Chapter 1
Introduction

1.1 Brief History

The idea or notion that light attenuation is proportional to the inverse square of the distance traveled can be traced to Kepler [1]. Its experimental verification was provided by Bouguer [2], who used the inverse square dependence to establish the exponential extinction law by studying the attenuation of light passing through translucent media. A mathematical foundation of radiometry was provided by Lambert [3], who used calculus to interpret experimental results and thereby develop appropriate mathematical models and physical theories. As pointed out by Mishchenko [4], although the first introduction of the radiative transfer equation (RTE) has traditionally been attributed to Schuster [5], the credit should go instead to Lommel [6], who derived an integral form of the RTE by considering the directional flow of radiant energy crossing a surface element; almost identical results were obtained independently by Chwolson [7].

The specific intensity (or radiance) is the most important quantity of classical radiative transfer theory (RTT). Planck [8] defined it by stating that the amount of radiant energy transported through a surface element in directions confined to a solid angle around the direction of propagation in a time interval is given by , where is the position vector of the surface element , and is the angle between and the normal to . This definition was adopted in the works of Milne [9], Hopf [10], and Chandrasekhar [11], and has since been used in many monographs [12-16] and textbooks [17-21] on RTT. To treat the polarization properties of radiation Stokes [22] introduced four parameters to describe the state of polarization. These so-called Stokes parameters were used by Chandrasekhar [11, 23] to replace the specific intensity with the four-element column vector to describe polarized radiation.

The heuristic derivation of the RTE adopted in Chapter 3 of this book for unpolarized as well as polarized radiation is based on classical RTT invoking the specific intensity and simple energy conservation arguments. Such a derivation is easy to understand and sufficient for our purpose. Mandel and Wolf [24] noted that a more fundamental derivation that can be traced to the Maxwell equations was desirable, and stated "In spite of the extensive use of the theory of radiative energy transfer, no satisfactory derivation of its basic equation. from electromagnetic theory. has been obtained up to now." Recently, however, much progress toward such a derivation has been made, as reported by Mishchenko [25].

1.2 What is Meant by a Coupled System?

In many applications, an accurate description is required of light propagation in two adjacent slabs of turbid media that are separated by an interface, across which the refractive index changes. Such a two-slab configuration will be referred to as a coupled system. Three important examples are atmosphere-water systems [26, 27], atmosphere-sea ice systems [28, 29], and air-tissue systems [30]. In each of these three examples, the change in the refractive index across the interface between the two media must be accounted for in order to model the transport of light throughout the respective coupled system correctly. In the second example, the refractive-index change, together with multiple scattering, leads to a significant trapping of light inside the strongly scattering, optically thick sea-ice medium [28, 29]. For imaging of biological tissues or satellite remote sensing of water bodies, an accurate radiative transfer (RT) model for a coupled system is an indispensable tool [31, 32]. In both cases, an accurate RT tool is essential for obtaining satisfactory solutions of retrieval problems through iterative forward/inverse modeling [33, 34].

In remote sensing of the Earth from space, one goal is to retrieve atmospheric and surface parameters from measurements of the radiation emerging at the top of the atmosphere (TOA) at a number of wavelengths [35, 36]. These retrieval parameters (RPs), such as aerosol type and loading and concentrations of aquatic constituents in an open ocean or coastal water area, depend on the inherent optical properties (IOPs) of the atmosphere and the water. If there is a model providing a link between the RPs and the IOPs, a forward RT model can be used to compute how the measured TOA radiation field should respond to changes in the RPs, and an inverse RT problem can be formulated and solved to derive information about the RPs [37, 38]. A forward RT model, employing IOPs that describe how atmospheric and aquatic constituents absorb and scatter light can be used to compute the multiply scattered light field in any particular direction (with specified polar and azimuth angles) at any particular depth level (including the TOA) in a vertically stratified medium, such as a coupled atmosphere-water system [34, 39]. Inorder to solve the inverse RT problem, it is important to have an accurate and efficient forward RT model. Accuracy is important in order to obtain reliable and robust retrievals, and efficiency is an issue because standard iterative solutions of the nonlinear inverse RT problem require executing the forward RT model repeatedly to compute the radiation field and its partial derivatives with respect to the RPs (the Jacobians) [37, 38].

1.3 Scope

While solutions to the scalar RTE, which involve only the first component of the Stokes vector (the radiance or intensity), are well developed, modern RT models that solve the vector RTE are capable of also accounting for polarization effects described by the second, third, and fourth components of the Stokes vector. Even if one's interest lies primarily in the radiance, it is important to realize that solutions of the scalar RTE, which ignores polarization effects, introduce errors in the computed radiances [40-42].

In this book, we will consider the theory and applications based on both scalar and vector RT models, which include polarization effects. There are numerous RT models available that include polarization effects (see Zhai et al. [43] and references therein for a list of papers), and the interest in applications based on polarized radiation is growing. There is also a growing interest in applications based on vector RT models that apply to coupled systems. Examples of vector RT modeling pertinent to a coupled atmosphere-water system include applications based on the doubling-adding method (e.g., Chowdhary [44], Chowdhary et al., [45-47], the successive order of scattering method (e.g., Chami et al., [48], Min and Duan [49], Zhai et al., [43]), the matrix operator method (e.g., Fisher and Grassl, [50], Ota et al., [51]), and Monte Carlo methods (e.g., Kattawar and Adams [40], Lotsberg and Stamnes [52]).

Chapter 2 provides definitions of IOPs including absorption and scattering coefficients as well as the normalized angular scattering cross section, commonly referred to as the scattering phase function, and the corresponding scattering phase matrix needed for vector RT modeling and applications. In several subsections basic scattering theory with emphasis on spherical particles (Mie-Lorenz theory) is reviewed, and IOPs for atmospheric gases and aerosols as well those for surface materials including snow/ice, liquid water, and land surfaces are discussed. The impact of a rough interface between the two adjacent slabs is also discussed.

In Chapter 3, an overview is given of the scalar RTE as well as the vector RTE applicable to a coupled system consisting of two adjacent slabs with different refractive indices. Several methods of solution are discussed: the successiveorder of scattering method, the discrete-ordinate method, the doubling-adding method, and the Monte Carlo method. In Chapter 4, we discuss forward RT modeling in coupled environmental systems based on the discrete-ordinate method, while Chapter 5 is devoted to a discussion of the inverse problem. Finally, in Chapter 6, a few typical applications are discussed including (i) how spectral redundancy can be exploited to reduce the computational burden in atmospheric RT problems, (ii) simultaneous retrieval of total ozone column amount and cloud effects from ground-based irradiance measurements, (iii) retrieval of aerosol and snow-ice properties in coupled atmosphere-cryosphere systems from space, (iv) retrieval of aerosol and aquatic parameters in coupled atmosphere-water systems from space, (v) vector RT in coupled systems, and (vi) how polarization measurements can be used to improve retrievals of atmospheric and surface parameters in coupled atmosphere-surface systems.

1.4 Limitations of Scope

We restrict our attention to scattering by molecules and small particles such as aerosols and cloud particles in an atmosphere, hydrosols in water bodies such as oceans, lakes, and rivers, and inclusions (air bubbles and brine pockets) embedded in ice. To explain the meaning of independent scattering, let us consider an infinitesimal volume element filled with small particles that are assumed to be randomly distributed within the volume element. Such infinitesimal volume elements are assumed to constitute the elementary scattering agents. Independent...