Introduction

The purpose of this book is to describe efficient methods to compute wave fields in an inhomogeneous medium. We consider wave fields of increasing complexity, viz., scalar waves, acoustic waves, and electromagnetic waves. The material media is considered to be isotropic. For scalar waves, the medium is a characterized by a single parameter, the wave speed . The scalar wave field is represented by the scalar quantity . For acoustic waves, we deal with two medium parameters: the mass density and compressibility . The wave field is represented by the scalar pressure and the particle-velocity vector . The acoustic wave speed is given by . For electromagnetic waves, the electric permittivity and magnetic permeability are the two medium parameters, while the electric-field vector and the magnetic-field vector represent the wave field. The electromagnetic wave speed is given by .

The subject area complies with the two-volume book on Scattering, edited by Pike and Sabatier [38]. This book yields an overwhelming theoretical overview of Scattering and Inverse Scattering in Pure and Applied Science, including some numerical results. Further, a comprehensive overview of the set of partial differential equations that govern the various wave-field descriptions can be found in De Hoop's Handbook of Radiation Scattering of Waves [14]. The emphasis is mainly on general principles and theorems that can serve to check numerical results rather than on highly specialized configurations for which more or less complicated analytical answers can be obtained. The system of differential equations customarily serves as the starting point for the computation of the wave field via a numerical discretization procedure applied to the pertaining differential equations, such as finite-difference and finite-element methods. The advantage of these local methods is that it leads to the inversion of a linear set of equations where the coefficient matrix is sparse. Presently, the finite-difference technique is very popular, because it admits an easy numerical implementation with the help of staggered rectangular grids in space.

De Hoop [14] replaced the partial differential equations by so-called source-type integral equations. The integrands are the products of known Green functions and unknown contrast sources. For scalar waves, the Green function is the field from a point source in an infinite domain with a suitable chosen wave speed . Evidently, the simplest choice is a homogeneous medium with constant wave speed. It is assumed that inside a bounded (scattering) domain , the wave speed differs from , while outside this domain the wave speed is equal to . Therefore, represents a contrasting object that disturbs the wave field in an embedding medium with wave speed . For convenience, the total wave field and the incident wave field are introduced. The total wave field is the actual wave field in whole space with wave speed , and the incident wave field is the wave field in whole space with wave speed . The difference between total and incident wave field is denoted as the scattered wave field. The contrast sources inside the scattering domain generate the scattered wave field in the whole space. The contrast source is defined as the product of the contrast function and the wave field. In fact, this relation serves as a constitutive relation. The contrast function is given by , and its nonzero value defines the scatterer. The contrast source vanishes outside the domain . Note that at this stage of the analysis, the contrast sources are unknown, since the wave field is yet unknown. When we take the observation points inside the scatterer, we arrive at an integral equation for the unknown wave field. Once the wave field is determined, we calculate the contrast-source distribution. Finally, the source-type integral representation provides the wave field in whole space. The major advantage is that the spatial support of the set of integral equations is limited to points where the contrast function does not vanish.

In this book, we consider a slightly different approach. Instead of solving the field integral equation, we multiply the field equation by the contrast, and we replace the product of field and contrast by the contrast source, both in the integrand and outside the integrand. We then end up with a contrast-source integral equation, where now the contrast source is the unknown function.

In general, the integral equation has to be solved with the aid of numerical methods. However, the integrand of both the field integral equation and the contrast-source integral equation is not defined at points where the Green function is singular. Usually, Galerkin methods are used to overcome this problem and convert the continuous integral equation into a discrete problem. In mathematics, it is referred to as a weak formulation of the continuous operator equation. In principle, it leads to the inversion of a linear set of equations where the coefficient matrix is fully filled. Obviously, this solution method is a global method.

This book is intended to provide an easy entry into the numerical methods for an effective solution of the contrast-source integral equations. Similar to the use of rectangular grids in the finite-difference technique, we also employ rectangular grids. On each grid point, our intention is to obtain a global wave function, where its value is replaced by its average over a small symmetrical domain around this grid point. This is achieved automatically when we replace the Green function by its mean (integrated) value over . Assuming that the width of each domain is small enough, we replace the value of the embedding medium inside a domain by its value at the grid point. The mean of the Green function, also denoted as the weak form of the Green function, can be calculated analytically. In this book, it is shown that this weak formulation results into second order accurate evaluations.

Since the embedding medium is homogeneous, the Green function only depends on the spatial distance between observation and integration point. The discrete integral operator becomes a discrete convolution and, for known contrast sources, it is calculated efficiently with a fast Fourier technique (FFT). Since the contrast source is unknown, the integral equation is conveniently solved with iterative methods.

A similar analysis applies in principle to acoustic and electromagnetic waves, but its becomes more complex due to the vector character of these wave fields. In addition, in the contrast-source integral equations, we have an extra term with spatial derivatives operating on the Green function.

Part I Forward Scattering Problem

In Part I of our book, we discuss the forward scattering problem based on the contrast-source integral equations. In this case, the contrast function is known. For a given incident wave, we show that the contrast sources inside the scatterer are the fundamental unknowns. After solving the integral equations for the contrast sources, the scattered wave field follows directly from its representation. The forward scattering problem is linear and uniquely solvable. In Chapters 1-3, we outline the theoretical formulation and numerical implementation for scalar, acoustic and electromagnetic waves, respectively.

Chapter 1

In this chapter, we discuss scalar waves. We start with the wave equation in space and time. We apply a Laplace transformation with respect to time, and then our analysis is performed in the complex -domain. Causality in time is taken into account by requiring that Re is positive. In Section 1.1, the theory of 3D scattering of waves by a bounded contrast is discussed. First, the radiation from a known source distribution is calculated, resulting into a source-type integral representation. This is the expression for the known incident field. Second, for an unknown contrast-source distribution in a bounded domain, the scattered field is written as a contrast-source integral representation. Third, the contrast-source integral equation is obtained by confining the position of observation to the interior of the scatterer. The unknown contrast-source distribution is the product of the wave-speed contrast and the interior field. In Section 1.2, we present 2D and 1D versions of the contrast-source representations. In Section 1.3, we discuss the numerical solution of the domain integral equations, for 1D, 2D, and 3D configurations. We define the mean of each wave function and we give the expressions of the weak form of the Green function, for all points in 1D, 2D, and 3D, respectively. We show that the discretized integral operators are circular convolutions in space, which can be carried out efficiently using FFTs. We discuss how the Conjugate Gradient (CG) method solves the discrete system of equations in an iterative way. In Section 1.4, we describe the Matlab input and output functions. The computations are carried out for a single frequency of operation and common time factor . In order to deal with causal signals, we assign a small positive real part of the Laplace parameter, viz. . In Sections 1.5 and 1.6, we present Matlab codes for solving the field integral equation and for solving the contrast-source integral equation, respectively. A performance analysis shows that the contrast-source method outperforms the field method. For the contrast-source method, it is further...