Models for Scattering from Rough Surfaces

Models for scattering of electromagnetic waves from random rough surfaces have been developed during the last two centuries and the scientific interest in the problem remains strong also today due to the importance of this phenomenon in diverse areas of science, such as measurements in optics, geophysics, communications and remote sensing of the Earth. Such models can be categorised into empirical models, analytical models and a combination of the two. Though very simple, empirical models are greatly dependent on the experimental conditions. In spite of their complexity, only theoretical models can yield a significant understanding of the interaction between the electromagnetic waves and the Earth’s surface, although an exact solution of equations governing this interaction may not always be available and approximate methods have to be used. The semi-empirical models, which are based on both physical considerations and experimental observations, can be set between these two kinds of models and can be easily inverted. In this survey, we will focus on the analytical models and we study more in detail the Kirchhoff Approximation (KA), the Small Perturbation Method (SPM) and the Integral Equation Method (IEM). The Kirchhoff Approximation and the Small Perturbation Methods represent early approaches to scattering which are still much used, whereas the Integral Equation Method represents a newer approach which has a larger domain of validity. These methods have been found to be the most common in the literature and many of the other methods are based or have much in common with these approaches. In section 2, we begin by giving a brief presentation of the scattering problem and introduce some concepts and results from the theory of electromagnetic fields which are often used in this context. We will also define the bistatic scattering coefficient, due to the importance of this type of measurement in many remote sensing applications, and in particular in the retrieval of soil moisture content. In section 3, we give a brief presentation on the Kirchhoff Approximation and its close variants, the Physical Optics (PO) and the Geometrical Optics (GO). In section 4, we give a brief presentation of the Small Perturbation Method and in section 5 we will present the Integral Equation Model.