The equivalent source method for electromagnetic scattering analysis and its geophysical application

The electromagnetic response of magnetically and electrically inhomogeneous media whose geometries are not amenable to conventional partial differential equation analysis are most readily analysed in an integral equation framework. The equivalent source concept of Green's theory affords a generalized manner of formulating static and time-varying electromagnetic problems; material property inhomogeneities are replaced by equivalent source distributions which satisfy a Fredholm integral equation of the second kind. -- For static field problems, the equivalent source method represents conductivity, permittivity and permeability variations in terms of current source, charge and "magnetic pole" density distributions. In this form, the problems have analogous mathematical forms and the equivalent source satisfies a scalar Fredholm equation. The formalism is readily related to the static field methods used in applied geophysics. -- The time-varying equivalent source formulation represents material property variations in terms of electric and magnetic current densities which satisfy a pair of coupled vector Fredholm equations. Analysis of the integral operators shows that the scattering operator is bimodal for many geophysical problems. This result leads to the analysis of scattering problems in terms of generalized eigenfunctions. The bimodal nature of the scattering operator often leads to highly ill-conditioned matrices when numerical methods are applied to geophysical problems. -- Approximate parametric solution methods of solving the time-varying electric scattering problem are considered. Approximation of the solution by a general functional form and applying minimum criteria reduce the integral equations to matrix equations. The least squares method is applied analysing magnetotelluric responses of 2-dimensional structures and the Galerkin formalism is used to find the eigenfunctions for a thin plate in a whole space. The results are compared with other available numerical and experimental results and assessments of tho methods are given.