Adjustment Model and Algorithm Based on Ellipsoid Uncertainty

doi:10.11947/j.JGGS.2020.0306

Journal of Geodesy and Geoinformation Science ›› 2020, Vol. 3 ›› Issue (3): 59-66.doi: 10.11947/j.JGGS.2020.0306

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Adjustment Model and Algorithm Based on Ellipsoid Uncertainty

Yingchun SONG^{1, 2}(), Yuguo XIA^{1, 2}, Xuemei XIE³

^1. Key Laboratory of Metallogenic Prediction of Nonferrous Metals and Geological Environment Monitoring Ministry of Education, School of Geoscience and Infophysics,Central South University, Changsha 410083, China
^2. School of Geosciences and Info Physics, Central South University, South Lushan Road, Changsha 410083, China
^3. School of Civil Engineering, Central South University of Forestry and Technology,Changsha 410004, China

Received:2019-11-14 Accepted:2020-05-14 Online:2020-09-20 Published:2020-09-30
About author:Yingchun SONG (1963—), male, Professor, PhD, majors in theory and method of measuring dataprocessing. E-mail: csusyc@csu.edu.cn; csusyc@qq.com
Supported by:
National Natural Science Foundation of China Nos(41674009);National Natural Science Foundation of China Nos(41574006);National Natural Science Foundation of China Nos(41674012)

Abstract

Abstract:

In surveying adjustment models, there is usually some uncertain additional information or prior information on parameters, which can constrain the parameters, and guarantee the uniqueness and stability of parameter solution. In this paper, we firstly use ellipsoidal sets to describe uncertainty, and establish a new adjustment model with ellipsoidal uncertainty. Furthermore, we give a new adjustment criterion based on minimization trace of an outer ellipsoid with two ellipsoid intersections, and analyze the propagation law of uncertainty. Correspondingly, we give a new algorithm for the adjustment model with ellipsoid uncertainty. Finally, we give three examples to test and verify the effectiveness of our algorithm, and illustrate the relation between our result and the weighted mixed estimation.

Key words: uncertain; ellipsoid uncertainty constraint; adjustment model; Ill-posed problem; set membership estimation

Yingchun SONG , Yuguo XIA, XuemeiXIE . Adjustment Model and Algorithm Based on Ellipsoid Uncertainty[J]. Journal of Geodesy and Geoinformation Science, 2020, 3(3): 59-66.

Figures/Tables 6

Fig.1

Fig.2

Fig.3

Tab.1

Tab.2

Tab.3

Algorithm comparison when normal equation matrix is ill-posed"

	True value	Least square estimation	Results of truncated singular value method	Ridge estimation (L curve method)	Our results
$X ˙$	-0.5230	-1.3473	-0.5350	-0.5527	-0.5107
	-2.7580	6.1625	-2.3729	-2.2639	-2.6774
	0.9020	10.4439	1.3585	1.3826	1.0340
	-0.5360	-0.3645	-0.5307	-0.4975	-0.5406
	-1.5600	2.8692	-1.3313	-1.3284	-1.4800
	2.5030	-5.9084	2.0671	2.0808	2.3682
	2.3340	-5.3319	1.9071	1.9355	2.1844
	-2.6650	-16.2141	-3.3873	-3.3289	-2.9072
F( $X ˙$ )	0.0043	1.6861×10^-5	0.0012	0.0044	0.3068
m	0	504.0441	1.3032	1.3089	0.1297

Tab.3

References 27

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Point name	True coordinates				Approximate coordinates
Point name	P₃	P₄	P₅	P₆		P₃	P₄	P₅	P₆
x	53743.151	48681.398	43767.234	40843.239	53743.674		48680.496	43768.794	40840.905
y	61003.810	55018.270	57968.590	64867.876	61006.568		55018.806	57966.087	64870.541

Side number	Edge length	Side number	Edge length
1	45.075	6	5731.756
2	5222.056	7	5438.383
3	5187.391	8	7493.316
4	7838.867	9	8884.603
5	5483.162	10	8839.687

Adjustment Model and Algorithm Based on Ellipsoid Uncertainty

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