Ellipsoid expansion method in establishing independent coordinate system in high-altitude area
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摘要: 分析工程控制网投影长度变形产生的原因及消除长度变形的方法，详述椭球膨胀法原理和计算椭球长半径变化量的4种方法。以高海拔地区某工程GNSS控制网为实例，采用平均曲率半径法、卯酉曲率半径法、平面解析法、广义微分法4种不同的椭球长半径变化量计算方法建立独立坐标系，比较分析4种方法计算得出的椭球长半径变化量、控制点的纬度及大地高变化量，全站仪实测距离的验证结果表明4种计算方法获得的高斯平面坐标精度基本一致，但平面解析法与广义微分法最严密，结论可为高海拔地区建立独立坐标系提供参考。Abstract: The causes of projection length deformation in engineering control networks and the methods to eliminate length deformation were analyzed. The principle of the ellipsoid expansion method and four methods for calculating the change in ellipsoid length radius were detailed. Taking the GNSS control network of a certain project in a high-altitude area as an example, four different methods for calculating the change in ellipsoid length radius, including the average curvature radius method, the Mao You curvature radius method, the plane analysis method, and the generalized differentiation method, were used to establish an independent coordinate system. The changes in ellipsoid length radius, latitude of control points, and ground height calculated by the four methods were compared and analyzed, and verified by the measured distance of the total station. It is believed that the Gaussian plane coordinate accuracy obtained by the four calculation methods is the same. Still, the plane analysis method and the generalized differentiation method are the most rigorous. The conclusion can provide a reference for establishing independent coordinate systems in high-altitude areas.
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表 1 椭球长半轴改变量
平均曲率半径法 卯酉曲率半径法 平面解析法 广义微分法 椭球长半轴改变量/m 3855.4846534 3846.2771717 3853.7264424 3853.7264317 
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表 2 测区中心点及5个控制点的纬度及大地高变化量
点号 变化量 平均曲率半径法 卯酉曲率半径法 平面解析法 广义微分法 中心点 纬度/（″） 0.18930877 0.18885667 0.18922244 0.18922244 大地高/m −1.75652156 7.44205677 −0.00001077 0.00000000 K1 纬度/（″） 0.18933703 0.18888487 0.18925069 0.18925069 大地高/m −1.75463028 7.44394353 0.00187965 0.00189042 K2 纬度/（″） 0.18932270 0.18887057 0.18923636 0.18923636 大地高/m −1.75558969 7.44298641 0.00092067 0.00093144 K3 纬度/（″） 0.18931672 0.18886460 0.18923038 0.18923038 大地高/m −1.75598989 7.44258717 0.00052066 0.00053143 K4 纬度/（″） 0.18929674 0.18884467 0.18921042 0.18921042 大地高/m −1.75732568 7.44125456 −0.00081453 −0.00080376 K5 纬度/（″） 0.18927062 0.18881861 0.18918430 0.18918430 大地高/m −1.75907177 7.43951265 −0.00255982 −0.00254905
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表 3 首级控制点的高斯平面坐标与CGCS2000下的高斯平面坐标差值
方法 K1 K2 K3 K4 K5 ΔX/m ΔY/m ΔX/m ΔY/m ΔX/m ΔY/m ΔX/m ΔY/m ΔX/m ΔY/m 平均曲率半径 2181.218 −2.065 2180.902 −2.026 2180.772 −1.490 2180.333 −1.606 2179.759 −1.730 卯酉圈半径法 2176.022 −2.060 2175.708 −2.021 2175.578 −1.486 2175.140 −1.602 2174.567 −1.725 平面解析法 2180.225 −2.064 2179.910 −2.024 2179.780 −1.489 2179.340 −1.605 2178.767 −1.728 广义微分法 2180.225 −2.064 2179.910 −2.024 2179.780 −1.489 2179.340 −1.605 2178.767 −1.728
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表 4 控制点间反算距离与实测距离对比表
距离起始点 不同坐标系下控制点间的反算距离/m S0（全站仪实测距离）/m (S1-S0)
/m(S2-S0)
/m(S3-S0)
/m(S4-S0)
/mS1(平均曲率半径法) S2(卯酉圈半径法) S3(平面解析法) S4(广义微分法) K1—K2 525.179 525.178 525.179 525.179 525.187 −0.008 −0.009 −0.008 −0.008 K2—K3 914.726 914.724 914.725 914.725 914.742 −0.016 −0.018 −0.017 −0.017 K3—K4 750.893 750.892 750.893 750.893 750.905 0.008 0.007 0.008 0.008 K4—K5 969.721 969.719 969.721 969.721 969.739 −0.018 −0.02 −0.018 −0.018
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