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.

Dynamics of the polar ionosphere

Chapter 5 described how the magnetosphere circulates as two regions, an inner one rotating daily with the Earth, and an outer one circulating under the influence of the solar wind. The polar ionosphere is connected by the geomagnetic field-lines to this outer region, and – since the field-lines are (almost) equipotentials – its circulation is essentially a projection of that of the outer magnetosphere.

F-region circulation

In the F region, where the ion–neutral collision frequency is small relative to the gyrofrequency, the plasma moves with the magnetic field-lines. Alternatively, we can say that the electric field which the solar wind generates across the magnetosphere (Section 5.5.3) is mapped into the F region along the equipotential field-lines. The polar-cap electric field so created (as measured by a stationary observer) then acts as the driving force for the F-region plasma (Sections 2.3.7 and 6.5.4). The integral of the electric field gives the total electric potential across the polar cap.

Vertical structure

Nomenclature of atmospheric vertical structure

The static atmosphere is described by the four properties, pressure (P), density (ρ), temperature (T) and composition. Between them these properties determine much of the atmosphere's behaviour. They are not independent, being related by the universal gas law which may be written in various forms (Equations 2.5–2.7). For our purposes the form

P = nkT, (Equation 2.7)

where n is the number of molecules per unit volume, is particularly useful. The quantity ‘n’ is properly called the concentration or the number density, but density alone is often used when the sense is clear.

The regions of the neutral atmosphere are named according to various schemes based in particular on the variations with height of the temperature, the composition, and the state of mixing. Figure 4.1 illustrates the most commonly used terms. The primary classification is according to the temperature gradient. In this system the regions are ‘spheres’ and the boundaries are ‘pauses’. Thus the troposphere, in which the temperature falls off at 10 K/km or less, is bounded by the tropopause at a height of 10–12 km. The stratosphere above was originally thought to be isothermal, but in fact is a region where the temperature increases with height. A maximum, due to heating by ultra-violet absorption in ozone, appears at about 50 km and this is the stratopause.

Introduction

The ionized part of the atmosphere, the ionosphere, contains significant numbers of free electrons and positive ions. There are also some negative ions at the lower altitudes. The medium as a whole is electrically neutral, there being equal numbers of positive and negative charges within a given volume. Although the charged particles may be only a minority amongst the neutral ones they exert a great influence on the medium's electrical properties, and herein lies their importance.

The first suggestions of electrified layers within the higher levels of the terrestrial atmosphere go back to the 19th century, but interest was regenerated with Marconi's well known experiments to transmit a radio signal from Cornwall in England to Newfoundland in Canada in 1901, and with the subsequent suggestions by Kennelly and by Heaviside (independently) that, because of the Earth's curvature, the waves must have been reflected from an ionized layer. The name ionosphere was coined by R. Watson-Watt in 1926, and came into common use about 1932.

Since that time the ionosphere has been extensively studied and most of its principal features, though not all, are now fairly well understood in terms of the physical and chemical processes of the upper atmosphere. Typical vertical structures are as shown in Fig. 6.1.

Introduction

Science and engineering are related activities with different objectives. The purpose of science is to gain knowledge, and the essence of scientific achievement is intellectual rather than practical. To a scientist the knowledge and the ideas are what matter most. Engineering, on the other hand, is all about practical things. The result of successful engineering endeavour is a machine, a device or a scheme for performing some specific task. What matters to the engineer is that the machine, device, etc., should work well, and he/she will draw on any area of knowledge or experience to achieve this. Some of that knowledge might be science based; some might not.

Having drawn the distinction, we should at the same time recognize that there are strong links between these activities. Science relies on instruments and computers, the products of the engineer, and it should be abundantly clear from Chapter 3 how much the progress of geospace has depended on the development of techniques. And although there is no law that engineering must be science based, it draws heavily on scientific knowledge in practice. It would be an unusual engineer who relied entirely on historical practice or intuition. It is the purpose of this chapter to consider the impact of geospace science on practical activities within the province of the engineer.