Three-Dimensional EM Forward Modeling Based On Integral Equation Method

Electromagnetic exploration has been playing an important role in mineralresources prospecting and crustal structure research since its start in1920s. The EMtheory has long been limited in1D regime, which has, however, showed greatlimitation for complicated geologic conditions. Integral equation (IE) method hasperformed great advantages in the3D numerical modeling of EM. For small sizeanomalies, IE need only grid the anomaly itself, which prevents the large storagerequirement of differential equation methods. But for large size anomalies, IE hasshowed its shortness. The coefficient matrix discreted from IE method is a densematrix, whose storage requirement would increase dramatically rapidly with the grids.Besides, it is usually not applicable for a huge matrix to get its inverse. Thus, aniterative solver is necessary for such problems. However, the matrix-vectormultiplication is very time-consuming for huge matrix. What’s more, the limitation ofdyadic Green’s functions has led to the inapplicability of IE method for complicatedgeologic conditions, which also limits the application of IE method. This paper has putforward some methods to solve these problems.By applying the Toeplitz properties of the coefficient matrix, the bottleneck ofstorage requirement in conventional IE method has been solved effectively. Accordingto the special properties of Toeplitz matrix, we applied FFT to the matrix-vectormultiplication in the iterative solver, which has speeding up the multiplication inconventional methods. The complicated geologic condition can be transformed intothe problems with multiple anomalies, we have presented a modeling approach bytaking the coupling effect between the anomalies into account. We have compared thisapproach with the conventional approach without coupling effect.Through the work in this paper, we have got the following conclusions:1) The discrete coefficient matrix from IE method can be decomposed into two parts by theirproperties, that is, the3D block-Toeplitz matrix and Hankel-(two level)-block-Toeplitz matrix, in which Hankel matrix can been transformed to the Toeplitz matrixthrough a permutation matrix.2) For its special properties, Toeplitz matrix can bestored as the first row and first column of each block, which would save a lotcomputer storage for huge number of grids.3)Teoplitz matrix is applicable for theadoption the FFT for the matrix-vector multiplication, which can get the resultseffectively and accurately.4) The coupling effect has little influence for the anomalies.