A 3-D FINITE-DIFFERENCE ALGORITHM FOR DC RESISTIVITY MODELLING USING CONJUGATE GRADIENT METHODS
A 3-D FINITE-DIFFERENCE ALGORITHM FOR DC RESISTIVITY MODELLING USING CONJUGATE GRADIENT METHODS
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DOI：
10.1111/j.1365-246x.1995.tb06897.x
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发表时间：
1995-12
影响因子：
2.8
通讯作者：
K. Spitzer
中科院分区：
文献类型：
--
作者：
K. Spitzer
本文提出了一种精确、高效的直流电阻率三维有限差分正演算法。电阻率问题的控制微分方程离散使用中心有限差分，来自二阶泰勒级数展开。电导率值可以任意分布在半空间内。网格点处的电导率通过分配给网格单元的电导率的体积加权算术平均值来计算。采用可变网格间距。该算法不限制源的数量和配置，尽管所有说明性示例都是使用表面处的两个电流电极来计算的。一般来说，这种离散化产生的线性方程组是非对称的，需要广义数值方程求解器。然而，在对称化矩阵方程之后，普通的共轭梯度法变得适用。它利用了矩阵的对称性，因而比一般的方法优越上级。一个有效的SSOR预条件（SSOR对称连续超松弛）提供快速收敛，通过减少矩阵的谱条件数，而不使用额外的内存。此外，紧凑的存储方案减少了存储器需求并加速了数学矩阵运算。五个不同的方程求解器的性能进行了研究的CPU时间。预处理共轭梯度法（CGPC）被证明是最有效的矩阵求解器，并能够解决大型方程组在中等时间（约21/2分钟的DEC阿尔法工作站上的网格与50 000个节点，48分钟为200000节点）。指出了容差值在迭代过程停止准则中的重要性。为了调查的准确性，数值计算结果进行了比较，分析或其他解决方案为三个不同的模型类，产生的最大偏差为3.5%或更少的视电阻率的计算值。总之，该算法为电阻率建模的实际应用提供了一个强大而灵活的工具。
SUMMARY An accurate and efficient 3-D finite-difference forward algorithm for DC resistivity modelling is developed. The governing differential equations of the resistivity problem are discretized using central finite differences that are derived by a second-order Taylor series expansion. Electrical conductivity values may be arbitrarily distributed within the half-space. Conductivities at the grid points are calculated by a volume-weighted arithmetic average from conductivities assigned to grid cells. Variable grid spacing is incorporated. The algorithm does not limit the number and configuration of the sources, although all illustrative examples are computed using two current electrodes at the surface. In general, the linear set of equations resulting from this kind of discretization is non-symmetric and requires generalized numerical equation solvers. However, after symmetrizing the matrix equations, the ordinary conjugate gradient method becomes applicable. It takes advantage of the matrix symmetry and, thus, is superior to the generalized methods. An efficient SSOR-preconditioner (SSOR symmetric successive overrelaxation) provides fast convergence by decreasing the spectral condition number of the matrix without using additional memory. Furthermore, a compact storage scheme reduces memory requirements and accelerates mathematical matrix operations. The performance of five different equation solvers is investigated in terms of cpu time. The preconditioned conjugate gradient method (CGPC) is shown to be the most efficient matrix solver and is able to solve large equation systems in moderate times (approximately 21/2 minutes on a DEC alpha workstation for a grid with 50 000 nodes, and 48 minutes for 200000 nodes). The importance of the tolerance value in the stopping criterion for the iteration process is pointed out. In order to investigate the accuracy, the numerical results are compared with analytical or other solutions for three different model classes, yielding maximum deviations of 3.5 per cent or much less for most of the computed values of the apparent resistivity. In conclusion, the presented algorithm provides a powerful and flexible tool for practical application in resistivity modelling.