10 800阶地形球谐系数模型构建的FFT块对角最小二乘法

摘要/Abstract

摘要：

针对10 800阶地形球谐系数模型构建中最小二乘法计算量大、法方程不易构建与解算等问题，本文利用三角函数的正交性构建了解算地形球谐系数的块对角最小二乘平差模型，推导了计算法方程自由项的快速傅里叶变换技术（FFT）表达式，构建了FFT块对角最小二乘平差模型，推导了构建地形球谐系数模型的FFT调和分析表达式。引入了X-数法计算完全规格化缔合勒让德函数（fnALFs），实现了次数m固定、阶数n变化计算球谐系数，降低了计算机内存的占用率。仿真试验表明，FFT块对角最小二乘法在计算精度上优于FFT调和分析法。本文利用Earth2014_TBI全球地形数据，通过块对角最小二乘法构建了10 800阶的地形球谐系数模型LS_10800.shc。试验表明：LS_10800.shc模型在全球范围的精度达到了9.31 m，优于Earth2014. TBI2014.degree10800.bshc模型10.15 m的精度，在我国及周边地区的精度达到了18.79 m，优于Earth2014. TBI2014.degree10800.bshc模型20.69 m的精度；与2000个GPS点相比，LS_10800.shc模型精度达到32.85 m，优于Earth2014. TBI2014.degree10800.bshc模型34.38 m的精度。

关键词: 地形球谐系数, FFT块对角最小二乘法, FFT调和分析法, 正交性

Abstract:

Regarding the large computational burden of the least square method and the difficulty in constructing and solving the normal equations in constructing spherical harmonic model of earth topography with degree 10 800, we utilized the orthogonality of trigonometric functions to construct a block-diagonal least square adjustment model for determining topography spherical harmonic coefficients, derived the FFT expression for calculating the free term of the normal equations, and established the FFT block-diagonal least square adjustment model. We also derived the FFT harmonic analysis method for computing the spherical harmonic coefficients of topography. The X-number method was introduced for calculating fully normalized associated Legendre functions (fnALFs), enabling the computation of spherical harmonic coefficients with fixed m and varying n, thereby reducing the memory usage of the computer. In simulation experiments, it has been verified that the FFT block diagonal least square method is superior to the FFT harmonic analysis method in terms of computational accuracy. Finally, we constructed a spherical harmonic coefficient model with degree 10 800 named LS_10800. shc using Earth2014_TBI global topography data through the block-diagonal least square method. Experiments show that the LS_10800. shc model achieves a global accuracy of 9.31 m, outperforming the Earth2014. TBI2014. degree10800. bshc model, which has a precision of 10.15 m. In China and its surrounding areas, the accuracy of the LS_10800. shc model reaches 18.79 m, outperforming the Earth2014. TBI2014. degree10800. bshc model, which has an accuracy of 20.69 m. Compared to 2000 GPS points, the LS_10800. shc model achieved an accuracy of 32.85 m, which is superior to the accuracy of 34.38 m achieved by the Earth2014. TBI2014. degree10800. bshc model.

Key words: spherical harmonic model of earth topography, FFT block-diagonal least square method, FFT harmonic analysis method, orthogonality

中图分类号:

P223

邢志斌, 李姗姗, 田苗, 孟瑶, 杨娜, 李倩, 常品要. 10 800阶地形球谐系数模型构建的FFT块对角最小二乘法[J]. 测绘学报, 2026, 55(1): 36-45.

Zhibin XING, Shanshan LI, Miao TIAN, Yao MENG, Na YANG, Qian LI, Pinyao CHANG. FFT block-diagonal least square method for constructing spherical harmonic model of earth topography with degree 10 800[J]. Acta Geodaetica et Cartographica Sinica, 2026, 55(1): 36-45.

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模型	最大值	最小值	平均值	RMS
Analysis_10800_simul.shc	514.7	－1 071.2	－0.04	11.8
LS_10800_simul.shc	2.3×10^－9	－2.4×10^－9	1.3×10^－12	4.2×10^－11

模型	最大值/m	最小值/m	平均值/m	std/m	≤std的比例/（%）	≤2std的比例/（%）	≤3std的比例/（%）
LS_10800.shc	1 698.60	－1 849.17	0.00	9.31	92.51	96.55	98.05
Earth2014TBI2014.shc	1 606.91	－1 837.77	－0.006	10.15	92.51	96.45	97.97

模型	误差最大值/m	误差最小值/m	误差平均值/m	误差std/m	≤std的比例/（%）	≤2std的比例/（%）	≤3std的比例/（%）
LS_10800.shc	354.98	－376.87	0.000 3	18.79	86.49	94.41	97.38
Earth2014TBI2014.shc	428.68	－409.46	0.000 9	20.69	86.52	94.34	97.34

模型	最大值/m	最小值/m	平均值/m	std/m	≤mean+std的比例/（%）	≤mean+2std的比例/（%）	≤mean+3std的比例/（%）
LS_10800.shc	334.51	－316.26	2.86	32.85	89.55	95.15	97.70
Earth2014TBI2014.shc	374.91	－309.83	3.27	34.38	89.55	95.20	97.65
Earth2014. TBI2014.1min.geod.grd	296.24	－268.22	2.28	29.53	89.70	95.40	97.55

模型	区域	最大值/m	最小值/m	平均值/m	std/m	≤mean+std的比例/（%）	≤mean+2std的比例/（%）	≤mean+3std的比例/（%）
LS_10800.shc	山区	334.51	－277.34	6.33	44.29	87.18	95.06	97.41
LS_10800.shc	平原	163.25	－316.26	0.29	20.32	89.65	96.70	98.09
Earth2014TBI2014.shc	山区	374.91	－281.42	7.23	46.60	87.76	95.06	97.18
Earth2014TBI2014.shc	平原	181.78	－309.83	0.34	20.78	89.04	96.43	98.09
Earth2014. TBI2014.1min.geod.grd	山区	296.24	－268.22	5.12	39.74	88.24	95.18	97.53
Earth2014. TBI2014.1min.geod.grd	平原	152.68	－260.57	0.17	18.42	88.61	96.17	98.09