Fast Realization of Ultra-high-degree Geopotential Model by Improved Least-squares Method

doi:10.11947/j.AGCS.2018.20170659

Abstract

Abstract: Using the least-squares method to solve the ultra-high-degree geopotential model is very difficult because this method needs to calculate massive data.In this paper,we first analyzed the characteristics of coefficient matrix and normal matrix based on the basic observation equation of geopotential model using the orthogonality of trigonometric function and surface coherence factor.Then,we reduced the matrix solution equation and designed the calculation and storage method of Legendre function based on normal matrix block diagonalization using the increment characteristic of "secondary M" during the process of solving normal matrix by coefficient matrix solution method.At last,the problems of storage and low computational efficiency of large matrix were solved by combining the equator symmetry characteristic of Legendre function,reduction of the matrix solution equation and design of the calculation and storage method of Legendre function.Through the experimental test,the improved method compared to traditional block diagonal method canimprove the efficiency by 300 times,by using this method we can solve ultra-high-degree geopotential model in ordinary PC,and the precision of the model has been improved by 5 orders of magnitude at least when compared with the numerical integral method.At the same time,the accuracy of the original data can be evaluated.

Key words: block-diagonal, least-squares, ultra-high-degree geopotential model, Legendre function

CLC Number:

P223

TIAN Jialei, LI Xinxing, WU Xiaoping, XING Zhibin. Fast Realization of Ultra-high-degree Geopotential Model by Improved Least-squares Method[J]. Acta Geodaetica et Cartographica Sinica, 2018, 47(11): 1437-1445.

References

[1] 刘晓刚. GOCE卫星测量恢复地球重力场模型的理论与方法[D]. 郑州:信息工程大学, 2011. LIU Xiaogang. Theory and Methods of the Earth's Gravity Field Model Recovery from GOCE Data[D]. Zhengzhou:Information Engineering University, 2011.
[2] 王正涛, 党亚民, 晁定波. 超高阶地球重力位模型确定的理论与方法[M]. 北京:测绘出版社, 2011. WANG Zhengtao, DANG Yamin, CHAO Dingbo. Theory and Methodology of Ultra-high-degree Geopotential Model Determination[M]. Beijing:Surveying and Mapping Press, 2011.
[3] 李新星. 超高阶地球重力场模型的构建[D]. 郑州:信息工程大学, 2013. LI Xinxing. Building of an Ultra-high-degree Geopotential Model[D]. Zhengzhou:Information Engineering University, 2013.
[4] HOFMANN-WELLENHOF B, MORITZ H. Physical Geodesy[M]. Vienna:Springer, 2006.
[5] 宁津生, 邱卫根, 陶本藻. 地球重力场模型理论[M]. 武汉:武汉测绘科技大学出版社, 1990. NING Jinsheng, QIU Weigen, TAO Benzao. Theory of Geopotential Model Determination[M]. Wuhan:Wuhan University Press, 1990.
[6] 陆仲连. 地球重力场理论与方法[M]. 北京:解放军出版社, 1996. LU Zhonglian. Theory and Method of Earth's Gravity[M]. Beijing:The People's Liberation Army Press, 1996.
[7] 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. Approximation Theory of the Earth Gravity Field and Determination of the Chinese Gravity Geoid Model 2000[M]. Wuhan:Wuhan University Press, 2003.
[8] 海斯卡涅W A, 莫里斯H. 物理大地测量学[M]. 卢福康, 胡国理, 译. 北京:测绘出版社, 1979. HEISKANEN W, MORITZ H. Physical Geodesy[M]. LU F K, HU G L, trans. Beijing:Surveying and Mapping Press, 1979.
[9] 许厚泽, 陆仲连. 中国地球重力场与大地水准面[M]. 北京:解放军出版社, 1997. XU Houze, LU Zhonglian. Chinese Earth Gravity Field and Geoid[M]. Beijing:The People's Liberation Army Press, 1997.
[10] GILARDONI M, REGUZZONI M, SAMPIETRO D. GECO:A Global Gravity Model by Locally Combining GOCE Data and EGM2008[J]. Studia Geophysica et Geodaetica, 2016, 60(2):228-247.
[11] HIRT C, REXER M, SCHEINERT M, et al. A New Degree-2190(10 km Resolution) Gravity Field Model for Antarctica Developed from GRACE, GOCE and Bedmap2 Data[J]. Journal of Geodesy, 2016, 90(2):105-127.
[12] 章传银, 郭春喜, 陈俊勇, 等. EGM2008地球重力场模型在中国大陆适用性分析[J]. 测绘学报, 2009, 38(4):283-289. ZHANG Chuanyin, GUO Chunxi, CHEN Junyong, et al. EGM 2008 and Its Application Analysis in Chinese Mainland[J]. Acta Geodaetica et Cartographica Sinica, 2009, 38(4):283-289.
[13] 束蝉方, 李斐, 郝卫峰. EGM2008模型在中国某地区的检核及适用性分析[J]. 武汉大学学报(信息科学版), 2011, 36(8):919-922. SHU Chanfang, LI Fei, HAO Weifeng. Evaluation of EGM 2008 and Its Application Analysis over a Particular Region of China[J]. Geomatics and Information Science of Wuhan University, 2011, 36(8):919-922.
[14] KOSTELECKY' J, KLOKOČNÍK J, BUCHA B, et al. Evaluation of Gravity Field Model EIGEN-6C4 by Means of Various Functions of Gravity Potential, and by GNSS/Levelling[J]. Geoinformatics FCE CTU, 2015, 14(1):7-28.
[15] PAVLIS N K, HOLMES S A, KENYON S C, et al. Correction to "The Development and Evaluation of the Earth Gravitational Model 2008(EGM2008)"[J]. Journal of Geophysical Research:Solid Earth, 2013, 118(5):2633.
[16] FÖRSTE C, BRUINSMA S L, ABRIKOSOV O, et al. EIGEN-6C4:The Latest Combined Global Gravity Field Model Including GOCE Data up to Degree and Order 1949 of GFZ Potsdam and GRGS Toulouse[C]//EGU General Assembly. Vienna, Austria:EGU, 2014.
[17] 郑伟, 许厚泽, 钟敏, 等. 地球重力场模型研究进展和现状[J]. 大地测量与地球动力学. 2010, 30(4):83-91. ZHENG Wei, XU Houze, ZHONG Min, et al. Progress and Present Status of Research on Earth's Gravitational Field Models[J]. Journal of Geodesy and Geodynamics, 2010, 30(4):83-91.
[18] 陈秋杰, 沈云中, 张兴福, 等. 基于GRACE卫星数据的高精度全球静态重力场模型[J]. 测绘学报, 2016, 45(4):396-403. DOI:10.11947/j.AGCS.2016.20150422. CHEN Qiujie, SHEN Yunzhong, ZHANG Xingfu, et al. GRACE Data-based High Accuracy Global Static Earth's Gravity Field Model[J]. Acta Geodaetica et Cartographica Sinica, 2016, 45(4):396-403. DOI:10.11947/j.AGCS.2016.20150422.
[19] 赫林, 李建成, 褚永海. 联合GRACE/GOCE重力场模型和GPS/水准数据确定我国85高程基准重力位[J]. 测绘学报, 2017, 46(7):815-823. DOI:10.11947/j.AGCS.2017.20160643. HE Lin, LI Jiancheng, CHU Yonghai. Evaluation of the Geopotential Value for the Local Vertical Datum of China Using GRACE/GOCE GGMs and GPS/Leveling Data[J]. Acta Geodaetica et Cartographica Sinica, 2017, 46(7):815-823. DOI:10.11947/j.AGCS.2017.20160643.
[20] 李新星, 吴晓平, 李姗姗, 等. 块对角最小二乘方法在确定全球重力场模型中的应用[J]. 测绘学报, 2014, 43(8):778-785. DOI:10.13485/j.cnki.11-2089.2014.0110. LI Xinxing, WU Xiaoping, LI Shanshan, et al. The Application of Block-diagonal Least-squares Methods in Geopotential Model Determination[J]. Acta Geodaetica et Cartographica Sinica, 2014, 43(8):778-785. DOI:10.13485/j.cnki.11-2089.2014.0110.
[21] 周浩, 罗志才, 钟波, 等. 利用最小二乘直接法反演卫星重力场模型的MPI并行算法[J]. 测绘学报, 2015, 44(8):833-839. DOI:10.11947/j.AGCS.2015.20140396. ZHOU Hao, LUO Zhicai, ZHONG Bo, et al. MPI Parallel Algorithm in Satellite Gravity Field Model Inversion on the Basis of Least Square Method[J]. Acta Geodaetica et Cartographica Sinica, 2015, 44(8):833-839. DOI:10.11947/j.AGCS.2015.20140396.
[22] 邢志斌, 李姗姗, 王伟, 等. 利用垂线偏差计算高程异常差法方程的快速构建方法[J]. 武汉大学学报(信息科学版), 2016, 41(6):778-783. XING Zhibin, LI Shanshan, WANG Wei, et al. Fast Approach to Constructing Normal Equation During the Time of Calculating Height Anomaly Difference by Using Vertical Deflections[J]. Geomatics and Information Science of Wuhan University, 2016, 41(6):778-783.
[23] 宋力杰, 欧阳桂崇. 超大规模大地网分区平差快速解算方法[J]. 测绘学报, 2003, 32(3):204-207. SONG Lijie, OUYANG Guichong. A Fast Method of Solving Partitioned Adjustment for Super Large-scale Geodetic Network[J]. Acta Geodaetica et Cartographica Sinica, 2003, 32(3):204-207.
[24] PAVLIS N K. Modeling and Estimation of a Low Degree Geopotential Model from Terrestrial Gravity Data[R]. Columbus:The Ohio State University, 1988.
[25] 梁伟. 基于块对角最小二乘方法构建超高阶重力场模型[D]. 武汉:武汉大学, 2016 LIANG Wei. The Determination of Ultra-high Gravity Field Model Based on Block-diagonal Least Squares Method[D]. Wuhan:Wuhan University, 2016.
[26] 张传定, 许厚泽, 吴星. 地球重力场调和分析中的"轮胎"问题[C]//2004年重力学与固体潮学术研讨会暨祝贺许厚泽院士70寿辰研讨会会议论文集. 武汉:中国地球物理学会, 中国测绘学会, 2004:302-314. ZHANG Chuanding, XU Houze, WU Xing. On the Torus Approach for Gobal Sherical Harmonic Computation[C]//Proceedings of Geodesy and Geodynamics Development. Wuhan:Publishing house of Hubei Science and Technology, 2004:302-314.