Abstract
An E-polarization problem in geo-electromagnetic induction would not normally be solved in terms of the two components of the magnetic field since it is much simpler to use the single component of the electric field which serves as a scalar potential function. When one is dealing with the two-dimensional limit at infinity of a three-dimensional structure, however, and when a finite difference solution of the full three-dimensional model is obtained in terms of the magnetic vector, then the unorthodox form of the problem emerges quite naturally on the boundaries of the three-dimensional grid. It is not difficult to write down finite difference equations in terms of the two magnetic components and the resistivity, but uncertainties arise when assigning resistivity values to the nodes because normally in two-dimensional E-polarization problems, the nodal values are defined as weighted averages of conductivities which are not the same as the reciprocals of the averaged resistivities. An alternative procedure, which avoids this ambiguity in the numerical model, is to obtain the finite difference equations by integrating over rectangular domains at each node. This approach leads to two 9-point equations in the magnetic components (rather than the usual single 5-point equation for the electric component) whose solution was initially found to be numerically unstable. A method of circumventing this instability is described and numerical solutions of E-polarization problems obtained using the magnetic components as variables are compared with solutions found by traditional finite difference methods.
