FFT block-diagonal least square method for constructing spherical harmonic model of earth topography with degree 10 800

Abstract

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

CLC Number:

P223

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