Introduction

The region of the spectrum in the vicinity of 10 μm wavelength is called the thermal infrared. It is important because many materials have strong vibrational absorption bands there (Chapter 3). In most remote-sensing measurements these bands can be detected only through their effects on the radiation that is thermally emitted by the planetary surface being studied. Many substances have overtone or combination bands at shorter wavelengths, and although the latter bands are observed in reflected light, their depths and shapes may be affected by the thermal radiation that is emitted by the material. Hence, even though the primary subject of this book is reflectance, it is important that the effects of thermal emission be discussed. It will be seen that most of the preceding discussions of reflectance also apply to emissivity at the same wavelength because of the complementary relation between the two quantities.

Figure 13.1 shows the spectrum of sunlight reflected from a surface with a diffusive reflectance of 10%, compared with the spectrum of thermal emission from a black body in radiative equilibrium with the sunlight, at various distances from the sun. Clearly, thermal emission can be ignored at short wavelengths, and reflected sunlight at long, but at intermediate wavelengths in the mid-infrared the radiance received by a detector viewing the surface includes both sources.

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 in 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 interpret quantitatively 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.

Introduction

Chapter 8 treated the bidirectional reflectance of an optically thick, plane-parallel particulate medium in which the particles were randomly oriented and could be regarded as embedded in a vacuum. In this chapter we will discuss the effects on the reflectance when each of these restrictions is removed.

Diffuse reflectance from a medium with a specularly reflecting surface

The upper surfaces of many particulate materials may be sufficiently smooth on a scale comparable to the wavelength that light is scattered both quasi-specularly from the surface and diffusely from below the surface. The specular component is known as regular reflection. Because a surface effectively becomes more optically smooth at large angles of incidence (Chapter 6), the regular component may become especially important at large phase angles.

The most familiar example of the combination of diffuse reflection and regular reflection is water containing suspended solids, as in rivers, lakes, and oceans. If a body of water is examined in the geometry for specular reflection from the surface, a bright glare is seen, which is the reflected image of the sun. However, if the same body is examined in an off-specular configuration, it looks dark and may be colored blue, brown, or green, depending on the nature of the suspended solids.

Introduction

One of the objectives of studying a planet by reflectance is to infer certain properties of the surface by inverting the remote measurement. In the laboratory, the objective of a reflectance measurement is usually to determine the spectral absorption coefficient of the material or, at least, some quantity proportional to it, by inversion of the reflectance.

There are at least three reasons why reflectance spectroscopy is a powerful technique for measuring the characteristic absorption spectrum of a particulate material. First, the dynamic range of the measurement is extremely large. Multiple scattering amplifies the contrast within very weak absorption bands in the light transmitted through the particles, while very strong bands can be detected by anomalous dispersion in radiation reflected from the particle surfaces. Hence, the measurement of a single spectrum can give information on the spectral absorption coefficient over a range of several orders of magnitude in α. Second, sample preparation is convenient and simply requires grinding the material to the desired degree of fineness and sieving it to constrain the particle size. Third, reflectance techniques are effective in the range k ∼10−3–10−1, where both transmission- and specular-reflection techniques are very difficult. By contrast, if α(λ) is measured by transmission, the sample must be sliced into a thin section that must then be polished on both sides; also, the range by which α(λ) can vary is limited to about one order of magnitude.

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 a planet with large-scale roughness.

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. To make the expressions to be derived as general as possible, it will be assumed that the surfaces are randomly rough. There is a large body of literature that treats shadowing on such surfaces — see, for example, Muhleman (1964), Wagner (1967), Saunders (1967), Hagfors (1968), Lumme and Bowell (1981), and Simpson and Tyler (1982), as well as the references cited in those papers — although many of those papers deal only with specular reflection, such as is involved in analyses of sea glitter or backscattered lunar radar signals.

Introduction

A fundamental interaction 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).

Introduction

In the equations for the reflectance and emissivity of a particulate medium developed in Chapters 7–12 it has been assumed that polarization can be neglected. For irregular particles that are large compared with the wavelength of the observation, this assumption is justified on the grounds that the light scattered by such particles is only weakly polarized. However, the polarization of the light scattered by a medium does contain information about the medium and thus is a useful tool for remote sensing. One of the advantages of using polarization is that it does not require absolute calibration of the detector, but only a measurement of the ratio of two radiances.

The discovery that sunlight scattered from a planetary regolith was polarized was made as early as 1811 by Arago, who noticed that moonlight was partially linearly polarized and that the dark lunar maria were more strongly polarized than the lighter highlands. Subsequent observations of planetary polarization were made by several persons, including Lord Rosse in Ireland. However, the quantitative measurement of polarization from bodies of the solar system was placed on a firm foundation in the 1920s by the classical studies of Lyot (1929). This work was later continued by Dollfus (1956) and his colleagues.

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, we summarize the various types of mechanisms by which light is absorbed.

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 − eEe, where e is the charge of an electron. Assume that Ee is parallel to the x axis.

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 or water. 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. Except in a few artificially simple cases, the exact solution of this class of problems is not possible today, even with the help of modern high-speed computers. Hence, we must resort to approximate methods whose underlying assumptions and degrees of validity must be judged by the accuracy with which they describe and predict observations.

Effective-medium theories

One such type of approximation is known as an effective-medium theory, which attempts to describe the electromagnetic behavior of a geometrically complex medium by a uniform dielectric constant that is a weighted average of the dielectric constants of all the constituents.

Introduction

The scattering of electromagnetic radiation by perfect, uniform, spherical particles was described in Chapter 5. However, such particles are rarely found in nature. Most pulverized materials, including planetary regoliths, volcanic ash, laboratory samples, and industrial substances, have particles that almost invariably are irregular in shape, have rough surfaces, and are not uniform in either structure or composition. Even the liquid droplets in clouds are not perfectly spherical, and they contain inclusions of submicroscopic particles around which the liquid has condensed, so that they are not perfectly uniform. At the present state of our computational and analytical capabilities it is not possible to find exact solutions of scattering by such particles, so that it is necessary to rely on approximate models.

The objective of any model of single-particle scattering is to relate the microscopic properties of the particle (its geometry and complex refractive index) to the macroscopic properties (the scattering and extinction efficiencies and the phase function) that, in principle, can be measured by an appropriate scattering experiment. This chapter describes a variety of models that have been proposed to account for the scattering of light by irregular particles. This is not an exhaustive survey; rather, it is a commentary on those models that are most often encountered in remote-sensing applications or that offer some particular insight into the problem.

Introduction

In Chapters 8, 9, and 10, exact expressions for several different types of reflectances and related quantities frequently encountered in remote sensing and diffuse reflectance spectroscopy will be given. Next, approximate solutions to the radiative-transfer equation will be developed in order to obtain analytic evaluations of these quantities. As we discussed in Chapter 1, even though such analytic solutions are approximate, 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 until Chapter 14. 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 (Chapter 6) (Liou and Scotland, 1971).