Computations of secondary potential for 3D DC resistivity modelling using an incomplete Choleski conjugate‐gradient method
Computations of secondary potential for 3D DC resistivity modelling using an incomplete Choleski conjugate‐gradient method
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DOI：
10.1046/j.1365-2478.2003.00392.x
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发表时间：
2003-11
影响因子：
2.6
通讯作者：
Xiaoping Wu;Yifei Xiao;Cheng Qi;Tong-Tsung Wang
中科院分区：
文献类型：
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作者：
Xiaoping Wu;Yifei Xiao;Cheng Qi;Tong-Tsung Wang
提出了一种高精度、高效率的三维有限差分直流电阻率正演算法。通常，FD计算最耗时的部分是求解大型线性方程组：AX=b，其中A是大型稀疏带对称矩阵。使用完全Choleski分解的直接方法相当慢，并且需要更多的计算机存储。我们引入了行索引稀疏存储模式来存储系数矩阵A，并引入了不完全Choleski共轭梯度(ICCG)方法来求解大型线性方程组。通过利用矩阵的对称性和稀疏性，ICCG方法的收敛速度更快，需要的计算机存储空间也更少。在一台533 MHz的奔腾计算机上，对于一个46020个节点的网格，大约需要15个S，比直接法快700倍，比对称逐次超松弛共轭梯度法快2.5倍。与用改进的ICCG求解器进行三维有限元电阻率模拟相比，我们的算法在迭代次数和计算机时间方面都有更高的效率。此外，我们还通过对FD方程的简单处理来求解三维直流电阻率模拟中的二次电势。两层模型和垂直接触的数值算例表明，该方法比直接求解相同网格节点下的总势具有更高的精度。此外，还对三维立方体进行了数值模拟，其偶极-偶极视电阻率与有限元和积分方程法的计算结果吻合较好。综上所述，多种技术的结合提供了一种快速、准确的3D FD正演模拟方法，这是3D电阻率反演的基础。
An accurate and efficient 3D finite‐difference (FD) forward algorithm for DC resistivity modelling is developed. In general, the most time‐consuming part of FD calculation is to solve large sets of linear equations: Ax=b, where A is a large sparse band symmetric matrix. The direct method using complete Choleski decomposition is quite slow and requires much more computer storage. We have introduced a row‐indexed sparse storage mode to store the coefficient matrix A and an incomplete Choleski conjugate‐gradient (ICCG) method to solve the large linear systems. By taking advantage of the matrix symmetry and sparsity, the ICCG method converges much more quickly and requires much less computer storage. It takes approximately 15 s on a 533 MHz Pentium computer for a grid with 46 020 nodes, which is approximately 700 times faster than the direct method and 2.5 times faster than the symmetric successive over‐relaxation (SSOR) conjugate‐gradient method. Compared with 3D finite‐element resistivity modelling with the improved ICCG solver, our algorithm is more efficient in terms of number of iterations and computer time. In addition, we solve for the secondary potential in 3D DC resistivity modelling by a simple manipulation of the FD equations. Two numerical examples of a two‐layered model and a vertical contact show that the method can achieve much higher accuracy than solving for the total potential directly with the same grid nodes. In addition, a 3D cubic body is simulated, for which the dipole–dipole apparent resistivities agree well with the results obtained with the finite‐element and integral‐equation methods. In conclusion, the combination of several techniques provides a rapid and accurate 3D FD forward modelling method which is fundamental to 3D resistivity inversion.