超高阶重力场模型最小二乘快速实现

doi:10.11947/j.AGCS.2018.20170659

测绘学报 ›› 2018, Vol. 47 ›› Issue (11): 1437-1445.doi: 10.11947/j.AGCS.2018.20170659

超高阶重力场模型最小二乘快速实现

田家磊^1,2, 李新星^1,3, 吴晓平¹, 邢志斌¹

1. 信息工程大学地理空间信息学院, 河南郑州 450001;
2. 63850部队, 吉林白城 137000;
3. 武汉大学测绘学院, 湖北武汉 430079

收稿日期:2017-11-18 修回日期:2018-05-14 出版日期:2018-11-20 发布日期:2018-11-29
通讯作者: 李新星 E-mail:minibad@126.com
作者简介:田家磊(1988-),男,博士生,研究方向为物理大地测量。E-mail:tianjialei3037@163.com
基金资助:
国家自然科学基金（41274029；41404020；41774018；41504018；41674082）；信息工程大学自立课题（2017503902；2016601002）

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

TIAN Jialei^1,2, LI Xinxing^1,3, WU Xiaoping¹, XING Zhibin¹

1. Institute of Geospatial Information, Information Engineering University, Zhengzhou 450001, China;
2. 63850 troops, Baicheng 137000, China;
3. School of Geodesy and Geomatics, Wuhan University, Wuhan 430079, China

Received:2017-11-18 Revised:2018-05-14 Online:2018-11-20 Published:2018-11-29
Supported by:
The National Natural Science Foundation of China (Nos. 41274029;41404020;41774018;41504018;41674082);The Self-reliance Project of Information Engineering University (Nos. 2017503902;2016601002)

摘要/Abstract

摘要： 采用最小二乘方法解算超高阶重力场模型，不可避免会遇到大型矩阵的计算，直接求解是难以实现的。本文从重力场模型的基本观测方程出发，利用正余弦函数和面球谐因子的正交性，分析系数矩阵及法矩阵的特点，在法矩阵块对角化的基础上，利用系数矩阵求解法矩阵时“次m”递增的特点，对法矩阵求解方程进行约化、对Legendre函数的计算和存储方式进行了设计，结合缔合Legendre函数关于赤道的对称性，解决了大型矩阵存储及计算效率低下的难题，实现了超高阶重力场模型最小二乘方法的小存储、高效率的解算。通过试验模拟，改进后的方法相比传统块对角方法效率提高300倍，利用此方法可以在普通PC机上快速、高精度地解算2160阶次超高阶重力场模型，算法精度相比数值积分方法至少提高了5个数量级，并且在一定程度上可以评估原始观测数据的精度。

关键词: 块对角矩阵, 最小二乘, 超高阶地球重力场模型, Legendre函数

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

中图分类号:

P223

田家磊, 李新星, 吴晓平, 邢志斌. 超高阶重力场模型最小二乘快速实现[J]. 测绘学报, 2018, 47(11): 1437-1445.

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.

参考文献

[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.

超高阶重力场模型最小二乘快速实现

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

PDF

可视化

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 15

编辑推荐

Metrics

本文评价

[1]	王珂, 刘国林, 付政庆, 王路遥. 基于Moore-Penrose广义逆及立体矩阵的可分离非线性最小二乘解算方法[J]. 测绘学报, 2022, 51(3): 340-350.
[2]	鄢咏折, 彭松, 马友青, 张烁, 亓晨, 温博, 李昊, 贾阳, 刘少创. 行星探测器导航相机着陆后参数标定方法[J]. 测绘学报, 2022, 51(3): 437-445.
[3]	闫凤池, 王振杰, 赵爽, 聂志喜, 孙振, 李伟嘉. 顾及双程声径的常梯度声线跟踪水下定位算法[J]. 测绘学报, 2022, 51(1): 31-40.
[4]	刘计洪, 胡俊, 李志伟, 朱建军. InSAR三维同震地表形变监测——窗口优化的SM-VCE算法[J]. 测绘学报, 2021, 50(9): 1222-1239.
[5]	范雕, 李姗姗, 欧阳永忠, 孟书宇, 陈成, 邢志斌, 张驰. 顾及海底地形非线性项的最小二乘配置反演方法[J]. 测绘学报, 2021, 50(7): 953-971.
[6]	郑凯, 李建胜, 王俊强, 欧阳文, 谷友艺, 张迅. DCLS-GAN:利用生成对抗网络的天绘一号卫星高原地区影像去云方法[J]. 测绘学报, 2021, 50(2): 248-259.
[7]	张兵兵, 牛继强, 王正涛, 徐丰, 田坤俊. Swarm系列卫星非差运动学厘米级精密定轨[J]. 测绘学报, 2021, 50(1): 27-36.
[8]	胡川, 方兴, 赵立都. 融合正交几何信息的非线性等式约束整体最小二乘平差及迭代算法[J]. 测绘学报, 2020, 49(7): 816-823.
[9]	解清华, 朱建军, 汪长城, 付海强, 张兵. 基于S-RVoG模型的PolInSAR森林高度非线性复数最小二乘反演算法[J]. 测绘学报, 2020, 49(10): 1303-1310.
[10]	张春森, 张奇源, 郭丙轩, 薛万唱. 倾斜影像自适应初始物方面元优化匹配算法[J]. 测绘学报, 2020, 49(1): 108-116.
[11]	郭锴, 刘焱雄, 徐文学, 何秀凤, 张子引, 刘海涛. 机载激光测深波形分解中LM与EM参数优化方法比较[J]. 测绘学报, 2020, 49(1): 117-131.
[12]	邓兴升, 彭思淳, 游扬声. 加权整体最小二乘EIO模型与算法[J]. 测绘学报, 2019, 48(7): 926-930.
[13]	王乐洋, 余航, 邹传义, 鲁铁定. 加权整体最小二乘EIV模型与算法——与“加权整体最小二乘EIO模型与算法”一文的讨论[J]. 测绘学报, 2019, 48(7): 931-937.
[14]	梁磊, 于锦海, 万晓云. 椭球谐和球谐系数之间一个简单的转换关系[J]. 测绘学报, 2019, 48(2): 185-190.
[15]	王志忠, 宋迎春, 何玲莉. 系数矩阵中部分有界不确定性的混合平差算法[J]. 测绘学报, 2018, 47(9): 1171-1178.