﻿ ﻿ 改进高斯-牛顿法的位场向下延拓
 文章快速检索 高级检索

1. 第二炮兵工程大学 907教研室，陕西 西安 710025； 2. 清华大学 信息科学技术学院自动化系，北京 100084； 3. 第二炮兵装备研究院，北京 100085

A Modified Gauss-Newton Method for Downward Continuation of Potential Field
ZENG Xiaoniu1,2LIU Daizhi1NIU Chao1,QI Wei3
1. Staff Room 907, The Second Artillery Engineering University, Xi’an 710025, China; 2. Department of Automation, Shool of Information Science and Technology, Tsinghua University, Beijing 100084, China; 3. The Second Artillery Equipment Academy, Beijing 100085, China
First author: ZENG Xiaoniu (1987—), male, PhD candidate, majors in geophysical data analysis and processing.E-mail： xiaoniuzeng@163.com
Abstract: For the ill-posed problem of potential field downward continuation, a novel improved iterative method is proposed based on the analysis of the basic principle and the filter characteristic of the filter function of the Gauss-Newton method. This novel improved method uses an incremental geometric regularization parameter adaptive choice method and the minimal residual criterion for the iterative step choice. The comparison analysis of gravity theoretical model and real geomagnetic data shows that the proposed iterative method has high downward continuation accuracy and good convergence.
Key words: potential field     Gauss-Newton method     downward continuation     regularization parameter     minimal residual criterion

1 引 言

2 位场正则向下延拓原理

3 位场改进高斯-牛顿法向下延拓原理

3.1 高斯-牛顿法基本原理

3.2 滤波函数对比分析

 图 1 Tikhonov和高斯-牛顿法滤波函数特性的对比 Fig. 1 The characteristic comparison of filter functions for Tikhonov and Guass-Newton methods

3.3 正则参数的选取

 图 2 奇异值σ2i和 Tikhonov滤波函数wλ(σ2i) Fig. 2 The singular value σ2i and the filter function wλ(σ2i) for Tikhonov method

3.4 残差最小步长准则

3.5 算法具体实现步骤

4 数值试验及结果比较

4.1 理论重力模型

 正则参数 RMSE/mGal RE/(%) Tikhonov 0.000 4 0.033 1 9.62 本文方法 0.000 4×1.8 k－1 0.026 2 4.85
1 mGal=1×10-5m/s

 图 3 加噪重力异常等值线图 Fig. 3 Contours of the gravity anomaly data with noise
 图 4 延拓结果在主剖面的对比 Fig. 4 The comparison of downward continuation results in main profile

4.2 航磁数据试验

 图 5 延拓均方误差随公比 q和迭代次数的变化 Fig. 5 Relationship between root mean square errors and iteration numbers for the method in this paper

 图 6 延拓相关性误差随公比 q迭代次数的变化 Fig. 6 Relationship between relative errors and iteration numbers for the method in this paper

 正则参数 RMSE/nT RE/(%) Tikhonov 0.000 5 12.59 8.38 本文方法 0.000 5×1.2 k－1 11.21 7.19
 图 7 原始航磁等值线 Fig. 7 Contours of the original aeromagnetic field

 图 8 向上延拓航磁异常加噪等值线 Fig. 8 Contours of upward continuation of aeromagnetic field with noise

 图 9 Tikhonov法向下延拓结果等值线 Fig. 9 Contours of the downward continued data for Tikhonov regularization method

 图 10 本文方法向下延拓结果等值线 Fig. 10 Contours of the downward continued data for the method in this paper

 图 11 本文方法均方误差随迭代次数的变化 Fig. 11 Relationship between root mean square errors and iteration numbers for the method in this paper

 图 12 本文方法相关性误差随迭代次数的变化 Fig. 12 Relationship between relative errors and iteration numbers for the method in this paper
5 结束语

 [1] WANG Yanfei. Computational Methods for Inverse Problems and Their Applications[M]. Beijing: Higher Education Press, 2007. (王彦飞. 反演问题的计算方法及其应用[M]. 北京: 高等教育出版社, 2007.) [2] WANG Xintao, SHI Pan, ZHU Feizhou. Regularization Methods and Spectral Decomposition for the Downward Continuation of Airborne Gravity Data[J]. Acta Geodaetica et Cartographica Sinica, 2004, 33(1): 33-38. (王兴涛, 石磐, 朱非洲. 航空重力测量数据向下延拓的正则化算法及谱分析[J]. 测绘学报, 2004, 33(1): 33-38.) [3] 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. (顾勇为, 归庆明. 航空重力测量数据向下延拓基于信噪比的正则化方法的研究[J]. 测绘学报, 2010, 39(5): 458-464.测绘学报, 2010, 39(5): 458-464.) [4] 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. (蒋涛, 李建成, 王正涛, 等. 航空重力向下延拓病态问题的求解[J]. 测绘学报, 2011, 40(6): 684-689.) [5] 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. (邓凯亮, 黄谟涛, 暴景阳, 等. 向下延拓航空重力数据的Tikhonov双参数正则化法[J]. 测绘学报, 2011, 40(6): 690-696.) [6] XU Shizhe. The Integral-iteration Method for Continuation of Potential Field[J]. Chinese Journal of Geophysics, 2006, 49(4): 1176-1182 (徐世浙. 位场延拓的积分-迭代法[J]. 地球物理学报, 2006, 49(4): 1176-1182.) [7] XU S Z, YANG J Y, YANG C F, et al. The Iteration Method for Downward Continuation of a Potential Field from a Horizontal Plane[J]. Geophsical Prospecting, 2007, 55(6), 883-889. [8] ZHANG Hui, CHENG Longwei, REN Zhixin, et al. Analysis on Convergence of Iteration Method for Potential Fields Downward Continuation and Research on Robust Downward Continuation Method[J]. Chinese Journal of Geophysics. 2009, 52(4): 1107-1113. (张辉, 陈龙伟, 任治新, 等. 位场向下延拓迭代法收敛性分析及稳健向下延拓方法研究[J]. 地球物理学报, 2009, 52(4): 1107-1113.) [9] LIU Dongjia, HONG Tianqiu, JIA Zhihai, et al. Wave Number Domain Iteration Method for Downward of Potential Fields and Its Convergence[J]. Chinese Journal of Geophysics. 2009, 52(6): 1599-1605. (刘东甲, 洪天求, 贾志海, 等. 位场向下延拓的波数域迭代法及其收敛性[J]. 地球物理学报, 2009, 52(6): 1600-1605.) [10] YU Bo, ZHAI Guojun, LIU Yanchun, et al. The Downward Continuation Method of Aeromagnetic Data to the Sea Level[J]. Acta Geodaetica et Cartographica Sinica, 2009, 38(3): 202-209. (于波, 翟国君, 刘雁春, 等. 利用航磁数据向下延拓得到海面磁场的方法[J]. 测绘学报, 2009, 38(3): 202-209.) [11] YU Bo, ZHAI Guojun, LIU Yanchun, et al. Analysis of Noise Effect on the Calculation Error of Downward Continuation with Iteration Method[J]. Chinese Journal of Geophysics, 2009, 52(8): 2182-2188. (于波, 翟国君, 刘雁春, 等. 噪声对向下延拓迭代法的计算误差影响分析[J]. 地球物理学报, 2009, 52(8): 2182-2188.) [12] ZENG Xiaoniu, LI Xihai, HAN Shaoqing, et al. A Comparison of Three Iteration Methods for Downward Continuation of Potential Fields[J]. Progress in Geophysics. 2011, 26(3): 908-915. (曾小牛, 李夕海, 韩绍卿, 等. 位场向下延拓三种迭代方法之比较[J]. 地球物理学进展, 2011, 26(3): 908-915.) [13] ZENG Xiaoniu, LI Xihai, LIU Daizhi, et al. Regularization Analysis of Integral-iteration Method and the Choice of Its Optimal Step-length[J]. Chinese Journal of Geophysics, 2011, 54(11): 2943-2950. (曾小牛, 李夕海, 刘代志, 等. 积分迭代法的正则性分析及其最优步长的选择[J]. 地球物理学报, 2011, 54(11): 2943-2950.) [14] YAO Changli, LI Hongwei, Zheng Yuanman et al. Research on Iteration Method Using in Potential Field Transformations[J]. Chinese Journal of Geophysics, 2012, 55(6): 2062-2078. (姚长利, 李宏伟, 郑元满, 等. 重磁位场转换计算中迭代法的综合分析与研究[J]. 地球物理学报, 2012, 55(6): 2062-2078.) [15] GAO Yuwen, LUO Yao. WEN Wu. The Compensation Method for Downward Continuation of Potential Field from Horizontal Plane and Its Application[J]. Chinese Journal of Geophysics, 2012, 55(8): 2717-2756. (高玉文, 骆遥, 文武. 补偿向下延拓方法研究及应用[J]. 地球物理学报, 2012, 55 (8): 2747-2756.) [16] CHEN Longwei, HU Xiaoping, WU Meiping, et al. Research on Spatial Domain Continuation Method for Potential Field[J]. Progress in Geophysics, 2012, 27(4): 1509-1518. (陈龙伟, 胡小平, 吴美平, 等. 空间域位场延拓新方法研究[J]. 地球物理学进展. 2012, 27(4): 1509-1518.) [17] YUAN Yaxiang, SUN Wenyu. Optimization Theory and Methods[M]. Beijing: Science Press, 1997. (袁亚湘, 孙文瑜. 最优化理论与方法[M]. 北京: 科学出版社, 1997.) [18] KING J T, CHILLINGWORTH D. Approximation Generalized Inverse by Iterated Regularization[J]. Numerical Function Analysis and Optimization. 1979, 1(5): 499-513. [19] RANGANATHAN A. The Levenberg-Marquardt Algorithm[EB/OL]. 2004[2012-03-08]. http://www. ananth.in/ Notes_files/lmtut.pdf, 2004. [20] PIANA M, BERTERO M. Projected Landweber Method and Preconditioning[J]. Inverse Problems, 1997, 13(2), 441-464. [21] HANKE M, GROETSCH C W. Nonstationary Iterated Tikhonov Regularization[J]. Journal of Optimization Theory and Applications, 1998, 98(1): 37-53. [22] PEREVERZEV S V, SCHOCK E. pakhage’s Implicit Iteration Method and the Information Complexity of Equations with Operators Having Closed Range[J]. Journal of Complexity, 1999, 15(3): 385-401. [23] GOLUB G H, HEATH M, WAHBA G. Generalized Cross-validation as a Method for Choosing a Good Ridge Parameter[J]. Technometrics, 1979, 21(2): 215-223. [24] HANSEN P C, O'LEARY D P. The Use of the L-curve in the Regularization of Discrete Ill-posed Problems[J]. SIAM Journal on Scientific Computing, 1993, 14(6): 1487-1503. [25] ZHOU J, SHI G G, GE Z L. Study of Iterative Regularization Methods for Potential Field Downward Continuation in Geophysical Navigation[J]. Journal of Astronautics, 2011, 32(4): 787-794.
http://dx.doi.org/10.13485/j.cnki.11-2089.2014.

0006

0

#### 文章信息

ZENG Xiaoniu, LIU Daizhi, NIU Chao,et al.

A Modified Gauss-Newton Method for Downward Continuation of Potential Field

Acta Geodaetica et Cartographica Sinica,2014,43(1):37-44.
http://dx.doi.org/10.13485/j.cnki.11-2089.2014.0006