Chapter 6 - Electromagnetic Fields in Inhomogeneous Media

Abstract

There are several techniques available for electromagnetic (EM) forward modeling in inhomogeneous media. They are based on numerical implementation of the differential equation (DE) approach (finite difference, FD, or finite element, FE, methods) or the integral equation (IE) approach. In this chapter, I discuss the principles of all these methods. The integral equation (IE) method is a powerful tool in electromagnetic (EM) modeling for geophysical applications. We derive the fundamental equations of the IE method in two and three dimensions and consider the methods of their solution in isotropic and anisotropic media. An effective approach to solve the system of integral equations is based on application of the contraction operator, which can be treated as an effective preconditioner of the original system of equations. We also consider a family of linear and nonlinear integral approximations of the EM fields, based on the IE formulation of Maxwell's equations. Another powerful group of methods of numerical modeling of EM fields uses differential equation methods. We discuss in detail the most important of these methods, the finite difference (FD) and finite element (FE) methods. The finite difference method provides a simple but effective tool for numerically solving the electromagnetic boundary-value problem. We present a discretization of Maxwell's equations using a staggered grid, and introduce a contraction preconditioner for a system of FD equations. Finally, this chapter concludes with the exposition of the most powerful technique for numerical modeling – the method of finite elements.