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).

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 fields contain a boundary separating regions of differing electric or magnetic constants, certain conditions on the continuity of the fields must be satisfied. It is shown in any textbook on electricity and magnetism that 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, it is found that these 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.

In this chapter we continue to study electromagnetic fluctuations in homogeneous, magnetized, collisionless plasmas. The new element here is that we consider the zeroth-order distribution function of each plasma component to be Maxwellian with drift velocity v0j parallel or antiparallel to B0 (Equation (3.1.3)). If two components have a relative drift v0 greater than some threshold, the corresponding free energy can lead to instability growth. Section 8.1 outlines the derivation of the dispersion equation for this case; Section 8.2 discusses electromagnetic ion/ion instabilities; Section 8.3 addresses electromagnetic electron/electron instabilities; Section 8.4 considers electromagnetic electron/ion instabilities; and Section 8.5 examines the consequences of electromagnetic effects on ion/ion instabilities that are electrostatic in the limit of zero β. Section 8.6 is a brief summary.

Space plasma heating and acceleration processes typically act on both species and are likely to give rise to beam/core distributions for both electrons and ions. However, in contrast to the case of T⊥ j > T‖j discussed in the previous chapter, the instabilities driven by beam/core free energies do not clearly separate into low frequency ion-driven and high frequency electron-driven modes. Thus, although we treat relative ion drifts and relative electron drifts separately in this chapter, this separation is due more to our desire to clarify the presentation than to any compelling physical arguments. Thus, in Sections 8.2 through 8.5, we consider a two-species, three-component plasma consisting of a relatively tenuous beam (denoted by subscript b), a relatively dense core (c), and a third component of the other species.

Every plasma is inhomogeneous to some extent, and the associated plasma gradients are sources of free energy that can drive plasma instabilities. In this chapter we consider the linear theory of drift instabilities, modes driven unstable by a plasma gradient perpendicular to B0.

In the direction parallel to a magnetic field, pressure gradients give rise to electric fields that lead to currents and bulk plasma motion; that is, such gradients do not correspond to a steady-state description under a macroscopic description of the plasma. However, pressure gradients perpendicular to a magnetic field can correspond to a steady-state situation; that is, ∇P in the momentum equation of a one-fluid description of the plasma can be balanced by the J × B0/c term. Nevertheless, such gradients do not correspond to an equilibrium plasma configuration; the zeroth-order distribution functions are non-Maxwellian and lead to the growth of plasma instabilities which act to dissipate the gradients. In this chapter we consider, as before, collisionless plasmas with a uniform zeroth-order magnetic field B0 = ẑB0. In Section 4.1 we discuss a model distribution function for density gradients perpendicular to a uniform magnetic field, examine the associated linear dispersion equation and discuss the two most popular density drift instabilities. Section 4.2 describes the instability properties that result when a plasma with a density gradient is subject to a uniform acceleration, and Section 4.3 briefly summarizes some properties of temperature drift instabilities.