超高阶全球地形球谐系数模型的构建方法与实现

doi:10.11947/j.AGCS.2023.20220003

测绘学报 ›› 2023, Vol. 52 ›› Issue (5): 748-759.doi: 10.11947/j.AGCS.2023.20220003

超高阶全球地形球谐系数模型的构建方法与实现

单建晨^1,2, 李姗姗¹, 范雕¹, 李新星¹, 黄炎¹, 邢志斌³

1. 信息工程大学, 河南郑州 450001;
2. 31201部队, 广东广州 510080;
3. 航天工程大学, 北京 102200

收稿日期:2022-01-07 修回日期:2022-12-15 发布日期:2023-05-27
作者简介:单建晨(1997-),男,硕士,研究方向为物理大地测量。E-mail:1170633968@qq.com
基金资助:
国家自然科学基金(42174007;42204009)

Design and implementation of ultra-high-degree spherical harmonic model of earth topography

SHAN Jianchen^1,2, LI Shanshan¹, FAN Diao¹, LI Xinxing¹, HUANG Yan¹, XING Zhibin³

1. Information Engineering University, Zhengzhou 450001, China;
2. Troops 31201, Guangzhou 510080, China;
3. University of Aerospace Engineering, Beijing 102200, China

Received:2022-01-07 Revised:2022-12-15 Published:2023-05-27
Supported by:
The National Natural Science Foundation of China(Nos. 42174007;42204009)

摘要/Abstract

摘要： 针对10 800阶地形球谐系数模型构建的计算精度、计算稳定性及超大规模运算量等问题,本文首先通过比较分析矩形离散积分方法、Gauss-Legendre积分方法和Driscoll/Healy积分方法,验证了Driscoll/Healy积分方法具有更高的计算精度。然后,给出了改进的Belikov公式,将完全正常化缔和勒让德函数在全球范围内以优于10^-12精度高效、稳定递推至10 800阶;提出了联合FFT和基于OpenMP的多核并行方法求解超高阶球谐系数的优化策略,显著提高了球谐系数模型构建的计算效率。最后,利用Earth2014_TBI与全球海底地形模型STO_IEU2020格网数据建立了全球10 800阶地形球谐系数模型sph.10 800_IEU,总体精度与Earth2014_TBI2014.shc相当,在试验海域的相对精度略优于模型Earth2014_TBI2014.shc。

关键词: 地形球谐系数模型, Gauss-Legendre积分法, Driscoll/Healy积分法, 矩形离散积分法, FFT, OpenMP

Abstract: Aiming at solving the problems of calculation accuracy, stability and the large scale operations in the construction of 10 800 degree spherical harmonic of topography, the experiments were carried out. For the first part, Driscoll/Healy quadrature was proved to have higher computational accuracy by comparing and analyzing rectangular discrete quadrature, Gauss-Legendre quadrature and Driscoll/Healy quadrature. For the second part, a modified Belikov formula was given, which could make sure that fnALF is recursed to order 10 800 with better accuracy than 10^-12 in the full latitude range. Also, an optimization strategy was proposed for computing ultra-high-degree spherical harmonic coefficients by combining FFT and multi-core parallel technology based on OpenMP, which significantly improved the computational efficiency. For the last part, the 10 800 degree spherical harmonic model of topography, sph. 10 800_IEU, whose overall accuracy was similar to Earth2014_TBI2014.shc, was established by using the grid data of model Earth2014_TBI and model STO_IEU2020, however, its relative precision in the experimental areas was proved slightly better than that of model Earth2014_TBI2014.shc.

Key words: spherical harmonic model of topography, Gauss Legendre quadrature, Driscoll/Healy quadrature, rectangular discrete quadrature, FFT, OpenMP

中图分类号:

P223

单建晨, 李姗姗, 范雕, 李新星, 黄炎, 邢志斌. 超高阶全球地形球谐系数模型的构建方法与实现[J]. 测绘学报, 2023, 52(5): 748-759.

SHAN Jianchen, LI Shanshan, FAN Diao, LI Xinxing, HUANG Yan, XING Zhibin. Design and implementation of ultra-high-degree spherical harmonic model of earth topography[J]. Acta Geodaetica et Cartographica Sinica, 2023, 52(5): 748-759.

参考文献

[1] HIRT C, FEATHERSTONE W E, MARTI U. Combining EGM2008 and SRTM/DTM2006.0 residual terrain model data to improve quasigeoid computations in mountainous areas devoid of gravity data[J]. Journal of Geodesy, 2010, 84(9):557-567.
[2] 李军, 欧阳明达, 李琦. 利用EGM2008+DTM2006.0模型精化区域似大地水准面[J]. 大地测量与地球动力学, 2018, 38(3):244-248. LI Jun, OUYANG Mingda, LI Qi. Refinement of regional quasi-geoid using the EGM2008+DTM2006.0 model[J]. Journal of Geodesy and Geodynamics, 2018, 38(3):244-248.
[3] HIRT C, REXER M, CLAESSENS S. Topographic evaluation of fifth-generation GOCE gravity field models-globally and regionally[J]. Newton's Bulletin, 2015.
[4] PAIL R, FECHER T, BARNES D, et al. Short note:the experimental geopotential model XGM2016[J]. Journal of Geodesy, 2018, 92(4):443-451.
[5] FENG Yan, JIANG Yong, JIANG Yi, et al. Spherical cap harmonic analysis of regional magnetic anomalies based on CHAMP satellite data[J]. Applied Geophysics, 2016, 13(3):561-569.
[6] KUHN M, HIRT C. Topographic gravitational potential up to second-order derivatives:an examination of approximation errors caused by rock-equivalent topography (RET)[J]. Journal of Geodesy, 2016, 90(9):883-902.
[7] JIN S G, JIN R, LI D. Assessment of BeiDou differential code bias variations from multi-GNSS network observations[J]. Annales Geophysicae, 2016, 34(2):259-269.
[8] 程娜. 基于多源数据的电离层异常监测及GNSS影响效应研究[J]. 测绘学报, 2021, 50(9):1277. DOI:10.11947/j.AGCS.2021.20200382. CHENG Na. Ionospheric anomaly monitoring based on multi-source data and study on GNSS effect[J]. Acta Geodaetica et Cartographica Sinica, 2021, 50(9):1277. DOI:10.11947/j.AGCS.2021.20200382.
[9] 梁伟, 徐新禹, 李建成, 等. 联合EGM2008模型重力异常和GOCE观测数据构建超高阶地球重力场模型SGG-UGM-1[J]. 测绘学报, 2018, 47(4):425-434. DOI:10.11947/j.AGCS.2018.20170269. LIANG Wei, XU Xinyu, LI Jiancheng, et al. The determination of an ultra-high gravity field model SGG-UGM-1 by combining EGM2008 gravity anomaly and GOCE observation data[J]. Acta Geodaetica et Cartographica Sinica, 2018, 47(4):425-434. DOI:10.11947/j.AGCS.2018.20170269.
[10] DRISCOLL J R, HEALY D M. Computing Fourier transforms and convolutions on the 2-sphere[J]. Advances in Applied Mathematics, 1994, 15(2):202-250.
[11] SNEEUW N, BUN R. Global spherical harmonic computation by two-dimensional Fourier methods[J]. Journal of Geodesy, 1996, 70(4):224-232.
[12] 吴星. 地球重力场调和分析方法研究[D]. 郑州:信息工程大学,2005. WU Xing. Research of methods of spherical harmonic analysis of the Earth's gravity field[D]. Zhengzhou:Information Engineering University, 2005.
[13] SUN Rong. New algorithms for spherical harmonic analysis of area mean values over blocks delineated by equiangular and Gaussian grids[J]. Journal of Geodesy, 2021, 95(5):47.
[14] 黄谟涛, 翟国君, 欧阳永忠, 等. 超高阶地球位模型的计算与分析[J]. 测绘学报, 2001, 30(3):208-213. HUANG Motao, ZHAI Guojun, OUYANG Yongzhong, et al. Analysis and computation of ultra high degree geopotential model[J]. Acta Geodaetica et Cartographic Sinica, 2001, 30(3):208-213.
[15] 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.
[16] FUKUSHIMA T. Numerical computation of spherical harmonics of arbitrary degree and order by extending exponent of floating point numbers[J]. Journal of Geodesy, 2012, 86(4):271-285.
[17] CHRISTIAN H. 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.
[18] XING Zhibin, LI Shanshan, TIAN Miao, et al. Numerical experiments on column-wise recurrence formula to compute fully normalized associated Legendre functions of ultra-high degree and order[J]. Journal of Geodesy, 2020, 94(1):2.
[19] RIZOS C. An efficient computer technique for the evaluation of geopotential from spherical harmonics[J]. Journal of Geodesy, 1979(11):161-169.
[20] COLOMBO O L. Numerical methods for harmonic analysis on the sphere[D]. Ohio:The Ohio State University, 1981.
[21] SNEEUW N. Global spherical harmonic analysis by least-squares and numerical quadrature methods in historical perspective[J]. Geophysical Journal International, 1994, 118(3):707-716.
[22] 刘聪. 地球重力场椭球谐展开的数值算法与实现[D]. 武汉:武汉大学, 2020. LIU Cong. Numerical algorithm and implementation of ellipsoid harmonic expansion of earth gravity field[D]. Wuhan:Wuhan University, 2020.
[23] REXER M, HIRT C. Ultra-high-degree surface spherical harmonic analysis using the Gauss-Legendre and the Driscoll/Healy quadrature theorem and application to planetary topography models of earth, Mars and Moon[J]. Surveys in Geophysics, 2015, 36(6):803-830.
[24] 黄炎,王庆宾,冯进凯,等. 局部地形改正快速计算的GPU并行的棱柱法[J]. 测绘学报, 2020, 49(11):1430-1437. DOI:10.11947/j.AGCS.2020.20170403. HUANG Yan, WANG Qingbin, FENG Jinkai, et al. Rapid calculation of local topographic correction based on GPU parallel prism method[J]. Acta Geodaetica et Cartographica Sinica, 2020,49(11):1430-1437. DOI:10.11947/j.AGCS.2020.20170403.
[25] 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. 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.
[26] 范雕. 卫星测高重力数据反演海底地形的理论和方法研究[D]. 郑州:信息工程大学, 2018. FAN Diao. Study on the theory and method of inversion of seabed topography from satellite altimetry gravity data[D]. Zhengzhou:Information Engineering University, 2018.

超高阶全球地形球谐系数模型的构建方法与实现

Design and implementation of ultra-high-degree spherical harmonic model of earth topography

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