Development of the software of three-dimensional interpretation for electromagnetic tomography

在研究有限元方法在三维电磁层析成像中的适用性时，发现当电导率对比度很高或使用频率较低时，其精度会显著下降。这是由于传统有限元法中违反了电场连续性的假设。因此，为了提高计算精度，我们采用了边缘单元有限元法和交错网格有限差分法，这两种方法都可以考虑到电场的不连续。结果表明，当频率在100 Hz以上时，高电导率对比度的三维模型可以获得较好的精度，而当频率较低时，方程组的病态性越强。研究还发现，本文主要采用的逐次过松弛法（SOR）在较低频率下不能很好地求解方程组，而带预处理的双共轭梯度法是很有前途的。我们计划在我们的公式中加入散度条件，这样可以减少系统物质的条件数，使解的迭代过程更加稳定。在完成三维建模程序后，无论使用的频率如何，都能提供所需的精度，我们将进行三维反演。

During the investigation of the applicability of the finite-element method (FEM) to 3-D modeling of electromagnetic tomography, it is found that the accuracy can be degraded significantly when the conductivity contrast is very high, or when the frequency used is low. This is due to the fact that the assumption on the continuity of the electric field is violated in the conventional FEM.Thus, in order to improve accuracy we used the edge element FEM and the staggered-grid finite difference method, both of which can take into account the discontinuity of the electric field. The modeling results show that good accuracy can be obtained for 3-D models with high conductivity contrast if the frequency is above about 100 Hz, and that the system of equations is more ill-conditioned as the frequency is lower. It is also found that successive overrelaxation (SOR) method mainly used in this study is not an adequate solver for the system of equations at lower frequencies, and that the biconjugate gradient method with a preconditioning is promising. We are planning to incorporate the divergence condition into our formulation, which is expected to reduce the condition numbers of the system materix and make the iterative process of the solution more stable. After completing 3-D modeling programs which give desired accuracy regardless of the frequency used, we are going to work on the 3-D inversion.