7

Scalar Radiative Transfer Theory Vector Radiative Transfer Theory Phase Matrix of Independent Scattering Extinction Matrix Emission Vector Boundary Conditions References and Additional Readings

260 269 269 272 275 283 286

2.1 2.2 2.3 2.4

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Radiative transfer theory is an important method to treat multiple scattering in medium consisting of random discrete scatteries. Classic books on the subject include Chandrasekhar [1960]; Ishimaru [1978]; Case and Zweifel [1967]. The theory has been extensively applied [Tsang et al., 1977; Tsang et al. 1985; Fung, 1994; Ulabyet al. 1990; Burke et al. 1979]. In this chapter, we derive the equation that governs the propagation of specific intensity in a medium containing random distribution of particles. The particles scatter and absorb the wave energy, and these characteristics should be included in a differential equation to be satisfied by the specific intensity. This equation is called the radiative transfer equation. There are three constituents of the radiative transfer equation. The extinction matrix describes the attenuation of specific intensity due to absorption and scattering. The phase matrix characterizes the coupling of intensities in two different directions due to scattering. The emission vector gives the thermal emission source of the specific intensity. Since the specific intensity is a four-element Stokes vector, the extinction and phase matrices are 4 x 4 matrices and the emission vector is a 4 X 1 column matrix. For spherical particles, the extinction matrix is diagonal and is a constant times the unit matrix. The emission vector has the first two elements equal and the last two elements equal to zero. For nonspherical particles, the extinction matrix is generally nondiagonal and the four elements of the emission vector are all nonzero. The resultant equations are called vector radiative transfer equations to distinguish them from the scalar transfer equation. We also define phase matrix as the averaged scattering cross section per unit volume of space where the volume is larger than many wavelengths and also large enough to include many particles. In this new definition, the coherent wave interactions among particles within wavelength scales can be included, and the collective scattering effects of particles in the neighborhood of each other are included. This is unlike the classical definition in which only the single-particle scattering characteristics are included in the phase matrix definition. Likewise, the extinction coefficient is defined as the average extinction cross section per unit volume of space, and the absorption coefficient is average absorption cross section per unit volume of space.

Radiative transfer equation is an integra-differential equation that governs the propagation of specific intensity.

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Consider a medium consisting of a large number of particles (Fig. 7.1.1). We have I(r, s) at all r and for all s due to scattering. We consider a "small" volume element dV = dAdl, and dl is along the direction s. The small volume element is centered at r. We consider the differential change in specific intensity 1(s) as it passes through dV (Fig. 7.1.1). The differential change of power in direction s is

(7.1.1)

The volume dV contains many particles that are randomly positioned. The volume dV is much bigger than ..\3 so that random phase prevails and the input and output relation of dV can be expressed in terms of intensities instead of fields. Thus dV, though called a "small" volume element, is really "not that small." There are three kinds of changes that will occur to 1(r, s) in the small volume element: (i) Extinction that contributes a negative change (ii) Emission by the particles inside the volume dV that contributes a positive change (iii) Bistatic scattering from direction Sf into direction s that contributes a positive change