Photon-Vegetation Interactions pp 45-109 | Cite as

# Discrete Ordinates Method for Photon Transport in Leaf Canopies

## Abstract

Energy in the radiation field is transported by point, massless particles called photons. The photon has a frequency ν such that its energy E is hν, where h is Planck’s constant. A massless particle has momentum E/c, where c is the vacuum speed of light. The position of the photon, at any time t, in the phase space is specified by three positional variables and three momentum variables. The three positional variables are denoted by the vector r⃗. In transport theory, it is conventional to use, rather than the three momentum variables, three equivalent variables; frequency ν and direction of photon travel Ω̱. Now, we can define the photon density distribution function χ(r⃗, ν, Ω̱, t), such that dn = χ(r⃗, ν, Ω̱, t)dr⃗ dν dΩ̱ is the number of photons (at time t) at a space point r⃗ in a differential volume dr⃗, with frequency ν in a frequency interval dν, and traveling in a direction Ω̱ in a solid angle element dΩ̱. In radiative transfer, it is conventional to use the specific intensity I(r⃗, ν, Ω̱, t) = chνχ(r⃗, ν, Ω̱, t), instead of the distribution function χ.

## Keywords

Vegetation Canopy Discrete Ordinate Leaf Canopy Scatter Phase Function Leaf Area Density- A
albedo of the leaf canopy

- A
_{s} albedo of the soil

- c
vacuum speed of light

- D
diffusion coefficient

- E
energy of a photon

- EQ
_{N} equal weight quadrature sets

- F
radiative flux

- F
_{d} incident flux density of diffuse sky radiation relative to a horizontal plane

- F
_{o} incident flux density of direct solar radiation ⊥ to Ω̱0

- F
_{r} Fresnel term

- F
_{s} scalar flux

- G
geometry factor

- g
_{L} probability density function of leaf normal orientation

- h
Planck’s constant

- I
specific intensity, radiance, angular flux

- I
_{d} diffuse sky intensity

- I
_{o} intensity of direct solar radiation incident in Ω̱0

- I
_{r} intensity reflected from the soil surface

- J
total source term

- K
correction factor for leaf specular reflection

- LAI
leaf area index

- n
refractive index

- P
normalized canopy scattering phase function

- p
probability

- p
_{j} joint probability

- Q
first collision source term

- q
emission density

- R
bidirectional reflectance factor

- r
_{LD} leaf hemispherical reflectance (for diffuse internal scattering)

- r
_{LS} leaf hemispherical reflectance (for specular reflection)

- r⃗
positional variables

- S
source term

- T
transmissivity

- t
time

- t
_{LD} leaf hemispherical transmittance

- U
energy density

- u
_{L} leaf area density function

- V
pressure tensor

- W
quadrature weight

- X
_{s}, Y_{s}, Z_{s} physical dimensions of a plant stand

- z
physical depth

- α
angle between the leaf normal and either the photon incident or exit direction

- Γ
area scattering phase function

- γ
_{L} leaf scattering phase function

- γ
_{LD} leaf scattering phase function for diffuse internal scattering

- γ
_{LS} leaf scattering phase function for specular reflection

- δ
Dirac delta function

- η
directional cosine with respect to the Y-coordinate

- θ
polar angle or zenith angle

- θ′
polar angle of the incident radiation beam

- θ
_{d} polar angle of incident diffuse sky radiation

- θ
_{L} polar angle of the leaf normal

- θ*
_{L} polar angle of the specularly reflecting leaf normal

- θ
_{O} polar angle of incident direct solar radiation

- κ
mean value of the projection factors of leaf hairs

- Λ
transport operator

- Λ
_{D} lower order diffusion operator

- μ,
directional cosine with respect to the Z-coordinate

- μ
cos θ; same as above

- μ′
cos θ′

- μ
_{d} cos θ

_{d}- μ
_{L} cos θ

_{L}- μ
_{o} cos θ

_{o}- ν
photon frequency

- ν′
incident photon frequency

- ξ,
directional cosine with respect to the X-coordinate

- ρ
_{s} bidirectional reflectance distribution function of the soil

- ϱ
rebalance factors

- ϱ
_{sys} global rebalance factor

- σ
total interaction coefficient

- σ
_{a} macroscopic absorption coefficient

- σ
_{s} differential scattering coefficient

- σ
_{s′}, scattering coefficient

- τ
optical depth

- ϒ
cross correlation function

- ϒ′
bidirectional indicator function

- Φ
area absorption coefficient

- φ
azimuthal angle

- φ′
azimuth of the incident radiation beam

- φ
_{d} azimuthal angle of diffuse sky radiation

- φ
_{L} azimuthal angle of the leaf normal

- φ*
_{L} azimuthal angle of the specularly reflecting leaf normal

- φ
_{o} azimuthal angle of direct solar radiation

- φ
_{t} boundary angle

- χ
photon density distribution function

- Ψ
area scattering coefficient

- ψ
finite element basis functions

- Ω̱
solid angle

- Ω̱′
solid angle containing the incident radiation beam

- Ω̱
_{d} solid angle containing the incident diffuse sky radiation

- Ω̱
_{L} solid angle containing the leaf normal

- Ω̱*
_{L} normal of specularly reflecting leaves

- Ω̱
_{o} solid angle containing the incident direct solar radiation

- Ω•∇⃗
streaming operator

- ω
single scattering albedo

- ω
_{L} leaf albedo

- ℵ
index of heliotropism

*ℓ*iteration index

- ℘
renormalization factors

- ℑ
azimuthally symmetric form of σs

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