Near-zone Direct and Indirect Topographic Effects Based on the Rectangular Prism and Surface Element

doi:10.11947/j.JGGS.2019.0302

Abstract

Abstract:

Helmert’s second method of condensation is an effective method for terrain reduction in the geoid and quasi-geoid determinations. Condensing the masses outside the geoid to a surface layer on the geoid produces several forms of topographic effects: direct effect on gravity, secondary indirect effect on gravity and indirect effects on the (quasi-) geoid, respectively. To strike a balance between computation accuracy and numerical efficiency, the global integration region of topographic effects is usually divided into near zone and far zone. We focus on the computation of near-zone topographic effects, which are functions of actual topographic masses and condensed masses. Since there have already been mature formulas for gravitational attraction and potential of actual topographic masses using rectangular prism model, we put forward surface element model for condensed masses. Afterwards, the formulas for near-zone direct and indirect effects are obtained easily by combining the rectangular prism model and surface element model. To overcome the planar approximation errors involved with the new formulas for near-zone topographic effects, the Earth’s curvature can be taken into account. It is recommended to apply the formulas based on the rectangular prism and surface element considering the Earth’s curvature to calculate near-zone topographic effects for high-accuracy demand to determine geoid and quasi-geoid.

Key words: helmert’s second method of condensation; direct effect; indirect effect; rectangular prism; surface element

Jian MA,Ziqing WEI,Zhenghui YANG,Xiaogang LIU,Jianfeng JI. Near-zone Direct and Indirect Topographic Effects Based on the Rectangular Prism and Surface Element[J]. Journal of Geodesy and Geoinformation Science, 2019, 2(3): 8-17.

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References 27

[1]	GUAN Zelin, GUAN Zheng, HUANG Motao , et al. Theory and Method of Regional Gravity Field Approximation[M]. Beijing: Surveying and Mapping Press, 1997.
[2]	HEISKANNEN W A, MORITZ H . Physical Geodesy[J]. Bulletin Géodésique, 1967,86(1):491-492. DOI: 10.1007/BF02525647
[3]	VANÍČEK P, HUANG J, NOVÁK P , et al. Determination of the Boundary Values for the Stokes-Helmert Problem[J]. Journal of Geodesy, 1999,73(4):180-192. DOI: 10.1007/s001900050235
[4]	LI Jiancheng . The Recent Chinese Terrestrial Digital Height Datum Model:Gravimetric Quasi-geoid CNGG2011[J]. Acta Geodaetica et Cartographica Sinica, 2012,41(5):651-660.
[5]	HUANG Jianliang, VÉRONNEAU M . Canadian Gravimetric Geoid Model 2010[J]. Journal of Geodesy, 2013,87(8):771-790. DOI: 10.1007/s00190-013-0645-0
[6]	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. DOI: 10.1007/s00190-011-0506-7
[7]	RONG Min . Stokes-Helmert Method for Geoid Determination[D]. Zhengzhou: Information Engineering University, 2015: 43-68. http://cdmd.cnki.com.cn/Article/CDMD-90005-1016058463.htm
[8]	SJÖBERG L E . Topographic Effects by the Stokes-Helmert Method of Geoid and Quasi-geoid Determinations[J]. Journal of Geodesy, 2000,74(2):255-268. DOI: 10.1007/s001900050284
[9]	SJÖBERG L E, NAHAVANDCHI H . On the Indirect Effect in the Stokes-Helmert Method of Geoid Determination[J]. Journal of Geodesy, 1999,73(2):87-93. DOI: 10.1007/s001900050222
[10]	NAHAVANDCHI H . The Direct Topographical Correction in Gravimetric Geoid Determination by the Stokes-Helmert Method[J]. Journal of Geodesy, 2000,74(6):488-496.DOI: 10.1007/s001900000110
[11]	NAHAVANDCHI H . Terrain Correction Computations by Spherical Harmonics and Integral Formulas[J]. Physics and Chemistry of the Earth, Part A:Solid Earth and Geodesy, 1999,24(1):73-78.DOI: 10.1016/S1464-1895(98)00013-1
[12]	LUO Zhicai, CHEN Yongqi, NING Jinsheng . Effect of Terrain on the Determination of High Precise Local Gravimetric Geoid[J]. Geomatics and Information Science of Wuhan University, 2003,28(3):340-344.
[13]	ZHANG Chuanyin, CHAO Dingbo, DING Jian , et al. Precision Topographical Effects for Any Kind of Field Quantities for Any Altitude[J]. Acta Geodaetica et Cartographica Sinica, 2009,38(1):28-34. DOI: 10.3321/j.issn:1001-1595.2009.01.006
[14]	GUO Chunxi, WANG Huimin, WANG Bin . Determination of High Resolution Grid Terrain and Isostatic Corrections in All China Area[J]. Acta Geodaetica et Cartographica Sinica, 2002,31(3):201-205. DOI: 10.3321/j.issn:1001-1595.2002.03.004
[15]	ZHANG Chuanyin, CHAO Dingbo, DING Jian , et al. Arithmetic and Characters Analysis of Terrain Effects for CM-order Precision Height Anomaly[J]. Acta Geodaetica et Cartographica Sinica, 2006,35(4):308-314. DOI: 10.3321/j.issn:1001-1595.2006.04.003
[16]	RONG Min, ZHOU Wei . Study on Topography Correction Based on Spherical Approximation[J]. Journal of Geodesy and Geodynamics, 2015,35(1):58-61.
[17]	MA Jian, WEI Ziqing, WU Lili , et al. The Bouguer Correction Algorithm for Gravity with Limited Range[J]. Acta Geodaetica et Cartographica Sinica, 2017,46(1):26-33.DOI: 10.11947/j.AGCS.2017.20160173
[18]	WEI Ziqing . Introduction to the Second Geodetic Boundary-value Problem with the Geocentric Reference Ellipsoidal Surface as the Boundary[J]. Geomatics Science and Engineering, 2015,35(1):1-6.
[19]	WANG Y M . Precise Computation of the Direct and Indirect Topographic Effects of Helmert’s 2nd Method of Condensation Using SRTM30 Digital Elevation Model[J]. Journal of Geodetic Science, 2011,1(4):305-312.
[20]	MAKHLOOF A A, ILK K H . Far-zone Effects for Different Topographic-compensation Models Based on a Spherical Harmonic Expansion of the Topography[J]. Journal of Geodesy, 2008,82(10):613-635. DOI: 10.1007/s00190-008-0214-0
[21]	NOVÁK P, NOVÁK P, VANÍČEK P, MARTINEC Z , et al. Effects of the Spherical Terrain on Gravity and the Geoid[J]. Journal of Geodesy, 2001,75(9-10):491-504. DOI: 10.1007/s001900100201
[22]	MARTINEC Z, VANÍČEK P . The Indirect Effect of Topography in the Stokes-Helmert Technique for a Spherical Approximation of the Geoid[J]. Manuscript Geodaetica, 1994(19):213-219.
[23]	NAGY D, GAPP G, BENEDEK J . The Gravitational Potential and Its Derivatives for the Prism[J]. Journal of Geodesy, 2000,74(7-8):552-560. DOI: 10.1007/s001900000116
[24]	FORSBERG R . A Study of Terrain Reductions, Density Anomalies and Geophysical Inversion Methods in Gravity Field Modelling[R]. Columbus: Ohio State University, 1984.
[25]	MA Jian, WEI Ziqing, SHEN Chen , et al. Transformation Relation Between the Topographic Correction and the Direct Topographic Effect[J]. Journal of Geomatics Science and Technology, 2017,34(3):245-250.
[26]	WICHIENCHAROEN C . The Indirect Effects on the Computation of Geoid Undulations[R]. [S.l.]: NASA, 1982.
[27]	MARTINEC Z . Boundary-value Problems for Gravimetric Determination of a Precise Geoid[M]. Berlin: Springer, 1998.

Topographic effects	Near-zone area	Min	Max	Mean	RMS
Direct effects /mGal	0.5°×0.5°	-64.680	61.066	-2.069	10.767
	1°×1°	-63.962	64.690	-0.672	10.548
	3°×3°	-63.452	66.896	0.439	10.545
Indirect effects /m	0.5°×0.5°	0.008	0.751	0.152	0.186
	1°×1°	0.007	0.756	0.151	0.186
	3°×3°	0.001	0.741	0.141	0.177

Formulas compared	Near-zone area	Min	Max	Mean	RMS
Formulas B and A	0.5°×0.5°	0.001	0.229	0.047	0.059
	1°×1°	0.001	0.237	0.048	0.061
	3°×3°	0.001	0.241	0.049	0.062
Formulas C and A	0.5°×0.5°	-10.660	9.713	-2.813	3.362
	1°×1°	-10.663	9.709	-2.814	3.364
	3°×3°	-10.662	9.710	-2.814	3.364

Formulas compared	Near-zone area	Min	Max	Mean	RMS
Formulas B and A	0.5°×0.5°	-0.241	-0.012	-0.070	0.082
	1°×1°	-0.445	-0.031	-0.145	0.168
	3°×3°	-1.146	-0.157	-0.499	0.556
Formulas C and A	0.5°×0.5°	-3.399	0.045	-0.703	0.899
	1°×1°	-2.992	0.132	-0.553	0.740
	3°×3°	-1.717	1.245	0.154	0.405

Topographic effects	Resolution	Min	Max	Mean	RMS
Direct effects /mGal	30″×30″	-54.342	90.365	8.240	14.518
Direct effects /mGal	3″×3″	-53.128	92.032	10.244	15.694
Indirect effects /m	30″×30″	0.007	0.764	0.151	0.187
Indirect effects /m	3″×3″	0.007	0.766	0.151	0.187

Topographic effects	Resolution	Min	Max	Mean	RMS
Direct effects /mGal	30″×30″	-0.568	5.304	3.837	4.040
Direct effects /mGal	3″×3″	-0.158	-0.015	-0.106	0.110
Indirect effects /cm	30″×30″	-0.187	0.686	0.048	0.161
Indirect effects /cm	3″×3″	-0.964	-0.060	-0.291	0.341