Three Dimensional Controlled Source Electromagnetic Numerical Simulation Based On Electric Field Double Curl Equation Using Finite Element Method

Due to the weakness and randomness of natural source in magnetotelluric method, a great deal of effort is spent in the field records and data acquisition. This problem can be solved by controlled source electromagnetic method due to artificial sources, and so it is widely used in nineral prospecting, oil and gas exploration, and so on. However, the currently controlled source electromagnetic research is limited to one-dimensional layered modeling or two-dimensional simulation for geologic body of unlimited extension along strike. Strictly speaking, the geophysical electromagnetic field problems should be discussed in three dimensional space. Under the condition of three-dimension, numerical simulation of controlled source is not a simple expansion of one dimension, two-dimension cases. But it also encounters many problems which do not appear in one-dimension and two-dimension cases. With the development of computer hardware technology, three dimensional controlled source electromagnetic forward modeling become available. So, in this paper, three dimensional controlled source electromagnetic problems are studied in depth.From the basic principles of controlled source electromagnetic, boundary value problem based on double electric field curl equation is derived, and then transformed into variational problem by generalized variational principle from which weak solutions form is obtained based on double curl equation of the electric field. In quasi-static conditions, the closed expressions far from source in the air and the earth is derived for electric dipole source, and by using it as the outer boundary conditions of the finite element method, difficulties in loading the boundary conditions is solved. The pseudo delta function in seismic method is used to simulate the electric dipole source in finite element simulation, In this paper, the author extend it into three-dimensional case from which singularity from source is eliminated, and also the the stability of the equation is remarkably. For sparse and large stiffness matrix in controlled source electromagnetic three-dimensional finite element simulation, full sparse compressed row storage method is used. And through it a great deal of storage space is saved by which large-scale modeling is becoming available in personal computers for hundreds of thousands of degrees of freedoms. Because of a huge condition number, large-scale, sparse complex coefficient equations is badly in-conditioned. In order to ensure the rapid convergence of the method, incomplete precondition LDLT technology is used. By using it, condition number of matrix is reduced from which the, convergence rate is accelerated.Under the guarantee of preconditions, Krylov subspace iteration method is introduced. The results show that Krylov subspace iteration combined with precondition, convergence is very quickly, and it is fast and efficient for solving electromagnetic finite element simulation of large-scale complex coefficient equations.Since the electric sources makes controlled source electromagnetic theory and numerical simulation very complex, The electric field double curl equation in three-dimensional finite element numerical simulation is adoped to avoids curling the electric dipole source by which the complexity of programming is reduced; Meanwhile, the introduction of divergence conditions ensures divergence to be zero for the electric field outside the source and avoid a spurious solution and make electromagnetic finite element method more complete in theory.