基于乘积型目标函数的电阻率法三维反演

doi:10.19657/j.geoscience.1000-8527.2022.074

摘要/Abstract

摘要：

为了减小地球物理反演的多解性,通常采用累加型目标函数,即数据拟合差项加上正则化项。基于累加型目标函数的反演存在如何优选正则化因子的问题,这个过程通常需要做大量的反演计算。本研究系统分析和实现了基于乘积型目标函数(数据拟合差项和正则化项相乘)的电阻率法三维反演,乘积型目标函数的反演不存在优选正则化因子的问题。三维正演采用非结构化有限单元法,三维反演采用有限内存拟牛顿方法。使用理论模型合成数据进行了三维反演,检验了基于乘积型目标函数的电阻率法三维反演的可行性与有效性。反演算例结果表明:基于乘积型目标函数的反演方法能够可靠恢复异常体的电阻率值、形态和位置。

关键词: 电阻率法, 正则化, 乘积型目标函数, 三维反演

Abstract:

To reduce the multiplicity of geophysical inversion solution, additive objective functions are usually used, i.e., the data term and regularization term. Inversion based on additive objective function has the problem of optimizing the regularization factor, which usually requires much inversion calculations. This study systematically analyzed and implemented the 3D resistivity inversion method, based on multiplicative objective function (i.e., multiplication of the data term and regularization term). The problem of optimizing regularization factor is absent in the inversion of multiplicative objective function. We used the unstructured finite element method for resolving the 3D forward problem, and the L-BFGS (Limited-memory Broyden-Flecher-Goldfarb-Shanno) method for 3D inversion. The feasibility and effectiveness of our approach is verified by using the synthetic data of the theoretical model. The results show that the inversion method (based on multiplicative objective function) can reliably recover the resistivity value, shape and position of anomalies.

Key words: resistivity method, regularization, multiplicative objective function, 3D inversion

中图分类号:

P631

谢海军, 谭捍东, 马逢群. 基于乘积型目标函数的电阻率法三维反演[J]. 现代地质, 2023, 37(01): 67-73.

XIE Haijun, TAN Handong, MA Fengqun. Three-dimensional Inversion for Resistivity Method Based on Multiplicative Regularization[J]. Geoscience, 2023, 37(01): 67-73.

图/表 9

图1 电阻率法三维模型示意图

Fig.1 Three-dimensional model of resistivity method

图2 非结构化网格示意图

Fig.2 Schematic diagram of unstructured grid

图3 二维非结构化网格模型向量相互作用关系 (中心深色网格代表当前需要光滑的模型单元,黑点表示各个网格单元中心点)

Fig.3 Vector interaction relationship of 2D unstructured mesh model

图4 水平方向双棱柱体模型和地表观测系统示意图

Fig.4 Horizontal double prismatic model and surface-surface observation system

图5 反演数据拟合差和模型粗糙度曲线

Fig.5 RMS misfit and model roughness inversion curves

图6 水平方向双棱柱模型的反演结果

Fig.6 Inversion results of the horizontal double prismatic model

图7 垂直方向双棱柱体模型和地表、地井、井地与井间观测系统示意图

Fig.7 Vertical double prismatic model and surface-surface, surface-borehole, borehole-surface and borehole-borehole observation system

图8 反演数据拟合差和模型粗糙度曲线

Fig.8 RMS misfit and model roughness inversion curves

图9 垂直方向双棱柱模型的反演结果

Fig.9 Inversion results of the vertical double prismatic model

参考文献 30

[1]	TIKHONOV A N. On the stability of inverse problems[J]. Proceedings of the USSR Academy of Sciences 1943, 39:195-198.
[2]	TIKHONOV A N, ARSENIN V Y. Solutions of ill-posed problems[J]. Mathematics of Computation, 1977, 32:491-491.
[3]	BAKUSHINSKII A B. The problem of the convergence of the iteratively regularized Gauss-Newton method[J]. Computational Mathematics and Mathematical Physics, 1992, 32:1353-1359.
[4]	BLASCHKE B, NEUBAUER A, SCHERZER O. On Convergence Rates for the Iteratively Regularized Gauss-Newton-Method[J]. IMA Journal of Numerical Analysis, 1997, 17(3):421-436. DOI URL
[5]	JIN Q N. On the iteratively regularized Gauss-Newton method for solving nonlinear ill-posed problems[J]. Mathematics of Computation, 2000, 69:1603-1623. DOI URL
[6]	VAN DEN BERG P M, BROEKHOVEN A L V, ABUBAKAR A. Extended contrast source inversion[J]. Inverse Problems, 1999, 15(5):1325-1344. DOI URL
[7]	VAN DEN BERG P M, ABUBAKAR A. Contrast source inversion method: state of art[J]. Journal of Electromagnetic Waves & Applications, 2001, 15(11):1503-1505.
[8]	VAN DEN BERG P M, ABUBAKAR A, FOKKEMA J.T. Multiplicative regularization for contrast profile inversion[J]. Radio Science, 2003, 38(2): VIC23.
[9]	RUDIN L I, OSHER S, FATEMI E. Nonlinear total variation based noise removal algorithms[J]. Physica D Nonlinear Phenomena, 1992, 60(1/4):259-268. DOI URL
[10]	CHAMBOLLE A. An algorithm for total variation minimization and applications[J]. Journal of Mathematical Imaging & Vision, 2004, 20(1/2):89-97.
[11]	GRIP N, SABOUROVA N, TU Y. Sensitivity-based model updating for structural damage identification using total variation regularization[J]. Mechanical Systems and Signal Processing, 2017, 84:365-383. DOI URL
[12]	DOICU A, SCHREIER F, HILGERS S, et al. An efficient inversion algorithm for atmospheric remote sensing with application to UV limb observations[J]. Journal of Quantitative Spectroscopy & Radiative Transfer, 2007, 103(1):193-208.
[13]	ROBERT S, MARTIN B, THORSTEN H. The iteratively regularized Gauss-Newton method with convex constraints and applications in 4Pi microscopy[J]. Inverse Problems, 2012, 28(1):1-5.
[14]	ENTEZAMI A, SHARIATMADAR H, SARMADI H. Structural damage detection by a new iterative regularization method and an improved sensitivity function[J]. Journal of Sound & Vibration, 2017, 399:285-307.
[15]	DINGY, LAW S S. Structural damping identification based on an iterative regularization method[J]. Journal of Sound and Vibration, 2011, 330(10):2281-2298. DOI URL
[16]	WANG L J, CAO H P, XIE Y X. An improved iterative tikhonov regularization method for solving the dynamic load identification problem[J]. International Journal for Computational Methods in Engineering Science and Mech, 2015, 16(5):292-300. DOI URL
[17]	ABUBAKAR A, VAN DEN BERG P M, MALLORQUI JJ. Imaging of biomedical data using a multiplicative regularized contrast source inversion method[J]. IEEE Trans Microwave Theory and Techniques, 2002, 50(7):1761-1771. DOI URL
[18]	ABUBAKAR A, VAN DEN BERG P M, HABASHY T M, et al. A multiplicative regularization approach for deblurring problems[J]. IEEE Trans Image Process, 2004, 13(11):1524-1532. DOI URL
[19]	HACINI M, HACHOUF F, DJEMAL K. A new speckle filtering method for ultrasound images based on a weighted multiplicative total variation[J]. Signal Processing, 2014, 103:214-229. DOI URL
[20]	BROWN T, JEFFREY I, MOJABI P. Multiplicatively regularized source reconstruction method for phaseless planar near-field antenna measurements[J]. IEEE Trans Antennas and Propagation, 2017, 65:2020-2031. DOI URL
[21]	HABASHY T M, ABUBAKAR A. A general framework for constraint minimization for the inversion of electromagnetic measurements[J]. Progress in Electromagnetics Research, 2004, 46:265-312. DOI URL
[22]	ZASLAVSKY M, DRUSKIN V, LIU J, et al. A Three-dimensional multiplicative-regularized non-linear inversion algorithm for cross-well electromagnetic and controlled-source electromagnetic applications[J]. SEG Technical Program Expanded Abstracts, 2008, 27:584-588.
[23]	ABUBAKAR A, HABASHY T M, LI M, et al. Inversion algorithms for large-scale geophysical electromagnetic measurements[J]. Inverse Problems, 2009, 25(12):1541-1548.
[24]	ALPAK F O, HABASHY T M, ABUBAKAR A, et al. A multiplicative regularized Gauss-Newton algorithm and its application to the joint inversion of induction logging and near-borehole pressure measurements[J]. Progress In electromagnetics Research B, 2011, 29(29):105-138. DOI URL
[25]	徐世浙. 地球物理中的有限单元法[M]. 北京: 科学出版社, 1994.
[26]	刘洋. 基于非结构网格的电阻率三维正反演及其应用研究[D]. 合肥: 中国科学技术大学, 2016.
[27]	梁鹏. 电阻率法三维各向异性正演与主轴各向异性反演研究[D]. 北京: 中国地质大学(北京), 2018.
[28]	ZHANG Y, KEY K. Parallel goal-oriented adaptive finite element modeling for 3D electromagnetic exploration[M]//SEG.SEG Technical Program Expanded Abstracts 2015, AGU Fall Meeting Abstracts. Washington:AGU, 2015:806-811.
[29]	KEY K. MARE2DEM: a 2-D inversion code for controlled-source electromagnetic and magnetotelluric data[J]. Geophysical Journal International, 2016, 207:571-588. DOI URL
[30]	王堃鹏. 张量CSAMT三维主轴各向异性正反演研究[D]. 北京: 中国地质大学(北京), 2017.