三维可控源电磁法非线性共轭梯度反演研究
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摘要
三维反演是电磁法解释的最有效手段,它可以克服一维和二维反演中存在的问题,反演出正确的地电结构。实现真正的完全三维反演是众多地球物理工作者的追求,但是由于其复杂性,三维反演技术尚未成熟,特别是可控源电磁法的三维反演,国外的研究还都停留在理论上,而国内的三维可控源电磁法研究还处于正演阶段,针对这一现状。本论文提出利用交错网格有限差分法来实现正演,然后利用非线性共轭梯度法完成可控源资料三维反演。
本论文首先给出了适用于全空间层状模型各种基本源的格林函数广义描述,其核心是获得过源平行于层界面的虚界面上矢量势谱振幅的系数。该方法首先避开具体激发源的物理描述,将格林函数统一分解为TE模式场和TM模式场,然后利用电场切向分量连续性和磁场切向分量在含源虚界面处的不连续性求出源所在虚界面上的初始振幅系数,最后根据TE场和TM场的传播理论递推获得各层场的振幅。通过本文理论获得的过源(虚)界面上矢量势初始振幅系数在形式上可以分解为全空间格林函数的振幅系数(源强度)和模型参数对场初始振幅的影响两个部分,并且对于不同的源只需要替换对应源的全空间格林函数振幅系数即可得到对应格林函数的一般表达式。在此基础上,引进了二级近似离散复镜像法来实现低频格林函数快速、精确的计算。通过数值分析,给出了在低频地球物理电磁场计算中该方法近似参数的选取原则。(1)、二级近似的积分区间里谱格林函数采样个数N为十倍的近似多项式的个数M；(2)、总的积分区间L2大小约为40/r,r为收发距离；(3)、两个积分区间的分界值L1为总的积分区间大小L2与第二个积分区间采样数N1的比值的Q倍；(4)、Q的最优值根据谱格林函数拟合相对误差随Q值变化的近似误差曲线与横轴的交点来确定。将该参数选取原则应用到有限长双极源电磁场计算中,计算得到的视电阻率和直接积分结果吻合,表明本论文提出的计算方案是可行的。
接着本论文利用基于二次场的交错网格有限差分方法来实现可控源电磁勘探的三维正演。计算过程中,Helmholtz方程右手源项中的基本场采用上述的数值计算技术进行。对均匀半空间近似模型的模拟结果与解析解基本吻合。对低阻异常和高阻异常体的不同频率的二次场和总场的形态特点进行了分析,发现：对于低阻体,其正上方的Ex总的电场强度减小,沿x轴方向在异常体两边Ex总的电场强度增加；对于高阻体,其变化规律与低阻体相反；对多个异常体的正演结果可以看出,本论文采用的长接地导线源对于低阻的敏感度要明显的高于对高阻的敏感度,这可以作为反演时选择背景电阻率的一个参考。
借助于基于交错网格的有限差分正演模拟,本论文对非线性共轭梯度法的可控源资料三维反演进行了研究。在进行反演计算时,非线性共轭梯度法不需要显式计算灵敏度矩阵,而只需要计算矩阵和向量的乘积,这只需要一次正演和一次伴随正演。再加上为了计算步长而进行的线性搜索需要的最少三次正演,一次迭代只需要6次左右的正演就可以完成。对简单异常体的反演结果表明,非线性共轭梯度法可以较准确的给出异常体的位置、大小和电阻率等信息。对于层状异常,反演只能正确的给出测区之内中间范围的地电结构。对于测区外的较小异常体模型反演表明,反演异常体的位置基本反映了真实异常,并对测区内反演的电阻率分布基本没有影响,基本消除了常规可控源音频大地电磁反演的所谓阴影效应。对野外数据的三维反演结果和二维反演结果基本一致,证明了本论文的三维非线性共轭梯度算法是完全可靠的。
Three dimensional inversion is the most effective method in data processing of electromagnetic method, which can get the correct geoelectric structure avoiding many problems in one dimensional inversion and two dimensional inversion. Many geophysics pursuit of full three dimensional inversion, but it's complexity make it still under development. Especially in controlled source electromagnetic method, the study is stay in theory abroad. Domestic three dimensional controlled source electromagnetic research is in forward modeling, and for such situation, this paper make use of staggered grid finite difference to achieve forward modeling, and nonlinear conjugate gradient method for three dimensional inversion.
Firstly, this paper introduces a method to generalize the description of Green's functions from various basic sources buried in stratified earths. The key of the theory is based on calculating the vector potential's amplitude coefficient on the virtual interface passing through the source and paralleling to the other interfaces. In order to get the generalized expression, we firstly separate Green's function into TE mode and TM mode, then a virtual interface parallel to layer interfaces is inserted through source. The continuity of the tangential electric field and the discontinuity of the tangential magnetic field on the virtual interface are then utilized to obtain the initial amplitude coefficient on the interface. Lastly, the amplitude coefficients in each layer will be recurred from the initial amplitudes using the propagation theory of TE mode and TM mode. The obtained initial amplitudes of vector potential on the virtual interface can be decomposed into the amplitude coefficient of whole space Green's function (source intensity) and the impact factor from model parameters; and for different source, to get the corresponding Green's function, only the amplitude coefficient of whole space is necessary to be changed. The methodology unifies Green's functions in electromagnetic exploration, simplifies the computation and gets better numerical stability. On this basis, this paper discusses how to applying the two-level approximate discrete complex image method to fast and accurately compute the low frequency Green's functions. Through numerical simulation, the approximate parameters in the compution of low frequency electromagnetc field is determined as follows:(1) Sampling number N of spectral green's function in the two-approximate integral interval are ten times of the number of approximate polynomial M; (2) The integral interval is about 40/r, where r is the offset; (3) The conjunction value L1 of the two level integral interval is Q times of the ratio of the total size of the integral interval L2 and the sampling number N1 of the second integral interval; (4) the optimal value of Q is decided by the cross point of the spectral Green's function approximate error plot about Q and the horizontal axis. The application of this principle to the calculation of the electromagnetic fields from a finite bipolar source shows that the calculated apparent resistivity agree with direct integral result very well, this reveals that the the proposed strategy is feasible.
Then, this paper use finite difference method based on the secondary field to achieve three dimensional modeling of controlled source electromagnetic. This paper analysis the secondary field and total field's morphological characteristics of low resistivity abnormal body and high resistivity abnormal with different frequences, and summed as:for low resistivity abnormal body, the total electric field Ex become small just above it, and on both sides of the abnormal body along the x-axis, the total electric field Ex become large; for the high resistivity abnormal body, its law to the contray. From the result of many abnormal bodys, it can be seen that the low resistivity abnormal body is easer to be distinguished then the high resistivity abnormal body, which can be a reference when select background resistivity in inversion.
Based on the staggered grid finite difference modeling, we use nonlinear conjugate gradient method to solve the problem of three dimensional inversion. When in the inversion calculation, nonlinear conjugate gradient method doesn't need to compute the sensitivity matrix explicitly, but only to calculate the product of matrix and vector, which just one forward modeling and one adjoint forward modeling. Although, another 3 or 4 forward problems should be done to evaluate the optimal step lenth, thus, there are totally about 5 time forward modeling in each iteration, the whole computation timen is much less than directly sensitivity matrix computation method. The inversion results of simple abnormal body show that nonlinear conjugate gradient method can get the correct position、size and resistivity of abnormal body. This paper also discusse the layered abnormal body and the abnormal body outside the surveyed area. For the layered abnormal body, we can only get the geoelectric structure in the surveyed area; for the little abnormal body outside, inversion give the right position of the abnormal body, and doesn't affect the inversional resistivity in the surveyed area. This shows that three dimensional inversion eliminate the shadow effect; field data inversion shows that three dimensional inversion give the same results as two dimensional inversion, which prove the algorithm of three dimensional nonlinear conjugate gradient method in this paper is reliable.
本论文首先给出了适用于全空间层状模型各种基本源的格林函数广义描述,其核心是获得过源平行于层界面的虚界面上矢量势谱振幅的系数。该方法首先避开具体激发源的物理描述,将格林函数统一分解为TE模式场和TM模式场,然后利用电场切向分量连续性和磁场切向分量在含源虚界面处的不连续性求出源所在虚界面上的初始振幅系数,最后根据TE场和TM场的传播理论递推获得各层场的振幅。通过本文理论获得的过源(虚)界面上矢量势初始振幅系数在形式上可以分解为全空间格林函数的振幅系数(源强度)和模型参数对场初始振幅的影响两个部分,并且对于不同的源只需要替换对应源的全空间格林函数振幅系数即可得到对应格林函数的一般表达式。在此基础上,引进了二级近似离散复镜像法来实现低频格林函数快速、精确的计算。通过数值分析,给出了在低频地球物理电磁场计算中该方法近似参数的选取原则。(1)、二级近似的积分区间里谱格林函数采样个数N为十倍的近似多项式的个数M；(2)、总的积分区间L2大小约为40/r,r为收发距离；(3)、两个积分区间的分界值L1为总的积分区间大小L2与第二个积分区间采样数N1的比值的Q倍；(4)、Q的最优值根据谱格林函数拟合相对误差随Q值变化的近似误差曲线与横轴的交点来确定。将该参数选取原则应用到有限长双极源电磁场计算中,计算得到的视电阻率和直接积分结果吻合,表明本论文提出的计算方案是可行的。
接着本论文利用基于二次场的交错网格有限差分方法来实现可控源电磁勘探的三维正演。计算过程中,Helmholtz方程右手源项中的基本场采用上述的数值计算技术进行。对均匀半空间近似模型的模拟结果与解析解基本吻合。对低阻异常和高阻异常体的不同频率的二次场和总场的形态特点进行了分析,发现：对于低阻体,其正上方的Ex总的电场强度减小,沿x轴方向在异常体两边Ex总的电场强度增加；对于高阻体,其变化规律与低阻体相反；对多个异常体的正演结果可以看出,本论文采用的长接地导线源对于低阻的敏感度要明显的高于对高阻的敏感度,这可以作为反演时选择背景电阻率的一个参考。
借助于基于交错网格的有限差分正演模拟,本论文对非线性共轭梯度法的可控源资料三维反演进行了研究。在进行反演计算时,非线性共轭梯度法不需要显式计算灵敏度矩阵,而只需要计算矩阵和向量的乘积,这只需要一次正演和一次伴随正演。再加上为了计算步长而进行的线性搜索需要的最少三次正演,一次迭代只需要6次左右的正演就可以完成。对简单异常体的反演结果表明,非线性共轭梯度法可以较准确的给出异常体的位置、大小和电阻率等信息。对于层状异常,反演只能正确的给出测区之内中间范围的地电结构。对于测区外的较小异常体模型反演表明,反演异常体的位置基本反映了真实异常,并对测区内反演的电阻率分布基本没有影响,基本消除了常规可控源音频大地电磁反演的所谓阴影效应。对野外数据的三维反演结果和二维反演结果基本一致,证明了本论文的三维非线性共轭梯度算法是完全可靠的。
Three dimensional inversion is the most effective method in data processing of electromagnetic method, which can get the correct geoelectric structure avoiding many problems in one dimensional inversion and two dimensional inversion. Many geophysics pursuit of full three dimensional inversion, but it's complexity make it still under development. Especially in controlled source electromagnetic method, the study is stay in theory abroad. Domestic three dimensional controlled source electromagnetic research is in forward modeling, and for such situation, this paper make use of staggered grid finite difference to achieve forward modeling, and nonlinear conjugate gradient method for three dimensional inversion.
Firstly, this paper introduces a method to generalize the description of Green's functions from various basic sources buried in stratified earths. The key of the theory is based on calculating the vector potential's amplitude coefficient on the virtual interface passing through the source and paralleling to the other interfaces. In order to get the generalized expression, we firstly separate Green's function into TE mode and TM mode, then a virtual interface parallel to layer interfaces is inserted through source. The continuity of the tangential electric field and the discontinuity of the tangential magnetic field on the virtual interface are then utilized to obtain the initial amplitude coefficient on the interface. Lastly, the amplitude coefficients in each layer will be recurred from the initial amplitudes using the propagation theory of TE mode and TM mode. The obtained initial amplitudes of vector potential on the virtual interface can be decomposed into the amplitude coefficient of whole space Green's function (source intensity) and the impact factor from model parameters; and for different source, to get the corresponding Green's function, only the amplitude coefficient of whole space is necessary to be changed. The methodology unifies Green's functions in electromagnetic exploration, simplifies the computation and gets better numerical stability. On this basis, this paper discusses how to applying the two-level approximate discrete complex image method to fast and accurately compute the low frequency Green's functions. Through numerical simulation, the approximate parameters in the compution of low frequency electromagnetc field is determined as follows:(1) Sampling number N of spectral green's function in the two-approximate integral interval are ten times of the number of approximate polynomial M; (2) The integral interval is about 40/r, where r is the offset; (3) The conjunction value L1 of the two level integral interval is Q times of the ratio of the total size of the integral interval L2 and the sampling number N1 of the second integral interval; (4) the optimal value of Q is decided by the cross point of the spectral Green's function approximate error plot about Q and the horizontal axis. The application of this principle to the calculation of the electromagnetic fields from a finite bipolar source shows that the calculated apparent resistivity agree with direct integral result very well, this reveals that the the proposed strategy is feasible.
Then, this paper use finite difference method based on the secondary field to achieve three dimensional modeling of controlled source electromagnetic. This paper analysis the secondary field and total field's morphological characteristics of low resistivity abnormal body and high resistivity abnormal with different frequences, and summed as:for low resistivity abnormal body, the total electric field Ex become small just above it, and on both sides of the abnormal body along the x-axis, the total electric field Ex become large; for the high resistivity abnormal body, its law to the contray. From the result of many abnormal bodys, it can be seen that the low resistivity abnormal body is easer to be distinguished then the high resistivity abnormal body, which can be a reference when select background resistivity in inversion.
Based on the staggered grid finite difference modeling, we use nonlinear conjugate gradient method to solve the problem of three dimensional inversion. When in the inversion calculation, nonlinear conjugate gradient method doesn't need to compute the sensitivity matrix explicitly, but only to calculate the product of matrix and vector, which just one forward modeling and one adjoint forward modeling. Although, another 3 or 4 forward problems should be done to evaluate the optimal step lenth, thus, there are totally about 5 time forward modeling in each iteration, the whole computation timen is much less than directly sensitivity matrix computation method. The inversion results of simple abnormal body show that nonlinear conjugate gradient method can get the correct position、size and resistivity of abnormal body. This paper also discusse the layered abnormal body and the abnormal body outside the surveyed area. For the layered abnormal body, we can only get the geoelectric structure in the surveyed area; for the little abnormal body outside, inversion give the right position of the abnormal body, and doesn't affect the inversional resistivity in the surveyed area. This shows that three dimensional inversion eliminate the shadow effect; field data inversion shows that three dimensional inversion give the same results as two dimensional inversion, which prove the algorithm of three dimensional nonlinear conjugate gradient method in this paper is reliable.
引文
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