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1. 信息工程大学 理学院，河南 郑州 450001；
2. 中船重工集团 第七一三研究所，河南 郑州 450015

Iterative Solution of Regularization to Ill-conditioned Problems in Geodesy and Geophysics
GU Yongwei1,GUI Qingming1,ZHANG Xuan2,Meng1
1. Institute of Science,Information Engineering University,Zhengzhou 450001,China;
2. 713th Research Institute of China Shipbuilding Industry Corporation,Zhengzhou 450015,China
First author: GU Yongwei (1965—),male,PhD,associate professor,majors in theory of surveying error and data processing.E-mail： gyw1019@sina.com
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-conditioned effects to the solutions must be overcome. By regularization ideas,the conjugate gradient method is improved. 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

1 引 言

2 数学模型及共轭梯度法解法

3 基于条件数控制的正则化迭代解法 3.1 干扰源向量与修正法矩阵

3.2 参数sβ1β2、…、βs的选取

(1) 求出ATA的特征值为0＜λ1＜λ2＜…＜λt,计算cond(ATA)，为修正法矩阵条件数的确定提供参考。

(2) 确定修正法矩阵条件数K。根据实际问题，将法矩阵的条件数降低几个数量级，即可达到明显的效果，如本文针对航空测量数据向下延拓问题，选500较合适。

(3) 计算条件指标[20]。按其是否大于K分为两类：，第一类的条件指标过大，相应的特征值λ1、λ2、…、λs过小，应作修改，由此确定s

(4) 对于过小的特征值λ1、λ2、…、λs，利用其对应的特征向量按照式(8)构造相应的干扰源向量，其中参数β1β2、…、βs的取法为

3.3 修正法方程与基于条件数控制的正则化迭代解法

4 算例及分析

 解法 x1 x2 x3 x4 x5 x6 x7 x8 x9 精度 真值 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

 图 1 重力异常真值 Fig. 1 True value of gravity anomaly

 图 2 重力异常的LS解 Fig. 2 LS solution of gravity anomaly

 图 3 重力异常的CG解 Fig. 3 CG solution of gravity anomaly

 图 4 重力异常Tikhonov正则化解 Fig. 4 Tikhonov solution of gravity anomaly

 图 5 重力异常的RBC正则化解 Fig. 5 RBC solution of gravity anomaly

 ms-2 估计方法 解的精度 与真值的最小差值 与真值的最大差值 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

5 结 论

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http://dx.doi.org/10.13485/j.cnki.11-2089.2014.0049

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#### 文章信息

GU Yongwei, GUI Qingming, ZHANG Xuan,et al.

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

Acta Geodaetica et Cartographica Sinica,2014,43(4):331-336.
http://dx.doi.org/10.13485/j.cnki.11-2089.2014.0049