Studies Of Three Dimensional Stagged-grid Finite Difference For Controlled Source Audio-frequency Magnetotelluric Numerical Simulation

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.