1 - Scalar-Wave Models in Electromagnetic Scattering

Summary

This chapter is concerned with the scalar forward models used in microwave (MW) imaging. These are mathematical models of varying degrees of accuracy that predict the field based on a known source of radiation in a known environment. They are called forward because they describe the causal (or forward-in-time) relationship in a phenomenon we could express as cause → effect. In imaging, the cause is described by the model parameters, i.e., (i) the parameters of the sources generating the field and (ii) the parameters of the environment where this field exists or propagates. The effect is described by the observation data, or simply, the data. These are signals acquired through measurements. Thus, in imaging, the forward model predicts the data, provided the model parameters are known.

The object of imaging, however, is the inverse problem, which, in contrast to the forward problem, is expressed as effect → cause. Finding what caused an effect is not an easy task. The second part of this book is dedicated to the mathematical methods used to accomplish this task. For now, it suffices to say that we first need to have a forward model of a phenomenon before we can start solving inverse problems based on this phenomenon. To illustrate this point, imagine that you are listening to a recording of a symphony; in order to tell which instruments play at any given time, you have first to have heard the sound of each instrument.

The phenomenon of interest in the forward models of MW imaging is the scattering of the high-frequency electromagnetic (EM) field by objects. The scattering objects are often referred to as targets, especially in radar, or as scatterers. In this chapter, we discuss the mathematical scalar models of scattering.

The EM field is a vectorial field fundamentally described by Maxwell's equations [1, 2, 3, 4, 5]. For a summary of Maxwell's equations, see Appendix A. However, to simplify the analysis, scalar approximations are often made, and here we start with these simpler models. The scalar-wave model is very useful as an intermediate step toward the understanding of the vectorial wave model. It can also serve as a bridge to understanding acoustic and elastic wave phenomena, which are widely used in imaging.