﻿ 病态不确定性平差模型的岭估计算法
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1. 东华理工大学测绘工程学院, 江西 南昌 330013;
2. 流域生态与地理环境监测国家测绘地理信息局重点实验室, 江西 南昌 330013;
3. 江西省数字国土重点实验室, 江西 南昌 330013;
4. 南昌航空大学, 江西 南昌 330063

Ridge estimation algorithm to ill-posed uncertainty adjustment model
LU Tieding1,2,3, WU Guangming1, ZHOU Shijian4
1. Faculty of Geomatics, East China University of Technology, Nanchang 330013, China;
2. Key Laboratory of Watershed Ecology and Geographical Environment Monitoring, NASMG, Nanchang 330013, China;
3. Jiangxi Province Key Lab for Digital Land, Nanchang 330013, China;
4. Nanchang Hangkong University, Nanchang 330063, China
Abstract: Uncertainties usually exist in the process of acquisition of measurement data, which affects the parameter estimation results. The solution method of uncertainty adjustment model can effectively improve the validity and reliability of parameter estimation. When the coefficient matrix of the observation equation has a singular value close to zero, the ridge estimation can effectively suppress the influence of the ill-posed state of the observation equation on the parameter estimation results.When the uncertainty adjustment model is ill-posed, it is more seriously affected by the error of the coefficient matrix and the observation, this paper applies ridge estimation method to ill-posed uncertainty adjustment model, derives an iterative algorithm to improve the stability and reliability of the result, and verifies it with two examples. The results show that the new method is effective and feasible.
Key words: ill-posed    uncertainty    adjustment model    ridge estimation

1 不确定性平差模型及平差准则

(1)

(2)

(3)

(4)

，将式(4)变为迭代计算形式

(5)

2 病态不确定性平差模型的岭估计

(6)

(7)

(8a)
(8b)
(8c)
(8d)
(8e)
(8f)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

3 病态不确定性平差模型岭估计解算步骤

(1) 在计算时先给出系数矩阵A、观测向量L、不确定度φβ

(2) 设定迭代初值，在迭代初值选择上，采用最小二乘估值或其他估值作为

(3) 在一范围[ab]内按一定步长Δd选择α

(4) 确定uμ，确定公式参考文献[10]的方法，计算公式为

(16a)
(16b)

(a)当uμ均大于0时

(b) 当uμ均小于等于0时

(c) 当u>0、μ≤0时

(d) 当u≤0、μ>0时

(5) 根据式(14)进行迭代计算

(17)

(ε为阈值)时，计算终止。重复步骤(3)、(4)、(5)直至α在[ab]内选择结束。

(6) 绘制的L-曲线图，确定拐点，得到此时的岭参数，并将此时的参数估值作为病态不确定性岭估计解。

4 算例及分析 4.1 算例1

 方案 真值 LS TLS ULS最优 R-LS R-TLS R-ULS最优 1 1.394 4 3.305 1 2.808 8 1.215 7 1.210 2 1.203 2 1 0.122 3 -2.804 8 -2.043 2 0.372 8 0.377 6 0.382 8 1 0.779 1 0.059 6 0.247 1 0.828 0 0.826 1 0.822 5 1 0.262 8 -3.589 4 -2.588 3 0.598 3 0.603 4 0.607 9 1 1.441 4 2.903 4 2.523 0 1.315 7 1.313 1 1.310 3 0 1.308 8 6.735 0 5.319 4 0.854 7 0.846 8 0.838 9

 图 1 各方法L-曲线图 Fig. 1 The L-curve of each methods

 图 2 不同区间结果 Fig. 2 Different interval results

 图 3 不同区间结果 Fig. 3 Different interval results

 图 4 φ、β(0，0.1]结果 Fig. 4 φ、β(0, 0.1) results

4.2 算例2

 方案 真值 LS TLS ULS R-LS R-TLS R-ULS最优 0 0.053 0 不收敛 不收敛 0.052 4 0.044 4 0.041 2 0 -0.084 6 -0.069 8 -0.022 8 -0.012 6 0 -0.805 3 -0.681 9 -0.163 7 -0.057 5 68 68.040 0 68.040 6 68.039 5 68.031 3 -26 -26.030 3 -25.905 3 -25.912 9 -25.930 5 9 8.511 3 8.947 2 8.961 6 8.964 4 14 14.007 2 14.007 8 14.006 7 14.008 4 41 40.808 0 40.945 3 40.965 4 40.968 1 -11 -11.585 7 -11.131 6 -11.028 4 -11.012 2 0 1.131 3 — — 0.711 6 0.204 8 0.116 1

 图 5 各方法L-曲线图 Fig. 5 The L-curve of each method

 图 6 不同区间结果 Fig. 6 Different interval results

 图 7 不同区间结果 Fig. 7 Different interval results

 图 8 φ、β(0，0.1]结果 Fig. 8 Results of φ、β(0, 0.1)

4.3 算例分析

5 结论

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http://dx.doi.org/10.11947/j.AGCS.2019.20180044

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#### 文章信息

LU Tieding, WU Guangming, ZHOU Shijian

Ridge estimation algorithm to ill-posed uncertainty adjustment model

Acta Geodaetica et Cartographica Sinica, 2019, 48(4): 403-411
http://dx.doi.org/10.11947/j.AGCS.2019.20180044