可控源音频大地电磁法三维交错采样有限差分数值模拟研究
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摘要
可控源音频大地电磁法(CSAMT)是在大地电磁法(MT)基础上发展起来的一种人工源频率域测深方法,该方法在矿产资源、油气、地热及工程勘查等领域得到了广泛的应用。当前针对CSAMT的研究主要集中在二维和2.5维,然而CSAMT的源和电磁场本质上是三维,因此CSAMT三维数值模拟是勘探地球物理领域迫切需要解决的前沿问题之一。本文将多重网格法与交错采样有限差分法结合,开展了CSAMT三维数值模拟研究。
从CSAMT满足的基本方程出发,推导了一次场和二次场所满足的麦克斯韦方程,得出了全空间下层状介质的解析解,通过典型模型数值计算确定了最佳汉克尔滤波系数。将三维交错采样有限差分应用到二次场计算中,推导了磁场分量与电场分量的关系表达式,将CSAMT三维数值模拟的二次场计算问题转换为大型线性方程组的求解,提出了各边界的二次场值为零的简洁边界条件。
多重网格法中采用标准粗化方法使粗网格单元值等于其对应的八个细网格单元值的加权平均。从磁场的三个分量出发,推导了限制算子和延拓算子,并在计算过程中保证限制算子比延拓算子至少高一阶代数精度。采用对称对角行存储格式,利用两个一维数组进行系数矩阵存储,使总体存储量达到最小。研究了三种磁场散度校正方法(直接磁场散度校正、磁场散度残差复校正、磁场散度残差实校正),并通过实际模型试算证明磁场散度残差实校正为最佳方法。通过多重网格法和两种经典的迭代方法的比较,证明多重网格法误差收敛速度与网格剖分尺度无关,剖分网格数对迭代次数基本没有影响,当网格剖分数大时多重网格法的迭代次数及计算时间均比不完全LU分解双共轭梯度迭代法少。通过研究多重网格层数对计算效率的影响,证明在网格剖分数相同情况下应尽可能增加多重网格层数。结合CSAMT三维数值模拟的特点,编制了基于MPI的三维数值模拟程序,利用相同模型比较了两种程序的计算速度,结果表明多核PC机的并行计算能大大提高计算效率。通过典型模型体的CSAMT三维数值模拟,分析了典型模型体CSAMT赤道装置和轴向装置的异常响应,并对场源附加效应和场源阴影效应进行了研究。
Controlled-source audio-frequency magnetotellurics (CSAMT) method, a frequency domain sounding method using artificial source which is developing from the magnetotellurics (MT) method, has been widely applied to mineral exploration, hydrocarbon exploration, geothermal exploration and engineering exploration. Most studies on CSAMT are focusing on the two dimensional and 2.5 dimensional resistivity models. While the source and the electromagnetic fields of CSAMT are essentially three dimension. Therefore, three dimensional CSAMT numerical simulation is one of the key scientific problems which are urgent to be solved in geophysical surveys. In this paper, the three dimensional CSAMT numerical simulation using multigrid method and staggered-grid finite difference method is studied in depth.
From the basic principles of the CSAMT, the Maxwell equation of the primary field and the secondary field are derived respectively. The analytical solution of layer model is worked out. The best Hankel filter coefficients are obtained by simulate some typical models. The three dimensional staggered-grid finite difference is applied to the secondary field numerical simulation. Based on the staggered-grid discretization, the relations betwen electrical components and magnetic components are derived. The secondary filed computation of three dimensional CSAMT numerical simulation is transformed to the solution of linear equations systems. The more explicit boundary conditions that the secondary field value is zero in each boundary are introduced.
In order to ensure a coarse grid filed value is equal to the weighted average of corresponding eight fine grids, a standard coarsening strategy is employed in multigrid method. The restriction operators and the interpolation operators are derived based on three magnetic field components. And the algebraic precision of restriction operators is one order higher than the interpolation operators at least. For sparse and large matrix in CSAMT three dimensional simulation, diagonally symmetrical row storage method is used. A great deal of storage space is saved by using two one dimensional arrays to store the matrix. Three divergence correction methods are presented from the characteristic that magnetic divergence in Maxwell equations is zero. Those methods are directly divergence correction of magnetic field (DDCM), divergence residual plural correction of magnetic field (RPCM) and divergence residua real correction of magnetic field (RRCM). Some geo-electrical models are simulated using the three divergence correction methods. The results show that RRCM is the best divergence correction method. Compared with the two traditional iteration methods, the convergence rate of multigrid method is nearly independent of meshing size. In large scale meshing grids, the computational time and iteration numbers of multigrid method is much fewer than ILU-BICG method. The result shows that more multigrid layers are better in the same meshing grids. According to the characteres of three dimensional CSAMT numerical simulation, we developed three dimensional CSAMT numerical modeling program using Message Passing Interface (MPI) technique. The computational speed of the two programs is compared with the same model. The result shows that the parallel algorithm based on multi-core PC can improve the computational efficiency greatly as well.
In the last part of this paper, some typical model’s responses of equatorial setting and axial setting are analyzed by CSAMT three dimensional numerical simulation. And the overprint effect and shadow effect of the source are also studied with three dimensional numerical simulation.
从CSAMT满足的基本方程出发,推导了一次场和二次场所满足的麦克斯韦方程,得出了全空间下层状介质的解析解,通过典型模型数值计算确定了最佳汉克尔滤波系数。将三维交错采样有限差分应用到二次场计算中,推导了磁场分量与电场分量的关系表达式,将CSAMT三维数值模拟的二次场计算问题转换为大型线性方程组的求解,提出了各边界的二次场值为零的简洁边界条件。
多重网格法中采用标准粗化方法使粗网格单元值等于其对应的八个细网格单元值的加权平均。从磁场的三个分量出发,推导了限制算子和延拓算子,并在计算过程中保证限制算子比延拓算子至少高一阶代数精度。采用对称对角行存储格式,利用两个一维数组进行系数矩阵存储,使总体存储量达到最小。研究了三种磁场散度校正方法(直接磁场散度校正、磁场散度残差复校正、磁场散度残差实校正),并通过实际模型试算证明磁场散度残差实校正为最佳方法。通过多重网格法和两种经典的迭代方法的比较,证明多重网格法误差收敛速度与网格剖分尺度无关,剖分网格数对迭代次数基本没有影响,当网格剖分数大时多重网格法的迭代次数及计算时间均比不完全LU分解双共轭梯度迭代法少。通过研究多重网格层数对计算效率的影响,证明在网格剖分数相同情况下应尽可能增加多重网格层数。结合CSAMT三维数值模拟的特点,编制了基于MPI的三维数值模拟程序,利用相同模型比较了两种程序的计算速度,结果表明多核PC机的并行计算能大大提高计算效率。通过典型模型体的CSAMT三维数值模拟,分析了典型模型体CSAMT赤道装置和轴向装置的异常响应,并对场源附加效应和场源阴影效应进行了研究。
Controlled-source audio-frequency magnetotellurics (CSAMT) method, a frequency domain sounding method using artificial source which is developing from the magnetotellurics (MT) method, has been widely applied to mineral exploration, hydrocarbon exploration, geothermal exploration and engineering exploration. Most studies on CSAMT are focusing on the two dimensional and 2.5 dimensional resistivity models. While the source and the electromagnetic fields of CSAMT are essentially three dimension. Therefore, three dimensional CSAMT numerical simulation is one of the key scientific problems which are urgent to be solved in geophysical surveys. In this paper, the three dimensional CSAMT numerical simulation using multigrid method and staggered-grid finite difference method is studied in depth.
From the basic principles of the CSAMT, the Maxwell equation of the primary field and the secondary field are derived respectively. The analytical solution of layer model is worked out. The best Hankel filter coefficients are obtained by simulate some typical models. The three dimensional staggered-grid finite difference is applied to the secondary field numerical simulation. Based on the staggered-grid discretization, the relations betwen electrical components and magnetic components are derived. The secondary filed computation of three dimensional CSAMT numerical simulation is transformed to the solution of linear equations systems. The more explicit boundary conditions that the secondary field value is zero in each boundary are introduced.
In order to ensure a coarse grid filed value is equal to the weighted average of corresponding eight fine grids, a standard coarsening strategy is employed in multigrid method. The restriction operators and the interpolation operators are derived based on three magnetic field components. And the algebraic precision of restriction operators is one order higher than the interpolation operators at least. For sparse and large matrix in CSAMT three dimensional simulation, diagonally symmetrical row storage method is used. A great deal of storage space is saved by using two one dimensional arrays to store the matrix. Three divergence correction methods are presented from the characteristic that magnetic divergence in Maxwell equations is zero. Those methods are directly divergence correction of magnetic field (DDCM), divergence residual plural correction of magnetic field (RPCM) and divergence residua real correction of magnetic field (RRCM). Some geo-electrical models are simulated using the three divergence correction methods. The results show that RRCM is the best divergence correction method. Compared with the two traditional iteration methods, the convergence rate of multigrid method is nearly independent of meshing size. In large scale meshing grids, the computational time and iteration numbers of multigrid method is much fewer than ILU-BICG method. The result shows that more multigrid layers are better in the same meshing grids. According to the characteres of three dimensional CSAMT numerical simulation, we developed three dimensional CSAMT numerical modeling program using Message Passing Interface (MPI) technique. The computational speed of the two programs is compared with the same model. The result shows that the parallel algorithm based on multi-core PC can improve the computational efficiency greatly as well.
In the last part of this paper, some typical model’s responses of equatorial setting and axial setting are analyzed by CSAMT three dimensional numerical simulation. And the overprint effect and shadow effect of the source are also studied with three dimensional numerical simulation.
引文
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