2.5D inversion of time domain airborne electromagnetic data using nonlinear conjugate gradients
-
摘要: 对于时间域航空电磁法二维和三维反演来说，最大的困难在于有效的算法和大的计算量需求.本文利用非线性共轭梯度法实现了时间域航空电磁法2.5维反演方法，着重解决了迭代反演过程中灵敏度矩阵计算、最佳迭代步长计算、初始模型选取等问题.在正演计算中，我们采用有限元法求解拉式傅氏域中的电磁场偏微分方程，再通过逆拉氏和逆傅氏变换高精度数值算法得到时间域电磁响应.在灵敏度矩阵计算中，采用了基于拉式傅氏双变换的伴随方程法，时间消耗只需计算两次正演，从而节约了大量计算时间.对于最佳步长计算，二次插值向后追踪法能够保证反演迭代的稳定性.设计两个理论模型，检验反演算法的有效性，并讨论了选择不同初始模型对反演结果的影响.模型算例表明：非线性共轭梯度方法应用于时间域航空电磁2.5维反演中稳定可靠，反演结果能够有效地反映地下真实电性结构.当选择的初始模型电阻率值与真实背景电阻率值接近时，能得到较好的反演结果，当初始模型电阻率远大于或远小于真实背景电阻率值时反演效果就会变差.
-
关键词:
- 时间域航空电磁法 /
- 2.5D瞬变电磁反演 /
- 伴随方程法 /
- 非线性共轭梯度 /
- 灵敏度矩阵
Abstract: Inversion of time domain airborne electromagnetic (AEM) data are known for the difficulties in large computational requirement and effective algorithms, especially for two and three-dimensional problems. We have developed a 2.5 dimensional (2.5D) inverse algorithm for the time domain AEM data using the nonlinear conjugate gradient method with improved accuracy and efficiency. This paper focuses on solving the computation of the sensitivity matrix, the optimal step length and the initial model selection in this algorithm. In the forward modeling, we employ the finite element method (FEM) to solve the Maxwell's equations in the Laplace and Fourier domains. The time domain responses are then obtained by the high-precision inverse Laplace and Fourier transforms. The sensitivity matrix is calculated by using the adjoint equation method with the Laplace and Fourier transforms, which requires only two forward modeling per iteration and reduces the time cost significantly. The backtracking method in the optimal step length computation ensures the stability of this iterative inverse algorithm. Then, we present two model studies and discuss the effects of different initial models. The synthetic studies demonstrate our inversion algorithm is stable and can yield reliable results, which reflect the underground electrical structure reasonably. It also turns out that the inversion result is good if the initial model is close to the true background. The inversion model becomes worse if the initial model is several times larger or smaller than the true values of the background resistivity. -
-
[1] Newman G A, Alumbaugh D L. 1995. Frequency-domain modelling of airborne electromagnetic responses using staggered finite differences. Geophysical Prospecting, 43(8): 1021-1042.
[2] Chen B, Mao L F, Liu G D. 2014. The estimated prospecting depth of CHTEM-I system by the method of diffusion electric field. Chinese J. Geophys. (in Chinese), 57(1): 303-309, doi: 10.6038/cjg20140126.
[3] Commer M, Newman G A. 2004. A parallel finite-difference approach for 3D transient electromagnetic modeling with galvanic sources. Geophysics, 69(5): 1192-1202.
[4] Commer M, Newman G A. 2009. Three-dimensional controlled-source electromagnetic and magnetotelluric joint inversion. Geophysical Journal International, 178(3): 1305-1316.
[5] Cox L H, Zhdanov M S. 2008. Advanced computational methods of rapid and rigorous 3-D inversion of airborne electromagnetic data. Communications in Computational Physics, 3(1): 160-179.
[6] Cox L H, Wilson G A, Zhdanov M S. 2012. 3D inversion of airborne electromagnetic data. Geophysics, 77(4): WB59-WB69.
[7] Farquharson C G. 1995. Approximate sensitivities for the multi-dimensional electromagnetic inverse problem [Ph. D. thesis]. Vancouver: University of British Columbia.
[8] Fraser D C. 1972. A new multicoil aerial electromagnetic prospecting system. Geophysics, 37(3): 518-537.
[9] Ghosh M K. 1972. Interpretation of airborne EM measurements based on thin sheet models [Ph. D. thesis]. Toronto: University of Toronto.
[10] Guillemoteau J, Sailhac P, Behaegel M. 2012. Fast approximate 2D inversion of airborne TEM data: Born approximation and empirical approach. Geophysics, 77(4): WB89-WB97.
[11] Haber E, Oldenburg D W, Shekhtman R. 2007. Inversion of time domain three-dimensional electromagnetic data. Geophysical Journal International, 171(2): 550-564.
[12] Kelbert A, Egbert G D, Schultz A. 2008. Non-linear conjugate gradient inversion for global EM induction: resolution studies. Geophysical Journal International, 173(2): 365-381.
[13] Li J H. 2011. 3D numerical simulation for transient electromagnetic field excited by large source loop based on vector finite element method [Ph. D. thesis] (in Chinese). Changsha: Central South University.
[14] Liu Y H, Yin C C. 2013. 3D inversion for frequency-domain HEM data. Chinese J. Geophys. (in Chinese), 56(12): 4278-4287, doi: 10.6038/cjg20131230.
[15] Long J B. 2014. Non-linear conjugate gradient inversion for 2.5D airborne transient electromagnetic data [Master's thesis] (in Chinese). Changsha: Central South University.
[16] Luo Y Z, Chang Y J. 2000. A rapid algorithm for G-S transform. Chinese J. Geophys. (in Chinese), 43(5): 684-690, doi: 10.3321/j.issn:0001-5733.2000.05.012.
[17] Luo Y Z, Zhang S Y, Wang W P. 2003. A research on one-dimension forward for aerial electromagnetic method in time domain. Chinese J. Geophys. (in Chinese), 46(5): 719-724.
[18] Nabighian M N. 1970. Quasi-static transient response of a conducting permeable sphere in a dipole field. Geophysics, 35(2): 303-309.
[19] Newman G A, Alumbaugh D L. 2000. Three-dimensional magnetotelluric inversion using non-linear conjugate gradients. Geophysical Journal International, 140(2): 410-424.
[20] Newman G A, Boggs P T. 2004. Solution accelerators for large-scale three-dimensional electromagnetic inverse problems. Inverse Problems, 20(6): S151-S170.
[21] Oldenburg D W, Haber E, Shekhtman R. 2013. Three dimensional inversion of multisource time domain electromagnetic data. Geophysics, 78(1): E47-E57.
[22] Qiang J K, Zhou J J, Man K F. 2015. Synthetic study of 2.5-D ATEM based on finite element method. Geophysical and Geochemical Exploration (in Chinese), 39(5): 1059-1062.
[23] Rodi W L, Mackie R L. 2001. Nonlinear conjugate gradients algorithm for 2-D magnetotelluric inversion. Geophysics, 66(1): 174-187.
[24] Singh S K. 1973. Electromagnetic transient response of a conducting sphere embedded in a conductive medium. Geophysics, 38(5): 864-893.
[25] Viezzoli A, Christiansen A V, Auken E, et al. 2008. Quasi-3D modeling of airborne TEM data by spatially constrained inversion. Geophysics, 73(3): F105-F113.
[26] Wait J R. 1969. Electromagnetic induction in a solid conducting sphere enclosed by a thin conducting spherical shell. Geophysics, 34(5): 753-759.
[27] Wang H J, Luo Y Z. 2003. Algorithm of a 2.5 dimensional finite element method for transient electromagnetic with a central loop. Chinese J. Geophys. (in Chinese), 46(6): 855-862.
[28] Wang Y H. 2013. The research on HTEM 2.5D forward modeling and curve analysis [Master's thesis] (in Chinese). Chengdu: Chengdu University of Technology.
[29] Weng A H, Liu Y H, Jia D Y, et al. 2012. Three-dimensional controlled source electromagnetic inversion using non-linear conjugate gradients. Chinese J. Geophys. (in Chinese), 55(10): 3506-3515, doi: 10.6038/j.issn.0001-5733.2012.10.034.
[30] Wilson G A, Raiche A P, Sugeng F. 2006. 2.5D inversion of airborne electromagnetic date. Exploration Geophysics, 37(4): 363-371.
[31] Xiong B, Luo Y Z, Qiang J K. 2004. Methods for calculating sensitivities for 2.5D transient electromagnetic inversion (Ⅰ). Progress in Geophysics (in Chinese), 19(3): 616-620.
[32] Xiong B, Luo Y Z. 2006. Finite element modeling of 2.5-D TEM with block homogeneous conductivity. Chinese J. Geophys. (in Chinese), 49(2): 590-597.
[33] Yin C C, Fraser D C. 2004. The effect of the electrical anisotropy on the response of helicopter-borne frequency-domain electromagnetic systems. Geophysical Prospecting, 52(5): 399-416.
[34] Yin C C, Hodges G. 2007. Simulated annealing for airborne EM inversion. Geophysics, 72(4): F189-F195.
[35] Yin C C, Huang X, Liu Y H, et al. 2014. Footprint for frequency-domain airborne electromagnetic systems. Geophysics, 79(6): E243-E254.
[36] Yin C C, Zhang B, Liu Y H, et al. 2015. 2.5-D forward modeling of the time-domain airborne EM system in areas with topographic relief. Chinese J. Geophys. (in Chinese), 58(4): 1411-1424, doi: 10.6038/cjg20150427.
[37] Yu X D. 2014. Inversion of 2.5 dimensional time domain helicopter airborne electromagnetic method [Master's thesis] (in Chinese). Chengdu: Chengdu University of Technology.
[38] Zaslavsky M, Druskin V, Abubakar A, et al. 2012. Large-scale Gauss-Newton inversion of transient controlled-source electromagnetic data using the model reduction approach.//SEG Technical Program Expanded Abstracts 2012. SEG: 1-6.
[39] Zhdanov M S. 2009. Geophysical Electromagnetic Theory and Methods. New York: Elsevier.
[40] Zhou J J. 2011. Research on airborne transient electromagnetic 2.5-D forward modeling [Master's thesis] (in Chinese). Changsha: Central South University.
[41] 陈斌, 毛立峰, 刘光鼎. 2014. 用扩散电场法估算CHTEM-I系统的探测深度. 地球物理学报, 57(1): 303-309, doi: 10.6038/cjg20140126.
[42] 李建慧. 2011. 基于矢量有限单元法的大回线源瞬变电磁法三维数值模拟[博士论文]. 长沙: 中南大学.
[43] 刘云鹤, 殷长春. 2013. 三维频率域航空电磁反演研究. 地球物理学报, 56(12): 4278-4287, doi: 10.6038/cjg20131230.
[44] 龙剑波. 2014. 2.5维航空瞬变电磁数据的非线性共轭梯度反演[硕士论文]. 长沙: 中南大学.
[45] 罗延钟, 昌彦君. 2000. G-S变换的快速算法. 地球物理学报, 43(5): 684-690, doi: 10.3321/j.issn:0001-5733.2000.05.012.
[46] 罗延钟, 张胜业, 王卫平. 2003. 时间域航空电磁法一维正演研究. 地球物理学报, 46(5): 719-724.
[47] 强建科, 周俊杰, 满开峰. 2015. 时间域航空电磁法2.5维有限元模拟. 物探与化探, 39(5): 1059-1062.
[48] 王华军, 罗延钟. 2003. 中心回线瞬变电磁法2.5维有限单元算法. 地球物理学报, 46(6): 855-862.
[49] 王宇航. 2013. 吊舱式时间域直升机航空电磁2.5维正演及响应曲线分析[硕士论文]. 成都: 成都理工大学.
[50] 翁爱华, 刘云鹤, 贾定宇等. 2012. 地面可控源频率测深三维非线性共轭梯度反演. 地球物理学报, 55(10): 3506-3515, doi: 10.6038/j.issn.0001-5733.2012.10.034.
[51] 熊彬, 罗延钟, 强建科. 2004. 瞬变电磁2.5维反演中灵敏度矩阵计算方法(Ⅰ). 地球物理学进展, 19(3): 616-620.
[52] 熊彬, 罗延钟. 2006. 电导率分块均匀的瞬变电磁2.5维有限元数值模拟. 地球物理学报, 49(2): 590-597.
[53] 殷长春, 张博, 刘云鹤等. 2015. 2.5维起伏地表条件下时间域航空电磁正演模拟. 地球物理学报, 58(4): 1411-1424, doi: 10.6038/cjg20150427.
[54] 余小东. 2014. 时间域直升机航空电磁法2.5维反演[硕士论文]. 成都: 成都理工大学.
[55] 周俊杰. 2011. 航空瞬变电磁法2.5维正演模拟研究[硕士论文]. 长沙: 中南大学.
-
计量
- 文章访问数: 2296
- PDF下载数: 1317
- 施引文献: 0
下载: