﻿ 等价条件平差模型的方差-协方差分量最小二乘估计方法
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1. 中国矿业大学环境与测绘学院, 江苏 徐州 221116;
2. 皇家墨尔本理工大学空间科学研究中心, 澳大利亚 维多利亚州 墨尔本 3001

Least-square variance-covariance component estimation method based on the equivalent conditional adjustment model
LIU Zhiping1, ZHU Dantong1, YU Hang1, ZHANG Kefei1,2
1. School of Environment Science and Spatial Informatics, China University of Mining and Technology, Xuzhou 221116, China;
2. Space Research Centre, RMIT University, Australia VIC 3001
Abstract: A VCE method termed the least-square variance-covariance component estimation method based on the equivalent conditional misclosure (LSV-ECM) is developed. Three steps are involved. First, the equivalent conditional misclosure is extracted using the projection matrix in the equivalent conditional adjustment model, of which the quadratic equations are established for variance-covariance component estimation. The quadratic equations in the form of matrix are then transformed to the linearized Gauss-Markov form using the half-vectorization operator. A simplified and generalized LSV-ECM method is derived using the least-square principle with an unbiased and optimal estimation.Furthermore, the equivalence between the LSV-ECM and the existing VCE methods is proven mathematically, and computational complexities of the LSV-ECM and the existing VCE methods are quantitatively analyzed and investigated in the indirect adjustment model. It is shown that the new method gives the highest computational efficiency. Finally, the performance and superiority of the new method is evaluated through an adjustment of a triangulateration network and an analysis of a coordinate time series of GNSS stations.
Key words: equivalent conditional adjustment model    variance-covariance component estimation    LSV-ECM method    triangulateration network    GNSS station coordinate time series

1 等价条件闭合差的方差分量最小二乘估计 1.1 等价条件平差模型

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1.2 方差-协方差分量最小二乘估计方法

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F = DW，根据式(4)、式(5)可得

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 参量 模型 条件平差 具有参数的条件平差 间接平差 附有限制的间接平差 B=0, C=0 C=0 A=-I, C=0 A=-I Qi AQiAT HAQiATHT HQiHT HcQiHcT W W HW HW HcW+HsZ DW ADLAT HADLATHT HDLHT HcDLHcT

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 VCE方法 矩阵求逆复杂度 加法乘法复杂度 通用Helmert[7] 4O(n3) k2(6n3－3n2+n－1)+k(6n2－n－1) LS-VCE[9, 11] 4O(n3) k2(6n3－5n2+n－1)+k(6n2－n－1) LSV-ECM 4O(r3) k2(6r3－3r2+r－1)+ k(6r2－r－1)

2 应用结果及分析 2.1 边角网平差

 结果 方案1 方案2 Helmert LS-VCE LSV-ECM Helmert LS-VCE LSV-ECM 3.64, 5.92 3.64, 5.92 3.64, 5.92 0.79, 3.07 0.79, 3.07 0.79, 3.07 -1.56, 0.98 -1.56, 0.98 -1.56, 0.98 -2.90, 0.53 -2.90, 0.53 -2.90, 0.53 0.89.1.03 0.89.1.03 0.89.1.03 0.09, 0.55 0.09, 0.55 0.09, 0.55 5.64, 1.64 5.64, 1.64 5.64, 1.64 3.48, 1.20 3.48, 1.20 3.48, 1.20 -12.38, 1.86 -12.38, 1.86 -12.38, 1.86 -17.80, 1.28 -17.80, 1.28 -17.80, 1.28 Tratio 1 96% 65% 1 97% 71%

2.2 GNSS站坐标时序建模

 图 1 CMONOC所选测站分布 Fig. 1 Geographical distribution of selected stations in the CMONOC network

GNSS站坐标时序的观测方程和随机模型分别为[11]

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 mm 站点 N E U Helmert LS-VCE LSV-ECM Helmert LS-VCE LSV-ECM Helmert LS-VCE LSV-ECM WN, FN WN, FN WN, FN WN, FN WN, FN WN, FN WN, FN WN, FN WN, FN ZHZC 0.80, 2.11 0.80, 2.11 0.80, 2.11 0.78, 2.98 0.78, 2.98 0.78, 2.98 3.43, 9.73 3.43, 9.73 3.43, 9.73 YANC 0.52, 1.67 0.52, 1.67 0.52, 1.67 0.45, 1.87 0.45, 1.87 0.45, 1.87 2.10, 6.73 2.10, 6.73 2.10, 6.73 XIAM 0.79, 3.57 0.79, 3.57 0.79, 3.57 0.93, 3.19 0.93, 3.19 0.93, 3.19 3.78, 13.13 3.78, 13.13 3.78, 13.13 WUHN 0.76, 3.23 0.76, 3.23 0.76, 3.23 0.77, 3.03 0.77, 3.03 0.77, 3.03 3.01, 13.03 3.01, 13.03 3.01, 13.03 TAIN 0.96, 2.31 0.96, 2.31 0.96, 2.31 0.92, 3.10 0.92, 3.10 0.92, 3.10 3.18, 8.77 3.18, 8.77 3.18, 8.77 QION 0.98, 5.20 0.98, 5.20 0.98, 5.20 1.27, 4.63 1.27, 4.63 1.27, 4.63 4.7, 15.01 4.7, 15.01 4.7, 15.01 LUZH 0.68, 2.19 0.68, 2.19 0.68, 2.19 0.62, 2.63 0.62, 2.63 0.62, 2.63 2.73, 12.19 2.73, 12.19 2.73, 12.19 KMIN 0.67, 3.73 0.67, 3.73 0.67, 3.73 0.65, 4.94 0.65, 4.94 0.65, 4.94 3.29, 14.57 3.29, 14.57 3.29, 14.57 JIXN 0.58, 2.52 0.58, 2.52 0.58, 2.52 0.61, 1.98 0.61, 1.98 0.61, 1.98 2.37, 6.75 2.37, 6.75 2.37, 6.75 HRBN 0.61, 3.15 0.61, 3.15 0.61, 3.15 0.33, 3.45 0.33, 3.45 0.33, 3.45 1.50, 11.83 1.50, 11.83 1.50, 11.83 HLAR 0.75, 2.97 0.75, 2.97 0.75, 2.97 0.70, 2.65 0.70, 2.65 0.70, 2.65 2.32, 11.50 2.32, 11.50 2.32, 11.50 GUAN 1.07, 3.41 1.07, 3.41 1.07, 3.41 1.21, 5.08 1.21, 5.08 1.21, 5.08 5.23, 16.22 5.23, 16.22 5.23, 16.22 DLHA 0.41, 2.50 0.41, 2.50 0.41, 2.50 0.51, 2.31 0.51, 2.31 0.51, 2.31 1.51, 10.49 1.51, 10.49 1.51, 10.49 CHUN 0.55, 3.09 0.55, 3.09 0.55, 3.09 0.44, 3.05 0.44, 3.05 0.44, 3.05 1.92, 13.27 1.92, 13.27 1.92, 13.27 BJSH 0.76, 1.99 0.76, 1.99 0.76, 1.99 0.76, 1.35 0.76, 1.35 0.76, 1.35 2.81, 7.45 2.81, 7.45 2.81, 7.45 BJFS 0.61, 3.49 0.61, 3.49 0.61, 3.49 0.62, 2.19 0.62, 2.19 0.62, 2.19 2.73, 7.31 2.73, 7.31 2.73, 7.31 Tratio(WN) 1 99.5% 74.0% 1 99.8% 73.8% 1 99.9% 72.1% Tratio(FN) 1 99.5% 74.0% 1 99.8% 73.8% 1 99.9% 72.1% 注：高斯白噪声(WN)的单位为mm；闪烁噪声(FN)的单位为mm/a0.25

 图 2 噪声分量大小与纬度的关系 Fig. 2 The relationship between noise components and latitude

 mm/a 站点 N E U 本文方法 CMONOC 本文方法 CMONOC 本文方法 CMONOC ZHNZ -11.35±0.07 -11.18±0.15 32.90±0.11 33.17±0.23 1.21±0.31 1.24±0.38 YANC -9.30±0.05 -8.69±0.14 32.57±0.10 32.46±0.05 1.06±0.22 1.02±0.13 XIAM -12.16±0.11 -12.48±0.17 32.47±0.06 32.82±0.17 1.4±0.42 0.78±0.36 WUHN -10.93±0.10 -11.08±-0.12 32.50±0.11 33.60±0.74 -0.79±0.42 0.47±0.41 TAIN -11.54±0.07 -11.58±0.37 30.98±0.10 31.38±0.16 0.92±0.28 1.21±0.30 QION -11.98±0.17 -10.24±0.78 31.57±0.10 31.94±0.15 -0.6±0.48 -0.46±0.32 LUZH -9.75±0.070 -9.61±0.13 34.96±0.15 35.76±0.28 0.46±0.39 0.35±0.30 KMIN -17.23±0.12 -16.18±0.87 33.09±0.08 31.13±0.53 -1.23±0.47 -0.48±0.31 JIXN -9.72±0.08 -10.35±0.06 29.13±0.16 28.66±0.11 1.79±0.22 1.53±0.17 HRBN -12.56±0.10 -12.38±-0.21 25.79±0.06 25.95±0.51 -0.49±0.38 -0.09±0.21 HLAR -10.51±0.09 -11.35±0.04 25.76±0.11 25.88±0.07 2.05±0.37 1.44±0.16 GUAN -11.11±0.11 -11.23±0.10 31.30±0.10 33.11±0.19 -0.33±0.52 -1.97±0.46 CHUN -11.58±0.10 -12.21±0.14 27.37±0.07 26.53±0.52 -1.81±0.42 -0.15±0.29 BJSH -11.45±0.06 -11.28±0.18 30.10±0.10 29.94±0.13 1.36±0.24 1.05±0.29 BJFS -9.94±0.11 -10.18±0.14 30.58±0.04 30.10±0.19 2.63±0.24 -0.11±0.63

3 结论

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http://dx.doi.org/10.11947/j.AGCS.2019.20180227

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#### 文章信息

LIU Zhiping, ZHU Dantong, YU Hang, ZHANG Kefei

Least-square variance-covariance component estimation method based on the equivalent conditional adjustment model

Acta Geodaetica et Cartographica Sinica, 2019, 48(9): 1088-1095
http://dx.doi.org/10.11947/j.AGCS.2019.20180227