBF model, the two-step approach yielded reductions of 0.036 m and 0.024 m in terms of STD and RMS, respectively, highlighting the superiority of the SRBF-based two-step cumulative approximation method. Moreover, the study area features a complex mountainous terrain, which can serve as a reference for constructing gravity field model in similar complex regions.

Key words: band-limited spherical radial basis function, two-step cumulative approximation method for local gravity field, GGM gravity data, airborne gravity data, terrestrial gravity data

CLC Number:

P312

Zhiwei MA, Shaofeng BIAN, Ruyi XU, Yongbing CHEN. Band-limited SRBF quadratic approximation method for local gravity fields in complex regions[J]. Acta Geodaetica et Cartographica Sinica, 2024, 53(5): 900-916.

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References 43

[1]	李建成, 陈俊勇, 宁津生, 等. 地球重力场逼近理论与中国2000似大地水准面的确定[M]. 武汉: 武汉大学出版社, 2003.
	LI Jiancheng, CHEN Junyong, NING Jinsheng, et al. The earth's gravitational field approximation theory and the determination of China 2000 quasi-geoid[M]. Wuhan: Wuhan University Press, 2003.
[2]	WU Yihao, LUO Zhicai, CHEN Wu, et al. High-resolution regional gravity field recovery from Poisson wavelets using heterogeneous observational techniques[J]. Earth, Planets and Space, 2017, 69(34): 1-15.
[3]	NEYMAN Y M, LI J, LIU Q. Modification of Stokes and Vening-Meinesz formulas for the inner zone of arbitrary shape by minimization of upper bound truncation errors[J]. Journal of Geodesy, 1996, 70(7): 410-418.
[4]	ABBAK R A, SJÖBERG L E, ELLMANN A, et al. A precise gravimetric geoid model in a mountainous area with scarce gravity data: a case study in central Turkey[J]. Studia Geophysica et Geodaetica, 2012, 56(4): 909-927.
[5]	FEATHERSTONE W E. Deterministic, stochastic, hybrid and band-limited modifications of Hotine's integral[J]. Journal of Geodesy, 2013, 87(5): 487-500.
[6]	WICHIENGAROEN C. A comparison of gravimetric undulations computed by the modified Molodensky truncation method and the method of least squares spectral combination by optimal integral kernels[J]. Bulletin Géodésique, 1984, 58(4): 494-509.
[7]	NAHAVANDCHI H, SJÖBERG L E. Precise geoid determination over Sweden using the Stokes-Helmert method and improved topographic corrections[J]. Journal of Geodesy, 2001, 75(2): 74-88.
[8]	LIU Yusheng, LOU Lizhi. Unified land-ocean quasi-geoid computation from heterogeneous data sets based on radial basis functions[J]. Remote Sensing, 2022, 14(13): 3015.
[9]	FORSBERG R, TSCHERNING C C. The use of height data in gravity field approximation by collocation[J]. Journal of Geophysical Research: Solid Earth, 1981, 86(B9): 7843-7854.
[10]	HWANG C, HSIAO Y, SHIH H, et al. Geodetic and geophysical results from a Taiwan airborne gravity survey: data reduction and accuracy assessment[J]. Journal of Geophysical Research: Solid Earth, 2007, 112(B4). https://doi.org/10.1029/2005JB004220.
[11]	MCCUBBINE J C, AMOS M J, TONTINI F C, et al. The New Zealand gravimetric quasigeoid model 2017 that incorporates nationwide airborne gravimetry[J]. Journal of Geodesy, 2018, 92(8): 923-937.
[12]	WILLBERG M, ZINGERLE P, PAIL R. Integration of airborne gravimetry data filtering into residual least-squares collocation: example from the 1 cm geoid experiment[J]. Journal of Geodesy, 2020, 94(8): 75. https://doi.org/10.1007/s00190-020-01396-2.
[13]	VARGA M, PITONˇÁK M, NOVÁK P, et al. Contribution of GRAV-D airborne gravity to improvement of regional gravimetric geoid modelling in Colorado, USA[J]. Journal of Geodesy, 2021, 95(5): 53. https://doi.org/10.1007/s00190-021-01494-9.
[14]	IŞIK M S, EROL B, EROL S, et al. High-resolution geoid modeling using least squares modification of Stokes and Hotine formulas in Colorado[J]. Journal of Geodesy, 2021, 95(5): 49. https://doi.org/10.1007/s00190-021-01501-2.
[15]	WU Yihao, ZHOU Hao, ZHONG Bo, et al. Regional gravity field recovery using the GOCE gravity gradient tensor and heterogeneous gravimetry and altimetry data[J]. Journal of Geophysical Research (Solid Earth), 2017, 122(8): 6928-6952.
[16]	FREEDEN W, GERVENS T, SCHREINER M. Constructive approximation on the sphere (with applications to geomathematics) [M]. Oxford: Clarendon Press, 1998.
[17]	SCHMIDT M, FENGLER M, MAYER-GÜRR T, et al. Regional gravity modeling in terms of spherical base functions[J]. Journal of Geodesy, 2007, 81(1): 17-38.
[18]	KLEES R, TENZER R, PRUTKIN I, et al. A data-driven approach to local gravity field modelling using spherical radial basis functions[J]. Journal of Geodesy, 2008, 82(8): 457-471.
[19]	WITTWER T. Local gravity field modelling with radial basis functions[D]. Delft: Delft University of Technology, 2009.
[20]	NAEIMI M, FLURY J, BRIEDEN P. On the regularization of regional gravity field solutions in spherical radial base functions[J]. Geophysical Journal International, 2015, 202(2): 1041-1053.
[21]	马志伟, 陆洋, 涂弋, 等. 利用Abel-Poisson径向基函数模型化局部重力场[J]. 测绘学报, 2016, 45(9): 1019-1027. DOI: 10.11947/j.AGCS.2016.20150519.
	MA Zhiwei, LU Yang, TU Yi, et al. Regional gravity field modeling with Abel-Poisson radial basis functions[J]. Acta Geodaetica et Cartographica Sinica, 2016, 45(9)1019-1027. DOI: 10.11947/j.AGCS.2016.20150519.
[22]	MA Zhiwei, YANG Meng, LIU Jie. Regional gravity field modeling using band-limited SRBFs: a case study in Colorado[J]. Remote Sensing, 2023, 15(18): 4515.
[23]	KLEES R, SLOBBE D C, FARAHANI H H. A methodology for least-squares local quasi-geoid modelling using a noisy satellite-only gravity field model[J]. Journal of Geodesy, 2018, 92(4): 431-442.
[24]	LIU Qing, SCHMIDT M, SÁNCHEZ L, et al. Regional gravity field refinement for (quasi-) geoid determination based on spherical radial basis functions in Colorado[J]. Journal of Geodesy, 2020, 94(10): 99. https://doi.org/10.1007/s00190-020-01431-2.
[25]	LI Xiaopeng. Using radial basis functions in airborne gravimetry for local geoid improvement[J]. Journal of Geodesy, 2018, 92(5): 471-485.
[26]	HEISKANENW A, MORITZ H. Physical geodesy[M]. San Francisco: W. H. Freeman, 1967.
[27]	KOCH K R, KUSCHE J. Regularization of geopotential determination from satellite data by variance components[J]. Journal of Geodesy, 2002, 76(5): 259-268.
[28]	BARTHELMES F. Definition of functionals of the geopotential and their calculation from spherical harmonic models[R]. Potsdam: GFZ German Research Centre for Geosciences, 2013
[29]	MORITZ H. Geodetic reference system 1980[J]. Journal of Geodesy, 2000, 74(1): 128-133.
[30]	HIRT C, REXER M. Earth2014: 1 arc-min shape, topography, bedrock and ice-sheet models-available as gridded data and degree-10, 800 spherical harmonics[J]. International Journal of Applied Earth Observation and Geoinformation, 2015, 39: 103-112.
[31]	FOROUGHI I, VANÍCˇEK P, KINGDON R W, et al. Sub-centimetre geoid[J]. Journal of Geodesy, 2019, 93(6): 849-868.
[32]	JIANG Tao, WANG Yanming. On the spectral combination of satellite gravity model, terrestrial and airborne gravity data for local gravimetric geoid computation[J]. Journal of Geodesy, 2016, 90(12): 1405-1418.
[33]	TORGE W. Gravimetry[M]. Berlin: Walter de Gruyter, 1989.
[34]	EICKER A. Gravity field refinement by radial basis functions from in-situ satellite data[D]. Bonn: Rheinische Friedrich-Wilhelms-Universität Bonn, 2008.
[35]	BENTEL K, SCHMIDT M, DENBY C. Artifacts in regional gravity representations with spherical radial basis functions[J]. Journal of Geographical Sciences, 2013, 3(3): 173-187.
[36]	NAEIMI M. Inversion of satellite gravity data using spherical radial base functions[D]. Hannover: Leibniz Universität Hannover, 2013.
[37]	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.
[38]	CHEN J, ZHANG X, CHEN Q, et al. Static gravity field recovery and accuracy analysis based on reprocessed GOCE level 1b gravity gradient observations[C]//Proceedings of the 24th EGU General Assembly, Vienna: IEEE, 2022.
[39]	SÁNCHEZ L, ÅGREN J, HUANG Jianliang, et al. Strategy for the realisation of the international height reference system (IHRS)[J]. Journal of Geodesy, 2021, 95(3): 33. https://doi.org/10.1007/s00190-021-01481-0.
[40]	BUCHA B, JANÁK J, PAPCˇO J, et al. High-resolution regional gravity field modelling in a mountainous area from terrestrial gravity data[J]. Geophysical Journal International, 2016, 207(2): 949-966.
[41]	REXER M, HIRT C, CLAESSENS S, et al. Layer-based modelling of the earth's gravitational potential up to 10-km scale in spherical harmonics in spherical and ellipsoidal approximation[J]. Surveys in Geophysics, 2016, 37(6): 1035-1074.
[42]	HIRT C, KUHN M, CLAESSENS S, et al. Study of the earth's short-scale gravity field using the ERTM2160 gravity model[J]. Computers & Geosciences, 2014, 73: 71-80.
[43]	WANG Y M, SALEH J, LI X, et al. The US gravimetric geoid of 2009 (USGG2009): model development and evaluation[J]. Journal of Geodesy, 2012, 86(3): 165-180.

模型	80阶	120阶	160阶	200阶	240阶	280阶	300阶
GOC6S	0.000	0.001	0.004	0.015	0.049	0.115	0.145
GGC16	0.001	0.002	0.006	0.020	0.058	0.122	-
TJ21S	0.000	0.000	0.001	0.004	0.012	0.026	0.032
WHU22S	0.000	0.001	0.003	0.009	0.025	0.054	0.066
TIM6	0.001	0.003	0.005	0.015	0.049	0.117	0.139
SPW5	0.013	0.021	0.027	0.033	0.061	0.111	0.134

参数类型	参数取值
参考重力场模型(GGM)	GOC6S、GGC16、TJ21S、WHU22S、TIM6、SPW5
核函数类型	Shannon、Blackman、Cubic polynomial (Cup)
航空重力数据的SRBF展开阶次	161～1500
GGM重力数据的SRBF展开阶次	161～280、161～300

展开阶次	GGM	核函数	Mean	STD	GGM	核函数	Mean	STD
280	GOC6S	Shannon	－0.087	0.111	GGC16	Shannon	－0.088	0.112
		Blackman	－0.100	0.115		Blackman	－0.101	0.116
		Cup	－0.099	0.120		Cup	－0.100	0.121
	TJ21S	Shannon	－0.088	0.112	WHU22S	Shannon	－0.093	0.111
		Blackman	－0.101	0.115		Blackman	－0.102	0.114
		Cup	－0.099	0.120		Cup	－0.100	0.120
	TIM6	Shannon	－0.087	0.111	SPW5	Shannon	－0.097	0.105
		Blackman	－0.100	0.115		Blackman	－0.100	0.115
		Cup	－0.099	0.120		Cup	－0.103	0.118
300	GOC6S	Shannon	－0.078	0.120	GGC16	Shannon	—	—
		Blackman	－0.116	0.136		Blackman	—	—
		Cup	－0.120	0.142		Cup	—	—
	TJ21S	Shannon	－0.074	0.121	WHU22S	Shannon	－0.079	0.119
		Blackman	－0.100	0.137		Blackman	－0.109	0.134
		Cup	－0.103	0.147		Cup	－0.113	0.146
	TIM6	Shannon	－0.081	0.120	SPW5	Shannon	－0.084	0.113
		Blackman	－0.118	0.132		Blackman	－0.104	0.128
		Cup	－0.122	0.146		Cup	－0.104	0.133

重力数据	Max	Min	Mean	STD
δg(x_air)	82.65	－80.02	－5.18	29.36
δg_res(x_air)	68.85	－63.48	0.16	20.49
δg(x_SPW5)	53.34	－65.04	－5.16	28.94
δg_res(x_SPW5)	47.63	－47.66	－0.95	18.86

模型	模型构成		Max	Min	Mean	STD
模型	参考场	剩余场	Max	Min	Mean	STD
GCM1	GOC6S(0～160)	Air(161～1500)＋GOC6S (161～280)	0.125	－0.320	－0.087	0.111
GAM1	GOC6S(0～160)	Air(161～1500)	0.196	－0.344	－0.070	0.128
GCM2	GGC16 (0～160)	Air (161～1500)＋GGC16 (161～280)	0.116	－0.312	－0.088	0.112
GAM2	GGC16(0～160)	Air(161～1500)	0.193	－0.346	－0.072	0.127
GCM3	TJ21S (0～160)	Air (161～1500)＋TJ21S (161～280)	0.145	－0.330	－0.088	0.112
GAM3	TJ21S (0～160)	Air(161～1500)	0.196	－0.342	－0.068	0.128
GCM4	WHU22S (0～160)	Air (161～1500)＋WHU22S (161～280)	0.140	－0.350	－0.093	0.111
GAM4	WHU22S (0～160)	Air(161～1500)	0.194	－0.342	0.070	0.127
GCM5	TIM6(0～160)	Air (161～1500)＋TIM6(161～280)	0.117	－0.311	－0.087	0.111
GAM5	TIM6(0～160)	Air(161～1500)	0.196	－0.340	－0.068	0.127
GCM6	SPW5(0～160)	Air (161～1500)＋SPW5(161～280)	0.102	－0.336	－0.097	0.105
GAM6	SPW5(0～160)	Air(161～1500)	0.196	－0.341	－0.068	0.127