﻿ 利用微分进化法确定海洋磁场向下延拓中的最优参数
 地球物理学报  2018, Vol. 61 Issue (8): 3278-3284 PDF

Determination of optimum parameters using a differential evolution algorithm in downward continuation of the marine geomagnetic field
LIU Qiang, BIAN Gang, YIN XiaoDong, JIN ShaoHua
Department of Military Oceanography and Hydrography & Cartography, Dalian Naval Academy, Dalian 116018, China
Abstract: The analytic continuation of potential fields is the main approach to realize the conversion of marine geomagnetic field at different altitudes, which is a key technique to construct the 3D marine geomagnetic field model. Focusing on the determination of the optimal regularization parameters and the number of iterations, the differential evolution algorithm (DE) is adopted in iterative downward continuation. Moreover, two kinds of optimum parameters can be selected effectively according to the minimum objective function by taking the entropy value of continuation results as the objective function, and the regularization parameter and times of iteration as individual species. Furthermore, the DE method has been applied to determine optimal parameters in several commonly used iteration methods with observed data. Compared with the optimum regularization parameters determined by the L-curve rule and the best number of iterations determined through multiple experiments, the optimum parameters determined by the DE method can achieve the best effect of continuation in different iterative methods, and the continuation results are closer to the real geomagnetic field. Hence, the DE method is recommended to be applied in downward continuation of the marine geomagnetic field.
Key words: Marine geomagnetic field    Downward continuation    Iteration method    Differential evolution algorithm    Optimal regularization parameter    Geomagnetic entropy
0 引言

1 向下延拓原理及最优参数确定 1.1 位场向下延拓迭代法

 (1)

 (2)

(a) 频域积分迭代法延拓算子

 (3)

(b) 频域迭代Tikhonov正则化法延拓算子

 (4)

(c) 频域Landweber迭代法延拓算子

 (5)

1.2 微分进化法确定最优参数

(1) 输入参数初值

 (6)

(2) 变异

 (7)

(3) 交叉

 (8)

(4) 地磁熵选择准则

 (9)

 (10)

 (11)

 图 1 微分进化法选择最优参数运算流程图 Fig. 1 Flow chart of choosing optimal parameters by the DE method
2 实测资料处理与对比分析

 (12)

 图 2 微分进化法目标函数曲线图 Fig. 2 Curves of the objective function in the DE method

 图 3 L-曲线图 Fig. 3 L-curve
 图 4 迭代次数与延拓误差关系曲线 Fig. 4 Relation curves of errors and iteration numbers of three methods

 图 5 实测值与延拓结果的对比(nT) (a) 0 m高度实际测量值；(b)积分迭代法延拓结果；(c)迭代Tikhonov正则化法延拓结果；(d) Landweber迭代法延拓结果. Fig. 5 Comparison of measured data and continuation results (in nT) (a) Measured data at 0 meter height; (b) The continuation result of integral-iteration method; (c) The continuation result of iterative Tikhonov regularization method; (d) The continuation result of Landweber iteration method.

3 结论

(1) 在迭代Tikhonov正则化法与Landweber迭代法计算中，微分进化法与传统L-曲线准则确定的最优正则化参数分别只差0.0003与0.0093，最佳迭代次数是一致的，延拓误差均只差0.01 nT，验证了微分进化法的有效性.

(2) 以延拓结果熵值最小为选择准则，确定的最优参数可使所采用的延拓方法达到最佳延拓效果，且只需进化二十几代即可搜索最优解，收敛速度快，自适应性强，可明显提高迭代延拓方法的计算效率与准确性.

References
 Chen L W, Zhang H, Zhang Z Q, et al. 2007. Technique of geomagnetic field continuation in underwater geomagnetic aided navigation. Journal of Chinese Inertial Technology, 15(6): 693-697. Liu D J, Hong T Q, Jia Z H, et al. 2009. Wave number domain iteration method for downward continuation of potential fields and its convergence. Chinese J. Geophys., 52(6): 1599-1605. DOI:10.3969/j.issn.0001-5733.2009.06.022 Liu X G, Li Y C, Xiao Y, et al. 2014. Optimal regularization parameter determination method in downward continuation of gravimetric and geomagnetic data. Acta Geodaetica et Cartographica Sinica, 43(9): 881-887. Ma T, Chen L W, Wu M P, et al. 2013. The selection of regularization parameter in downward continuation of potential field based on L-curve method. Progress in Geophys., 28(5): 2485-2494. DOI:10.6038/pg20130527 Ministry of Land and Resources of the People's Republic of China. 2010. DZ/T 0142-2010. Criterion of aeromagnetic survey (in Chinese). Beijing: China Standard Publishing House. Song W Q, Gao Y K, Zhu H W. 2013. The differential evolution inversion method based on Bayesian theory for micro-seismic data. Chinese J. Geophys., 56(4): 1331-1339. DOI:10.6038/cjg20130427 Storn R, Price K. 1996. Minimizing the real functions of the ICEC'96 contest by differential evolution. //Proceeding of the IEEE Conference on Evolutionary Computation. Nagoya, Japan: IEEE, 842-844. Sun W, Wu X P, Wang Q B, et al. 2014. Wave number domain iterative Tikhonov regularization method for downward continuation of airborne gravity data. Acta Geodaetica et Cartographica Sinica, 43(6): 566-574. Wang T Y, Yang J, Yan T J, et al. 2014. The differential evolution algorithm in geophysical inversion. Geology and Exploration, 50(5): 971-975. Wang W J. 2010. Regularization algorithms for solving ill-posed matrix equations arising from geophysical inversions[Ph. D. thesis] (in Chinese). Chengdu: Chengdu University of Technology. Wang Y F. 2007. Computational Methods for Inverse Problems and Their Applications. Beijing: Higher Education Press: 76-84. Wang Y G, Zhang F X, Wang Z W, et al. 2011. Taylor series iteration for downward continuation of potential field. OGP, 46(4): 657-662. Xu S Z. 2006. The integral-iteration method for continuation of potential fields. Chinese J. Geophys., 49(4): 1176-1182. Xu X S, Zhang Y Q. 2008. Application of modified terrain entropy algorithm in terrain aided navigation. Journal of Chinese Inertial Technology, 16(5): 595-598. Yao C L, Li H W, Zheng Y M, et al. 2012. Research on iteration method using in potential field transformations. Chinese J. Geophys., 55(6): 2062-2078. DOI:10.6038/j.issn.0001-5733.2012.06.028 Zeng X N, Li X H, Han S Q, et al. 2011. A comparison of three iteration methods for downward continuation of potential fields. Progress in Geophys., 26(3): 908-915. DOI:10.3969/j.issn.1004-2903.2011.03.016 Zeng X N, Li X H, Niu C, et al. 2013. Regularization-integral iteration in wave number domain for downward continuation of potential fields. OGP, 48(4): 643-650. 陈龙伟, 张辉, 郑志强, 等. 2007. 水下地磁辅助导航中地磁场延拓方法. 中国惯性技术学报, 15(6): 693-697. 刘东甲, 洪天求, 贾志海, 等. 2009. 位场向下延拓的波数域迭代法及其收敛性. 地球物理学报, 52(6): 1599-1605. DOI:10.3969/j.issn.0001-5733.2009.06.022 刘晓刚, 李迎春, 肖云, 等. 2014. 重力与磁力测量数据向下延拓中最优正则化参数确定方法. 测绘学报, 43(9): 881-887. 马涛, 陈龙伟, 吴美平, 等. 2013. 基于L曲线法的位场向下延拓正则化参数选择. 地球物理学进展, 28(5): 2485-2494. DOI:10.6038/pg20130527 宋维琪, 高艳珂, 朱海伟. 2013. 微地震资料贝叶斯理论差分进化反演方法. 地球物理学报, 56(4): 1331-1339. DOI:10.6038/cjg20130427 孙文, 吴晓平, 王庆宾, 等. 2014. 航空重力数据向下延拓的波数域迭代Tikhonov正则化方法. 测绘学报, 43(6): 566-574. 王天意, 杨进, 颜廷杰, 等. 2014. 地球物理反演中的差分进化算法. 地质与勘探, 50(5): 971-975. 王文娟. 2010. 地球物理反演中病态矩阵方程正则化解算方法研究[博士论文]. 成都: 成都理工大学. 王彦飞. 2007. 反演问题的计算方法及其应用. 北京: 高等教育出版社: 76-84. 王彦国, 张风旭, 王祝文, 等. 2011. 位场向下延拓的泰勒级数迭代法. 石油地球物理勘探, 46(4): 657-662. 徐世浙. 2006. 位场延拓的积分-迭代法. 地球物理学报, 49(4): 1176-1182. 徐晓苏, 张逸群. 2008. 改进的地形熵算法在地形辅助导航中的应用. 中国惯性技术学报, 16(5): 595-598. 姚长利, 李宏伟, 郑元满, 等. 2012. 重磁位场转换计算中迭代法的综合分析与研究. 地球物理学报, 55(6): 2062-2078. DOI:10.6038/j.issn.0001-5733.2012.06.028 曾小牛, 李夕海, 韩绍卿, 等. 2011. 位场向下延拓三种迭代方法之比较. 地球物理学进展, 26(3): 908-915. DOI:10.3969/j.issn.1004-2903.2011.03.016 曾小牛, 李夕海, 牛超, 等. 2013. 位场向下延拓的波数域正则-积分迭代法. 石油地球物理勘探, 48(4): 643-650. 中华人民共和国国土资源部. 2010. DZ/T 0142-2010. 航空磁测技术规范. 北京: 中国标准出版社.