Iterative Solution of Regularization to Ill-conditioned Problems in Geodesy and Geophysics

doi:10.11947/j.JGGS.2019.0107

Abstract

Abstract:

In geodesy and geophysics, many large-scale over-determined linear equations need to be solved which are often ill-conditioned. When the conjugate gradient method is used, their ill-conditioning effects to the solutions must be overcome, which is studied in this paper. Through the regularization ideas, the conjugate gradient method is improved, and the regularization iterative solution based on controlling condition number is put forward. Firstly by constructing the interference source vector, a new equation is derived with ill-condition diminished greatly, which has the same solution to the original normal equation. Then the new equation is solved by conjugate gradient method. Finally the effectiveness of the new method is verified by some numerical experiments of airborne gravity downward to the earth surface. In the numerical experiments the new method is compared with LS, CG and Tikhonov methods, and its accuracy is the highest.

Key words: ill-condition; regularization; condition number; interference source vector; iteration

Yongwei GU,Qingming GUI,Xuan ZHANG,Songhui HAN. Iterative Solution of Regularization to Ill-conditioned Problems in Geodesy and Geophysics[J]. Journal of Geodesy and Geoinformation Science, 2019, 2(1): 59-65.

Figures/Tables 7

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References 20

[1]	ZHENG Wei, XU Houze, ZHONG Min , et al. Accurate and Rapid Determination of the GRACE Earth's Gravitational Field Using the Improved Pre-conditioned Conjugate-gradient Approach and Three-dimensional Interpolation Method[J]. Progress in Geophysics, 2011,26(3):805-812.
[2]	LIU C S, ATLURI S N . An Iterative Method Using an Optimal Descent Vector, for Solving an Ill-conditioned System Bx=b, Better and Faster Than the Conjugate Gradient Method[J]. CMES: Computer Modeling in Engineering & Sciences, 2011,80(3):275-298.
[3]	XU Shufang, GAO Li, ZHANG Pingwen. Numerical Linear Algebra[M]. Beijing: Beijing University Press, 2004.
[4]	XU Peiliang, FUKUDA Y, LIU Yumei . Multiple Parameter Regularization: Numerical Solutions and Applications to the Determination of Geopotential from Precise Satellite Orbits[J]. Journal of Geodesy, 2006,80(1):17-27.
[5]	LIU C S, HONG H K, ATLURI S N . Novel Algorithms Based on the Conjugate Gradient Method for Inverting Ill-conditioned Matrices, and a New Regularization Method to Solve Ill-posed Linear Systems[J]. CMES: Computer Modeling in Engineering & Sciences, 2010,60(3):279-308.
[6]	OU Jikun . Uniform Expression of Solutions of Ill-posed Problems in Surveying Adjustment and the Fitting Method by Selection of the Parameter Weights[J]. Acta Geodaetica et Cartographica Sinica, 2004,33(4):283-288. DOI: 10.3321/j.issn:1001-1595.2004.04.001.
[7]	WANG Zhenjie. Regularization of Ill-posed Problems in Surveying[M]. Beijing: Science Press, 2006.
[8]	WANG Xingtao, XIA Zheren, SHI Pan , et al. A Comparison of Different Downward Continuation Methods for Airborne Gravity Data[J]. Chinese Journal of Geophysics, 2004,47(6):1017-1022.
[9]	LUO Xiaowen, OU Jikun, YUAN Yunbin . An Improved Regularization Method for Estimating Near Real-time Systematic Errors Suitable for Medium-long GPS Baseline Solutions[J]. Earth, Planets and Space, 2008,60(8):793-800.
[10]	GU Yongwei, GUI Qingming, BIAN Shaofeng , et al. Comparison Between Tikhonov Regularization and Truncated SVD in Geophysics[J]. Geomatics and Information Science of Wuhan University, 2005,30(3):238-241.
[11]	GU Yongwei, GUI Qingming . Study of Regularization Based on Signal-to-Noise Index in Airborne Gravity Downward to the Earth Surface[J]. Acta Geodaetica et Cartographica Sinica, 2010,39(5):458-464.
[12]	JIANG Tao, LI Jiancheng, WANG Zhengtao , et al. Solution of Ill-posed Problem in Downward Continuation of Airborne Gravity[J]. Acta Geodaetica et Cartographica Sinica, 2011,40(6):684-689.
[13]	GE Xuming, WU Jicang . Generalized Regularization to Ill-posed Total Least Squares Problem[J]. Acta Geodaetica et Cartographica Sinica, 2012,41(3):372-377.
[14]	SAVE H, BETTADPUR S, TAPLY B D . Reducing Errors in the GRACE Gravity Solutions Using Regularization[J]. Journal of Geodesy, 2012,86(9):695-711.
[15]	XU Xinyu, LI Jiancheng, WANG Zhengtao , et al. The Simulation Research on the Tikhonov Regularization Applied in Gravity Field Determination of GOCE Satellite Mission[J]. Acta Geodaetica et Cartographica Sinica, 2010,39(5):465-470.
[16]	DENG Kailiang, HUANG Motao, BAO Jingyang , et al. Tikhonov Two-parameter Regulation Algorithm in Downward Continuation of Airborne Gravity Data[J]. Acta Geodaetica et Cartographica Sinica, 2011,40(6):690-696.
[17]	SCHAFFRIN B . Minimum Mean Squared Error (MSE) Adjustment and the Optimal Tykhonov-Phillips Regularization Parameter via Reproducing Best Invariant Quadratic Uniformly Unbiased Estimates (repro-BIQUUE)[J]. Journal of Geodesy, 2008,82(2):113-121.
[18]	XU Chang, DENG Chengfa . Solving Multicollinearity in Dam Regression Model Using TSVD[J]. Geo-Spatial Information Science, 2011,14(3):230-234.
[19]	SHEN Yunzhong, XU Peiliang, LI Bofeng . Bias-corrected Regularized Solution to Inverse Ill-posed Models[J]. Journal of Geodesy, 2012,86(8):597-608.
[20]	GUI Qingming, YAO Shaowen, GU Yongwei , et al. A New Method to Diagnose Multicollinearity Based on Condition Index and Variance Decomposition Proportion (CIVDP)[J]. Acta Geodaetica et Cartographica Sinica, 2006,35(3):210-214. DOI: 10.3321/j.issn:1001-1595.2006.03.003.

method	x₁	x₂	x₃	x₄	x₅	x₆	x₇	x₈	x₉	accuracy
true	1	1	1	1	1	1	1	1	1	0
LS	1.1188	0.8820	1.6802	0.5235	1.0156	0.9874	0.8618	0.6628	1.4762	0.3460
CG	0.5661	0.7222	0.7257	0.8861	0.7474	0.8469	0.7489	1.2521	1.1196	0.2543
Tikhonov	0.9149	0.9190	0.4762	0.7613	1.0108	0.9969	0.9230	1.2557	1.2346	0.2289
RBC	0.7243	0.8084	0.8167	0.9034	0.8261	0.8824	0.8267	1.1096	1.0379	0.1644

method	accuracy	the minimum difference to the true value	the maximum difference to the true value
LS	2.5277×10^-4	2.1373×10^-6	7.8019×10^-4
CG	6.2631×10^-5	3.0000×10^-7	3.1600×10^-5
Tikhonov	1.1933×10^-5	3.5650×10^-9	3.6385×10^-5
RBC	8.8814×10^-6	1.5000×10^-7	1.5800×10^-5