1. Radiative Transfer Equation for Vegetation Canopies2. Vegetated Surfaces Reflectance3. Boundary Conditions4. Decomposition of the Boundary Value Problem for RadiativeProblem SetsReferencesFurther ReadingsChapter 4 Radiative Transfer in Vegetation Canopies 1. Radiative Transfer Equation for Vegetation Canopies ............................................................... 1 2. Vegetated Surfaces Reflectance.................................................................................................. 3 3. Boundary Conditions .................................................................................................................. 5 4. Decomposition of the Boundary Value Problem for Radiative Transfer Equation.................. 10 Problem Sets ................................................................................................................................. 17 References..................................................................................................................................... 18 Further Readings........................................................................................................................... 19 1. Radiative Transfer Equation for Vegetation Canopies Solar radiation scattered from a vegetation canopy and measured by satellite sensors results from interaction of photons traversing through the foliage medium, bounded at the bottom by a radiatively participating surface. Therefore to estimate the canopy radiation regime, three important features must be carefully formulated. They are (1) the architecture of individual plant and the entire canopy; (2) optical properties of vegetation elements (leaves, stems) and soil; the former depends on physiological conditions (water status, pigment concentration); and (3) atmospheric conditions which determine the incident radiation field [Ross, 1981]. We idealize a vegetation canopy as a medium filled with small planar elements of negligible thickness. We ignore all organs other than green leaves. In addition, we neglect the finite size of vegetation canopy elements. Thus, the vegetation canopy is treated as a gas with non-dimensional planar scattering centers, i.e., a turbid medium. In other words, one cuts leaves residing in an elementary volume at a given spatial point r into “dimensionless pieces” and uniformly distributes them within the elementary volume. Three variables, the leaf area density distribution function )r(uL, the leaf normal distribution, ),r(gLL!, and the leaf scattering phase function, ),,r(LL!!"!#$ (Chapter 3) are used in the theory of radiative transfer in vegetation canopies to convey “information” about the total leaf area, leaf orientations and leaf optical properties in the elementary volume at r before “converting the leaves into the gas.” It should be emphasized that the turbid medium assumption is a mathematical idealization of canopy structure, which ignores finite size of leaves. In reality, finite size scatters can cast shadows. This causes a very sharp peak in reflected radiation about the retro-solar direction. This phenomenon is referred to as the “hot spot” effect. It is clear that point scatters cannot cast shadows and thus the turbid medium concept in its original formulation [Ross, 1981] fails to 1predict or duplicate experimental observation of exiting radiation about the retro-illumination direction. Zhang et al. [2002] showed that if the solution to the radiative transfer equation is treated as a Schwartz distribution, then an additional term must be added to the solution of the radiative transfer equation. This term describes the hot spot effect. This result justifies the use of the transport equation as the basis to model canopy radiation regime. Here we will follow classical radiative transfer theory in vegetation canopies proposed by Ross [1981]. For the mathematical theory of Schwartz distributions applicable to the transport equation, the reader is referred to Germogenova [1986], Choulli and Stefanov [1996] and Antyufeev [1996]. In addition to canopy structure and its optics a domain V in which the radiative transfer process is studied should be specified. In remote sensing application, a parallelepiped of horizontal dimensions XS, YS, and height ZS is usually taken as the domain V. The top , bottom tV%bV%, and lateral surfaces of the parallelepiped form the canopy boundary lV%lbtVVVV%&%&%'% . The height of a tallest plant in V can be taken as ZS. The dimension of the upper boundary tV% coincides with a footprint of the imagery. The function characterizing the radiative field in V is the specific intensity introduced in Chapter 2. Under condition of the absence of polarization, frequency shifting interaction, and emission processes within the canopy, the monochromatic specific intensity ),r(I !( is given by the stationery radiative transfer equation (Chapter 2, Eq. (24)) with 0),r(q '!(. Substituting vegetation-specific coefficients (Chapter 3, Eqs (13) and (15)) into the transport equation (Chapter 2, Eq. (24)), one obtains the radiative transfer equation for a vegetation canopy occupying the domain V, namely, .!d)!,r()I!!,r("#)r(u)!,r()Ir()u!,rG()!,r(I!4#$$L$L$##"#'&)*+ (1) The boundary condition for the radiative transfer problem is given by ,),r(B)!,r(Ib$b$!' ,Vrb%, .0)r(nb-!* (2) Here ),r(Bb$! is the intensity of radiation entering the domain V through a point Br on the boundary in the direction V%!. Directions along which photons can enter the vegetation canopy through the point br satisfy the inequality 0)r(nb-!* where )r(nb is an outward normal vector at br. The solution of the boundary value problem, Eqs (1)-(2), i.e., the monochromatic specific intensity ),r(I !(, depends on wavelength, (, location r, and direction !. Here, the position vector r denotes the triplet (x,y,z) with (0<x<XS), (0<y<YS) and (0<z<ZS) and is expressed in Cartesian coordinates with its origin, O=(0,0,0), at the top of the vegetation canopy and the Z axis directed down into the vegetation canopy. The unit vector ),(~./! has an azimuthal angle measured in the (XY) plane from the positive X axis in