Introduction

Virtually every natural and artificial material encountered in our environment is optically nonuniform on scales appreciably larger than molecular. The atmosphere is a mixture of several gases, submicroscopic aerosol particles of varying composition, and larger cloud particles. Sands and soils typically consist of many different kinds and sizes of mineral particles separated by air, water, or vacuum. Living things are made of cells, which themselves are internally inhomogeneous and are organized into larger structures, such as leaves, skin, or hair. Paint consists of white scatterers, typically TiO2 particles, held together by a binder containing the dye that gives the material its color. These examples show that if we wish to interpret the electromagnetic radiation that reaches us from our surroundings quantitatively, it is necessary to consider the propagation of light through nonuniform media.

Although the equations for this problem can be formally written down (Ishimaru, 1978), their solution to produce useful, practical answers is another matter. The exact solution of Maxwell's equations for this class of problems is possible today only for ensembles of a small number of particles of relatively simple shapes, even with the help of modern high-speed computers. Several persons have obtained direct solutions of Maxwell's equations for a few interacting spheres (e.g., Liang and Lo, 1967; Bruning and Lo, 1971a, b; Fuller and Kattawar, 1988a, b; Xu, 1995). Lumme et al (1997) used the discrete dipole approximation to synthesize scattering by several particles.

Introduction

The opposition effect is a sharp surge observed in the reflected brightness of a particulate medium around zero phase angle. Its name derives from the fact that the phase angle is zero for solar-system objects at astronomical opposition when the Sun, the Earth, and the object are aligned. Depending on the material the angular width of the peak can range from about 1° to more than 20°. It has many names including the heiligenschein (literally “holy glow”), hot spot, bright shadow and backscatter peak. We have already encountered the opposition effect peak in Figures 8.12 and 8.18. and it is further illustrated in Figures 9.1–9.3. It should not be confused with the glory in the phase function of a sphere (Chapter 5), which is also often called the heiligenschein.

The opposition effect is a nearly ubiquitous property of particulate media, including vegetation (Hapke et al., 1996), laboratory powders (Hapke andVan Horn, 1963; Oetking, 1966; Egan and Hilgeman, 1976; Montgomery and Kohl, 1980; Nelson et al., 1998, 2000), and regoliths of the Moon (Gehrels et al., 1964; Whitaker, 1969; Wildey, 1978; Buratti et al., 1996), Mars (Thorpe, 1978), asteroids (Gehrels et al., 1964; Bowell and Lumme, 1979; Belskaya and Shevchenko, 2000), satellites of the outer planets (Brown and Cruikshank, 1983; Domingue et al., 1991), and the rings of Saturn (French et al., 2007). On a clear day you can see it as a glow around the shadow of your head when your shadow falls on grass or soil.

Introduction

The differential reflection and scattering of light as a function of wavelength form the basis of the science of reflectance spectroscopy. This chapter discusses the absorption of electromagnetic radiation by solids and liquids. The classical descriptions of absorption and dispersion are derived first, followed by a brief discussion of these processes from the point of view of quantum mechanics and modern physics. Finally, the various types of mechanisms by which light is absorbed are summarized.

Classical dispersion theory

Conductors: the drude model

The simplest model for absorption and dispersion by a solid is that of Drude (1959). This model assumes that some of the electrons are free to move within the lattice, while the ions are assumed to remain fixed. These approximate the conditions within a metal. The average electric-charge density associated with the semifree electrons is equal to the average of that associated with the lattice ions, so that the total electric-charge density ρe = 0. Because the quantum-mechanical wave functions of the conduction electrons are not localized in a metal, the local field Eloc seen by the electrons is equal to the macroscopic field Ee. Thus, the force on each electron is –e0Ee, where e0 is the charge of an electron. Assume that Ee is parallel to the x -axis.

In addition to the electric field, there is a force due to collisions of each electron with the lattice, resulting in nonradiative loss of energy.

Introduction

In the following chapters a variety of expressions for several different types of reflectances and related quantities frequently encountered in remote sensing and diffuse reflectance spectroscopy will be given, including empirical formulae, and solutions to the equation of radiative transfer. Approximate analytic solutions to the radiative-transfer equation will be developed. As was discussed in Chapter 1, even though such analytic solutions are not exact, they are useful because there is little point in doing a detailed, exact calculation of the reflectance from a medium when the scattering properties of the particles that make up the medium are unknown and the absolute accuracy of the measurement is not high. In most of the cases encountered in remote sensing an approximate analytic solution is much more convenient and not necessarily less accurate than a numerical computer calculation.

In keeping with this discussion, polarization will be ignored in the derivations. This neglect is justified because most of the applications of interest involve the interpretation of remote-sensing or laboratory measurements in which the polarization of the incident irradiance is usually small. Although certain particles, such as Rayleigh scatterers or perfect spheres, may polarize the light strongly at some angles, the particles encountered in most applications are large, rough, and irregular, and the polarization of the light scattered by them is relatively small (Liou and Schotland, 1971; see also Chapter 6). Hence, to first order, both the incident radiation and scattered radiation may be assumed to be unpolarized.

Introduction

The expressions for reflectance developed in previous chapters of this book implicitly assume that the apparent surface of the particulate medium is smooth on scales large compared with the particle size. Although that assumption may be valid for surfaces in the laboratory, it is certainly not the case for planetary regoliths. In this chapter the expressions that were derived in Chapters 8–10 to describe the light scattered from a planet with a smooth surface will be modified so as to be applicable to surfaces with large-scale roughness. By “large-scale roughness” is meant that areas of the surface larger than the particle size but smaller than the detector footprint are tilted with an irregular distribution of slopes. Persons uninterested in the details of the derivation may wish to jump directly to the Summary Section 12.D, after reading this introductory section.

In calculations of this type we are immediately faced with the problem of choosing an appropriate geometric model to describe roughness. Some authors have chosen specific shapes, such as hemispherical cups (Van Diggelen, 1959; Hameen-Anttila, 1967), that approximate impact craters on the surface of a planet. However, such models may not be applicable to other geometries such as hills or dunes. To make the expressions to be derived as general as possible, it will be assumed that the surfaces are randomly rough.

Scientific rationale

The subject of this book is remote sensing, that is, seeing “with clearer eyes.” In particular, it is concerned with how light is emitted and scattered by media composed of discrete particles and what can be learned about such a medium from its scattering properties.

If you stop reading now and look around, you will notice that most of the surfaces you see consist of particulate materials. Sometimes the particles are loose, as in soils or clouds. Sometimes they are embedded in a transparent matrix, as in paint, which consists of white particles in a colored binder. Or they may be fused together, as in rocks, or tiles which consist of sintered ceramic powder. Even vegetation is a kind of particulate medium in which the “particles” are leaves and stems. These examples show that if we wish to quantitatively interpret the electromagnetic radiation that reaches us, rather than simply form an image from it, it is necessary to consider the scattering and propagation of light within nonuniform media.

One of the first persons to use remote sensing to learn about the surface of a planet was Galileo Galilei. Galileo (1638) noticed that the full Moon, as it rose over his garden wall opposite the setting Sun, was darker than the sunlit wall.

Introduction

In this chapter the specular or mirror-like reflection that occurs when a plane electromagnetic wave encounters a plane surface separating two regions with different refractive indices is discussed quantitatively, along with the accompanying transmission, or refraction, through the interface. Specular reflection is important to the topic of this book for several reasons. First, it is an important tool for investigating properties of materials in the laboratory. Second, it occurs in remote-sensing applications when light is reflected from smooth parts of a planetary surface, such as the ocean. Third, it is one of the mechanisms by which light is scattered from a particle whose size is large compared with the wavelength, so that an understanding of this phenomenon is necessary to an understanding of diffuse reflectance from planetary regoliths.

Boundary conditions in electromagnetic theory

Whenever a volume contains a boundary separating regions of differing electric or magnetic constants, the components of De and Bm perpendicular to the surface and the components of Ee and Hm tangential to the surface must be continuous across the boundary. If the fields constitute an electromagnetic wave propagating through the surface from one medium to another, the amplitudes of the fields are different within the two regions. Therefore, the continuity conditions cannot be satisfied unless there is another wave propagating backward from the surface into the first medium, in addition to the wave propagating forward from the surface into the second medium.

Introduction

One of the fundamental interactions of electromagnetic radiation with a particulate medium is scattering by individual particles, and many of the properties of the light diffusely reflected from a particulate surface can be understood, at least qualitatively, in terms of single-particle scattering. This chapter considers scattering by a sphere. Although perfectly spherical particles are rarely encountered in the laboratory and never in planetary soils, they are found in nature in clouds composed of liquid droplets. For this reason alone, spheres are worth discussing. Even more important, however, is the fact that a sphere is the simplest three-dimensional object whose interaction with a plane electromagnetic wave can be calculated by exact solution of Maxwell's equations. Therefore, in developing various approximate methods for handling scattering by nonuniform, nonspherical particles, the insights afforded by uniform spheres are invaluable.

In the first part of this chapter some of the quantities in general use in treatments of diffuse scattering are defined. Next, the theory of scattering by a spherical particle is described qualitatively, and conclusions from the theory are discussed in detail. Finally, an analytic approximation to the scattering efficiency that is valid when the radius is large compared with the wavelength is derived.

Concepts and definitions

Radiance

In a radiation field where the light is uncollimated, the amount of power at position r crossing unit area perpendicular to the direction of propagation Ω, traveling into unit solid angle about Ω, is called the radiance and will be denoted by I (r, Ω). Radiance is often also called specific intensity, or simply intensity, or brightness.