Introduction

In the radiative-transport equations for the reflectance and emittance of a particulate medium developed in Chapter 7 it was 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 relatively weakly polarized (e.g., Figure 6.7b). However, even though it may be small, the polarization of the light scattered by a medium does contain information about the medium and, thus, is potentially 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, 1998) and his colleagues. For a more detailed historical account, the reviews by Dollfus (1961, 1962) and Gehrels (1974) should be consulted.

Introduction

The basic expression for the bidirectional reflectance of a semi-infinite medium of isotropically scattering particles was derived in Chapter 8 and is given by equation (8.48). This expression was further refined to include an approximate correction for anisotropic scatterers (8.60) and considerations of the effects of porosity (8.70) and the opposition effect in Chapter 9 (9.47). Variants in different geometries were discussed in Chapter 10. In this chapter we will derive expressions that involve integration of the bidirectional reflectance over one or both hemispheres. The remote sensing of bodies of the solar system has its own nomenclature because of the special problems of astronomical observation. Expressions for quantities commonly encountered in planetary spectrophotometry will be derived. Only media for which the porosity factor K = 1 will be treated. Approximate expressions when K > 1 may be readily obtained by substituting one of the linear approximations (8.54 or 8.55) for the H function in the integrand.

Integrated reflectances

Biconical reflectances

If the source and detector do not occupy negligibly small solid angles as seen from the surface, appropriate expressions for the reflectances may be found by numerically integrating one of the above equations over the angular distribution of the radiance from the source and the angular distribution of the response of the detector. In general, such reflectances will be biconical. However, because they would be specific to each particular system, it would not be particularly useful to discuss biconical reflectances in detail.

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 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 possible to find exact solutions of scattering by such particles only by the expenditure of considerable computer time and memory (to say nothing of the effort of writing detailed programs), so that approximate models remain extremely useful.

The objective of any model of single-particle scattering is to relate the microscopic properties of the particle (its structure 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 describe 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.

The organization of this chapter is as follows. First, the quantities defined in Chapter 5 for spherical particles are extended to particles that are nonspherical in shape.

Introduction

One of the objectives of studying a planet by reflectance is to infer various properties of the surface by inverting a set of remote measurements. 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 spectral reflectance.

For several reasons reflectance spectroscopy is a powerful technique for measuring the characteristic absorption spectrum of a particulate material. The dynamic range of the measurement is extremely large. Multiple scattering amplifies the contrast within weak absorption bands in the light transmitted through the particles, while strong bands can be detected by anomalous dispersion in the 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. This is especially true for the range ni = 10-3 - 10-1, where both transmission and reflection techniques are difficult. Another advantage of reflectance methods is that sample preparation is convenient and simply requires grinding the material to the desired degree of fineness and sieving it to constrain the particle size. 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, and the range by which α(λ) can vary is not much more than about one order of magnitude.

Introduction

The temperature distribution in the regolith of a body of the solar system is important for several reasons. First, the mean temperature is one of the fundamental properties of an object; second, the regolith is the part of the planet studied by remote-sensing methods; and third, the temperature distribution in the upper layers constitutes the boundary condition for thermal models of the deeper interior. The temperatures in the regolith are governed by three energy-transport processes that interact within the medium: visible and near-IR solar radiation, thermal radiation, and heat conduction. However, most models treat the energy as being carried entirely by thermal conduction inside the medium, the interaction with radiation being treated only as a boundary condition. In this chapter all three processes occurring in the medium are treated simultaneously. In doing so it will be seen that radiative conductivity of heat and the solid-state greenhouse effect appear intrinsically without any ad hoc assumptions.

In Section 16.2 the time-dependent, one-dimensional radiative and heat conduction equations are introduced. The time-independent equations are solved analytically for two important cases: a layer of isotropically scattering particles heated from below and radiating into a vacuum, simulating conditions often used in laboratory IR and thermal measurements, and a semi-infinite medium of isotropic scatterers heated by incident visible light and radiating into a vacuum, simulating a planetary regolith in equilibrium with sunlight. In the final section some time-dependent problems are discussed briefly.

Introduction

The region of the electromagnetic spectrum in the vicinity of 10-μm wavelength is referred to as the mid-infrared, but is also called the thermal infrared because objects at the temperature of the Earth's surface emit radiation strongly there. It is important because many materials have strong vibrational absorption bands at these wavelengths (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 combinations of these bands at shorter wavelengths; although they can be observed in reflected light, their depths and shapes may be affected by emitted thermal radiation. It will be seen that there are complementary relations, known as Kirchhoff's laws, between reflectance and emissivity at the same wavelength. Hence, much of the preceding discussions of reflectance can also be applied to emissivity.

Figure 15.1 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 the radiance received by a detector viewing the surface includes both sources.

In this chapter, expressions will be derived for the radiant power received by a detector viewing a particulate medium, such as a powder in the laboratory or a planetary regolith, when either or both reflected sunlight and thermally emitted radiation are present.

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.