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Approximate Analytical Methods for Calculating the Reflection Functions of Leaf Canopies in Remote Sensing Applications

  • T. Nilson

Abstract

The turbid medium concept (Ross 1981; Ross and Nilson 1975) has appeared to be one of the most rigorous approaches to radiative transfer in plant canopies. This concept leads to the integro-differential radiative transfer equation, whose main advantage lies in the use of the mathematical methods of solving the transfer equation that were developed in atmospheric and neutron physics (Chandrasekhar 1960; Sobolev 1972; Davison 1958; several chapters in the present Vol., etc.). The equation of transfer in plant canopies has some specific features which make it more difficult to solve with traditional methods because numerical algorithms consume too much computer time. The transfer theory has also been extended by interpreting the interaction cross-sections of the transfer equation as random variables (Anisimov and Menzhulin 1983). This method allows us to derive higher moments of the spatial distribution of radiance (variance, skewness, etc.) in addition to mean radiance. In spite of evident success, there still exist some theoretical problems in motivating the use of the transfer equation in such a complicated optical medium as vegetation canopies (Ross and Nilson 1975). There has also been some criticism of the turbid medium concept inspired by the use of Maxwell equations (Kozoderov 1982).

Keywords

Multiple Scattering Solar Zenith Angle Reflectance Model Reflectance Factor Canopy Reflectance 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Symbols

BDRF

bidirectional reflectance factor

NIR

near infrared

R

canopy reflectance factor

R1C

reflectance factor of single scattering on canopy elements

R1S

reflectance factor of single reflectance from the soil

RM

multiple scattering reflectance

z

depth in the canopy

H

total depth (height) of the canopy

Ω̱ ~ (θ, φ)

the view direction

θ

the polar angle

φ

the azimuth

μ

cos θ

Ω̱0 ~ (θ0, φ0)

the direction of solar radiation

θ0

the solar zenith angle

φ0

the solar azimuth

Μ0

cos θ0

Ω̱L ~ (θL, φL)

the direction of the foliage normal

θL

normal’s zenith angle

φL

azimuth

uL(z)

foliage area volume density at the depth z

gL(Ω̱L)/2π

foliage normal’s distribution function

γL(Ω̱L, Ω̱0 → Ω̱)

the scattering phase function of a canopy element with normal direction Ω̱L

Γ(Ω̱0 → Ω̱)

the scattering phase function for the canopy medium

p(z, z′, z″

the bidirectional probability of the simultaneous free lines of sight between the depths z and z′ in the direction Ω′ and between the depths z′ and z” in the direction Ω, when viewed from the same point at the height z′

Ω̱′, Ω̱)

the bidirectional probability of the simultaneous free lines of sight between the depths z and z′ in the direction Ω̱′ and between the depths z′ and z″ in the direction Ω̱, when viewed from the same point at the height z′

a(z, Ω̱′)

the gap probability in the canopy viewed from the depth z in the direction Ω̱

CHS

the hot spot correction factor

G(Ω̱)

the projection of a unit foliage area on the plane perpendicular to the direction Ω̱

γLD

the diffuse scattering phase function of leaves

γLS

the specular reflection phase function of leaves

Γs

the specular reflection phase function for the canopy medium

ΓD

the diffuse scattering phase function for the canopy medium

K(α0)

the reduction factor for the specular reflection

α0

the angle between leaf normal and incident radiation direction

k

hair area index on leaves

γs

s-component of the specular reflection

γp

p-component of the specular reflection

n

the leaf refraction index

γLP

the portion of linearly polarized radiation reflected from the leaf

Ω̱L*=L*, φL*)

the direction of the leaf normal specularly reflecting radiation

Rsoil

the soil reflectance factor

rL

leaf reflection coefficient

tL

leaf transmission coefficient

ω = rL + tL

leaf scattering coefficient (albedo of single scattering)

I(n)1 (z, Ω̱)

the radiance of downward radiation at the depth z in the direction Ω̱, n times scattered within the plant canopy

I(n)2(z, Ω̱)

the radiance of upward radiation n times scattered

M, M′, M″

points in the canopy

pl, p2,…, p5

various probabilities

A

albedo of the plant canopy

An

albedo of n-th order scattering

Asoil

the soil albedo

F1

the downward flux density of radiation at least once scattered in the canopy

F2

the upward flux density of radiation

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References

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© Springer-Verlag Berlin Heidelberg 1991

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  • T. Nilson

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