瞬变电磁虚拟波场Born近似算法研究
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摘要
根据瞬变电磁场所满足的扩散方程与波动场所满足的波动方程之间的数学关系,运用扫时波场变换技术,得到稳定的虚拟波场.在虚拟波场的基础上,借用地震勘探中的Born近似逆散射技术实现对电性界面进行成像的目的.在理论推导中,直接从时间域波动方程出发,运用格林定理、引入均匀半空间的Green函数建立散射场关于速度扰动量的表达式;在数值求解中,运用奇异值分解的方法实现对第一类Fredholm积分方程的求解.通过对理论模型的计算以及实例应用,均表明瞬变电磁虚拟波场Born近似算法对地下电性界面具有良好的分辨能力.
According to the mathematical relation between the diffusion equation satisfied by the transient electromagnetic field and the wave equation satisfied by the wave field, the technology of "sweep time" and "correlation superposition" is introduced to obtain the stable pseudo wave-field. Based on the pseudo wave-field, the technique of Born approximate inverse scattering in seismic exploration is used to image the electrical interface. In the theoretical derivation, the expression of velocity perturbation of scattering field is obtained by Green's theorem and Green function of uniform half-space. In the process of numerical solution, the first class of Fredholm integral equation is solved by singular value decomposition. Through the calculation of the theoretical model and practical application, it is shown that the Born approximation algorithm of Transient Electromagnet(TEM)pseudo wave-field has a good resolution capability for the underground electrical interface.
According to the mathematical relation between the diffusion equation satisfied by the transient electromagnetic field and the wave equation satisfied by the wave field, the technology of "sweep time" and "correlation superposition" is introduced to obtain the stable pseudo wave-field. Based on the pseudo wave-field, the technique of Born approximate inverse scattering in seismic exploration is used to image the electrical interface. In the theoretical derivation, the expression of velocity perturbation of scattering field is obtained by Green's theorem and Green function of uniform half-space. In the process of numerical solution, the first class of Fredholm integral equation is solved by singular value decomposition. Through the calculation of the theoretical model and practical application, it is shown that the Born approximation algorithm of Transient Electromagnet(TEM)pseudo wave-field has a good resolution capability for the underground electrical interface.
引文
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Bleistein N, Cohen J K, Stockwell J W, et al. 2001. Mathematics of Multidimensional Seismic Imaging, Migration, and Inversion [J]. Applied Mechanics Reviews, 54(5): B94-B96, doi: 10.1007/978-1- 4613- 0001- 4.
Cohen J K, Bleistein N. 1979. Velocity inversion procedure for acoustic waves [J]. Geophysics, 44(6): 1077-1087, doi: 10.1190/1.1440996.
Huang L J, Yang W C. 1991. Approximate method for inverse scattering inversion of acoustic equation [J]. Chinese Journal of geophysics (in Chinese), 34(05): 626- 634, doi: 10.3321/j.issn:0001-5733.1991.05.011.
Lee K H, Liu G, Morrison H F. 1989. A New Approach to Modeling the Electromagnetic Response of Conductive Media [J]. Geophysics, 54(9): 1180-1192, doi: 10.1190/1.1442753.
Lee K H, Xie G Q. 1993. A New Approach to Imaging with Low-frequency Electromagnetic Fields [J]. Geophysics, 58(6): 780-796, doi: 10.1190/1.1443464.
Li X, Qi Z P, Xue G Q, et al. 2010. Extension imaging of 3d curved surface of transient electromagnetic virtual wave field [J]. Chinese Journal of geophysics (in Chinese), 53(12): 3005-3011, doi: 10.3969/j.issn.0001-5733.2010.12.025.
Li X, Xue G Q, Guo W B. 2007. Research progress in TEM pseudo-seismic imaging [J]. Progress in Geophysics (in Chinese), 22(3): 811-816, doi: 10.3969/j.issn.1004-2903.2007.03.023.
Li X, Xue G Q, Liu Y A, et al. 2012. A research on TEM imaging method based on synthetic-aperture technology [J]. Chinese Journal of geophysics (in Chinese), 55(01): 333-340, doi: 10.6038/j.issn.0001-5733.2012.01.034.
Li X, Xue G Q, Song J P, et al. 2005. An optimize method for transient electromagnetic field-wave field conversion [J]. Chinese Journal of geophysics (in Chinese), 48(05): 1185-1190, doi:10.3321/jivssn:0001-5733.2005.05.029.
Qi Z P, Li X, Wu Q, et al. 2013. A new algorithm for full-time-domain wave-field transformation based on transient electromagnetic method [J]. Chinese Journal of geophysics (in Chinese), 56(10): 3581-3595, doi: 10.6038/cjg20131033.
Xie G Q, Li J H. 1988. New feature iteration method for solving inverse scattering potential problems of three-dimensional wave equation [J]. Chinese science (series A mathematical science, astronomy and technology science) (in Chinese), A(04): 359-365.
Xue G Q, Li X, Guo W B, et al. 2006. Equivalent transformation from TEM field sounding data to plane wave electromagnetic sounding data [J]. Chinese Journal Of Geophysics (in Chinese), 49(5): 1539-1545, doi: 10.3321/j.issn:0001-5733.2006.05.036.
Xue G Q, Li X, Qi Z P, et al. 2011. Study of sharpening the wave-form of TEM pseudo-seismic [J]. Chinese Journal Of Geophysics (in Chinese), 54(5): 1384-1390, doi: 10.3969/j.issn.0001-5733.2011.05.028.
Xue G Q, Li X, Song J P, et al. 2004. Theoretical analysis and numerical calculation of transient electromagnetic imaging of loop source [J]. Chinese Journal of geophysics (in Chinese), 47(02): 338-343, doi: 10.3321/j.issn:0001-5733.2004.02.023.
Yang W C. 1995. Inverse scattering and iterative linearization inversion of wave equation [J]. Petroleum geophysics (in Chinese), 34(03): 114-121.
Zhang J, Li X, Zhao Y, et al. 2011. A technology research of high-resolation imaging for the transient electromagnetic pseudo wave field [J]. Progress in geophysics (in Chinese), 26(03): 1077-1084, doi: 10.3969/j.issn.1004-2903.2011.03.038.
Zhu H W, Li X, Zhang J, et al. 2010. Information collecting technology in 3-D pseudo-seismic imaging of transient electromagnetics [J]. Progress in geophysics (in Chinese), 25(05): 1648-1656, doi: 10.3969/j.issn.1004-2903.2010.05.017.
黄联捷, 杨文采. 1991. 声波方程逆散射反演的近似方法[J]. 地球物理学报, 34(05): 626- 634, doi: 10.3321/j.issn:0001-5733.1991.05.011.
李貅, 戚志鹏, 薛国强, 等. 2010. 瞬变电磁虚拟波场的三维曲面延拓成像[J]. 地球物理学报, 53(12): 3005-3011, doi: 10.3969/j.issn.0001-5733.2010.12.025.
李貅, 薛国强, 郭文波. 2007. 瞬变电磁法拟地震成像研究进展[J]. 地球物理学进展, 22(03): 811-816, doi: 10.3969/j.issn.1004-2903.2007.03.023.
李貅, 薛国强, 刘银爱, 等. 2012. 瞬变电磁合成孔径成像方法研究[J]. 地球物理学报, 55(01): 333-340, doi: 10.6038/j.issn.0001-5733.2012.01.034.
李貅, 薛国强, 宋建平, 等. 2005. 从瞬变电磁场到波场的优化算法[J]. 地球物理学报, 48(05): 1185-1190, doi: 10.3321/j.issn:0001-5733.2005.05.029.
戚志鹏, 李貅, 吴琼, 等. 2013. 从瞬变电磁扩散场到拟地震波场的全时域反变换算法[J]. 地球物理学报, 56(10): 3581-3595, doi: 10.6038/cjg20131033.
谢干权, 李建华. 1988. 解三维波方程逆散射势问题的新特征迭代法[J]. 中国科学(A辑数学理学天文学技术科学), A(04): 359-365.
薛国强, 李貅, 郭文波, 等. 2006. 从瞬变电磁测深数据到平面电磁波场数据的等效转换[J]. 地球物理学报, 49(5): 1539-1545, doi: 10.3321/j.issn:0001-5733.2006.05.036.
薛国强, 李貅, 戚志鹏, 等. 2011. 瞬变电磁拟地震子波宽度压缩研究[J]. 地球物理学报, 54(5): 1384-1390, doi: 10.3969/j.issn.0001-5733.2011.05.028.
薛国强, 李貅, 宋建平, 等. 2004. 回线源瞬变电磁成像的理论分析及数值计算[J]. 地球物理学报, 47(02): 338-343, doi: 10.3321/j.issn:0001-5733.2004.02.023.
杨文采. 1995. 波动方程的逆散射与迭代线性化反演[J]. 石油物探, 34(03): 114-121.
张军, 李貅, 赵莹, 等. 2011. 瞬变电磁虚拟波场高分辨成像技术研究[J]. 地球物理进展, 26(03): 1077-1084, doi: 10.3969/j.issn.1004-2903.2011.03.038.
朱宏伟, 李貅, 张军, 等. 2010. 瞬变电磁法三维拟地震成像信息提取技术[J]. 地球物理学进展, 25(05): 1648-1656, doi: 10.3969/j.issn.1004-2903.2010.05.017.
Bleistein N, Cohen J K, Stockwell J W, et al. 2001. Mathematics of Multidimensional Seismic Imaging, Migration, and Inversion [J]. Applied Mechanics Reviews, 54(5): B94-B96, doi: 10.1007/978-1- 4613- 0001- 4.
Cohen J K, Bleistein N. 1979. Velocity inversion procedure for acoustic waves [J]. Geophysics, 44(6): 1077-1087, doi: 10.1190/1.1440996.
Huang L J, Yang W C. 1991. Approximate method for inverse scattering inversion of acoustic equation [J]. Chinese Journal of geophysics (in Chinese), 34(05): 626- 634, doi: 10.3321/j.issn:0001-5733.1991.05.011.
Lee K H, Liu G, Morrison H F. 1989. A New Approach to Modeling the Electromagnetic Response of Conductive Media [J]. Geophysics, 54(9): 1180-1192, doi: 10.1190/1.1442753.
Lee K H, Xie G Q. 1993. A New Approach to Imaging with Low-frequency Electromagnetic Fields [J]. Geophysics, 58(6): 780-796, doi: 10.1190/1.1443464.
Li X, Qi Z P, Xue G Q, et al. 2010. Extension imaging of 3d curved surface of transient electromagnetic virtual wave field [J]. Chinese Journal of geophysics (in Chinese), 53(12): 3005-3011, doi: 10.3969/j.issn.0001-5733.2010.12.025.
Li X, Xue G Q, Guo W B. 2007. Research progress in TEM pseudo-seismic imaging [J]. Progress in Geophysics (in Chinese), 22(3): 811-816, doi: 10.3969/j.issn.1004-2903.2007.03.023.
Li X, Xue G Q, Liu Y A, et al. 2012. A research on TEM imaging method based on synthetic-aperture technology [J]. Chinese Journal of geophysics (in Chinese), 55(01): 333-340, doi: 10.6038/j.issn.0001-5733.2012.01.034.
Li X, Xue G Q, Song J P, et al. 2005. An optimize method for transient electromagnetic field-wave field conversion [J]. Chinese Journal of geophysics (in Chinese), 48(05): 1185-1190, doi:10.3321/jivssn:0001-5733.2005.05.029.
Qi Z P, Li X, Wu Q, et al. 2013. A new algorithm for full-time-domain wave-field transformation based on transient electromagnetic method [J]. Chinese Journal of geophysics (in Chinese), 56(10): 3581-3595, doi: 10.6038/cjg20131033.
Xie G Q, Li J H. 1988. New feature iteration method for solving inverse scattering potential problems of three-dimensional wave equation [J]. Chinese science (series A mathematical science, astronomy and technology science) (in Chinese), A(04): 359-365.
Xue G Q, Li X, Guo W B, et al. 2006. Equivalent transformation from TEM field sounding data to plane wave electromagnetic sounding data [J]. Chinese Journal Of Geophysics (in Chinese), 49(5): 1539-1545, doi: 10.3321/j.issn:0001-5733.2006.05.036.
Xue G Q, Li X, Qi Z P, et al. 2011. Study of sharpening the wave-form of TEM pseudo-seismic [J]. Chinese Journal Of Geophysics (in Chinese), 54(5): 1384-1390, doi: 10.3969/j.issn.0001-5733.2011.05.028.
Xue G Q, Li X, Song J P, et al. 2004. Theoretical analysis and numerical calculation of transient electromagnetic imaging of loop source [J]. Chinese Journal of geophysics (in Chinese), 47(02): 338-343, doi: 10.3321/j.issn:0001-5733.2004.02.023.
Yang W C. 1995. Inverse scattering and iterative linearization inversion of wave equation [J]. Petroleum geophysics (in Chinese), 34(03): 114-121.
Zhang J, Li X, Zhao Y, et al. 2011. A technology research of high-resolation imaging for the transient electromagnetic pseudo wave field [J]. Progress in geophysics (in Chinese), 26(03): 1077-1084, doi: 10.3969/j.issn.1004-2903.2011.03.038.
Zhu H W, Li X, Zhang J, et al. 2010. Information collecting technology in 3-D pseudo-seismic imaging of transient electromagnetics [J]. Progress in geophysics (in Chinese), 25(05): 1648-1656, doi: 10.3969/j.issn.1004-2903.2010.05.017.
黄联捷, 杨文采. 1991. 声波方程逆散射反演的近似方法[J]. 地球物理学报, 34(05): 626- 634, doi: 10.3321/j.issn:0001-5733.1991.05.011.
李貅, 戚志鹏, 薛国强, 等. 2010. 瞬变电磁虚拟波场的三维曲面延拓成像[J]. 地球物理学报, 53(12): 3005-3011, doi: 10.3969/j.issn.0001-5733.2010.12.025.
李貅, 薛国强, 郭文波. 2007. 瞬变电磁法拟地震成像研究进展[J]. 地球物理学进展, 22(03): 811-816, doi: 10.3969/j.issn.1004-2903.2007.03.023.
李貅, 薛国强, 刘银爱, 等. 2012. 瞬变电磁合成孔径成像方法研究[J]. 地球物理学报, 55(01): 333-340, doi: 10.6038/j.issn.0001-5733.2012.01.034.
李貅, 薛国强, 宋建平, 等. 2005. 从瞬变电磁场到波场的优化算法[J]. 地球物理学报, 48(05): 1185-1190, doi: 10.3321/j.issn:0001-5733.2005.05.029.
戚志鹏, 李貅, 吴琼, 等. 2013. 从瞬变电磁扩散场到拟地震波场的全时域反变换算法[J]. 地球物理学报, 56(10): 3581-3595, doi: 10.6038/cjg20131033.
谢干权, 李建华. 1988. 解三维波方程逆散射势问题的新特征迭代法[J]. 中国科学(A辑数学理学天文学技术科学), A(04): 359-365.
薛国强, 李貅, 郭文波, 等. 2006. 从瞬变电磁测深数据到平面电磁波场数据的等效转换[J]. 地球物理学报, 49(5): 1539-1545, doi: 10.3321/j.issn:0001-5733.2006.05.036.
薛国强, 李貅, 戚志鹏, 等. 2011. 瞬变电磁拟地震子波宽度压缩研究[J]. 地球物理学报, 54(5): 1384-1390, doi: 10.3969/j.issn.0001-5733.2011.05.028.
薛国强, 李貅, 宋建平, 等. 2004. 回线源瞬变电磁成像的理论分析及数值计算[J]. 地球物理学报, 47(02): 338-343, doi: 10.3321/j.issn:0001-5733.2004.02.023.
杨文采. 1995. 波动方程的逆散射与迭代线性化反演[J]. 石油物探, 34(03): 114-121.
张军, 李貅, 赵莹, 等. 2011. 瞬变电磁虚拟波场高分辨成像技术研究[J]. 地球物理进展, 26(03): 1077-1084, doi: 10.3969/j.issn.1004-2903.2011.03.038.
朱宏伟, 李貅, 张军, 等. 2010. 瞬变电磁法三维拟地震成像信息提取技术[J]. 地球物理学进展, 25(05): 1648-1656, doi: 10.3969/j.issn.1004-2903.2010.05.017.