Photon-Vegetation Interactions pp 161-190 | Cite as

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

## 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

Corn Photosynthesis Refraction Azimuth Remote Sensing### Symbols

- BDRF
bidirectional reflectance factor

- NIR
near infrared

- R
canopy reflectance factor

- R
^{1}_{C} reflectance factor of single scattering on canopy elements

- R
_{1S} reflectance factor of single reflectance from the soil

- R
^{M} 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

- Ω̱
_{0}~ (θ_{0}, φ_{0}) the direction of solar radiation

- θ
_{0} the solar zenith angle

- φ
_{0} the solar azimuth

- Μ
_{0} cos θ

_{0}- Ω̱
_{L}~ (θ_{L}, φ_{L}) the direction of the foliage normal

- θ
_{L} normal’s zenith angle

- φ
_{L} azimuth

- u
_{L}(z) foliage area volume density at the depth z

- g
_{L}(Ω̱_{L})/2π foliage normal’s distribution function

- γ
_{L}(Ω̱_{L}, Ω̱_{0}→ Ω̱) the scattering phase function of a canopy element with normal direction Ω̱

_{L}- Γ(Ω̱
_{0}→ 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

- C
_{HS} the hot spot correction factor

- G(Ω̱)
the projection of a unit foliage area on the plane perpendicular to the direction Ω̱

- γ
_{LD} the diffuse scattering phase function of leaves

- γ
_{LS} the specular reflection phase function of leaves

- Γ
_{s} the specular reflection phase function for the canopy medium

- Γ
_{D} the diffuse scattering phase function for the canopy medium

- K(α
_{0}) the reduction factor for the specular reflection

- α
_{0} the angle between leaf normal and incident radiation direction

- k
hair area index on leaves

- γ
_{s} s-component of the specular reflection

- γ
_{p} p-component of the specular reflection

- n
the leaf refraction index

- γ
_{LP} the portion of linearly polarized radiation reflected from the leaf

- Ω̱
_{L}^{*=}(θ_{L}^{*}, φ_{L}^{*}) the direction of the leaf normal specularly reflecting radiation

- R
_{soil} the soil reflectance factor

- r
_{L} leaf reflection coefficient

- t
_{L} leaf transmission coefficient

- ω = r
_{L}+ t_{L} 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

- p
_{l,}p_{2},…, p_{5} various probabilities

- A
albedo of the plant canopy

- A
_{n} albedo of n-th order scattering

- A
_{soil} the soil albedo

- F
_{1} the downward flux density of radiation at least once scattered in the canopy

- F
_{2} the upward flux density of radiation

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### References

- Anisimov OA, Menzhulin GV (1983) On statistical properties of radiative transfer in non-homogeneous plant canopies. Meteorol Gidrol 7:61–66 (in Russian)Google Scholar
- Breece HT, Holmes RJ (1971) Bidirectional scattering characteristics of healthy green soybean and corn leaves in vivo Appl Opt 10:119–127Google Scholar
- Card DH (1987) A simplified derivation of leaf normal spherical coordinates. IEEE Trans Geosci Remote Sens GE-25, 6:884–885CrossRefGoogle Scholar
- Chandrasekhar S (1960) Radiative transfer. Dover, New YorkGoogle Scholar
- Davison B (1958) Neutron transport theory. Oxford Univ Press, LondGoogle Scholar
- Gerstl SAW, Simmer C, Powers BJ (1986) The canopy hot spot as crop identifier. In: Damen MCJ et al. (eds) Remote Sens Res Dev Environ Mgmt Proc 7th Int Symp Enschede, The Netherlands, ISPRS, 26:261–263Google Scholar
- Goel NS, Strebel DE (1984) Simple beta distribution representation of leaf orientation in vegetation canopies. Agron J 76:800–802CrossRefGoogle Scholar
- Kozoderov VV (1982) Application of electromagnetic field equations in the case of radiation interaction with natural formations. Earth Res Space 3:69–76 (in Russian, English translation in Sov J Remote Sens)Google Scholar
- Kuusk A (1983) The hot spot effect of a uniform vegetative cover. Earth Res Space 4:90–99Google Scholar
- Kuusk A (1983) The hot spot effect of a uniform vegetative cover. (in Russian, English translation in Sov J Remote Sens 1985, 3(4):645–658)Google Scholar
- Kuusk A (1987) Direct sunlight scattering by the crown of a tree. Earth Res Space 2:106–111 (in Russian, to be translated in Sov J Remote Sens)Google Scholar
- Kuusk A, Nilson T (1989) The reflectance of shortwave radiation from multilayer plant canopies. Estonian Acad Sci Sect Phys Astron, Preprint A-l, Tallinn, 71 ppGoogle Scholar
- Kuusk A, Anton J, Nilson T (1984) Reflection indicatrices of vegetation covers. Earth Res Space 5:68–75.Google Scholar
- Kuusk A, Anton J, Nilson T (1984) Reflection indicatrices of vegetation covers.(in Russian, English translation in Sov J Remote Sens 1985, 4(5): 802–813)Google Scholar
- Moldau H (1965) On the use of polarized radiation to analyse the reflection indicatrixes of leaves. Investigations on Atmospheric Physics, 7: Questions on Radiation Regime of Plant Stand. Acad Sci ESSR Inst Phys Astron, Tartu, pp 96–101 (in Russian)Google Scholar
- Nilson T (1968) The calculation of spectral fluxes of shortwave radiation in plant communities. Investigations on Atmospheric Physics, 11: Solar radiation regime in plant stand. Acad Sci ESSR Inst Phys Astron, Tartu, pp 55–80 (in Russian)Google Scholar
- Nilson T (1971) A theoretical analysis of the frequency of gaps in plant stands. Agric Meteorol 8:25–38CrossRefGoogle Scholar
- Nilson T (1977) A theory of radiation penetration into non-homogeneous plant canopies. The penetration of solar radiation into plant canopies. Tartu, Acad Sci ESSR, pp 5–70 (in Russian)Google Scholar
- Nilson T (1990) A reflectance model for forests. Earth Res Space 3:67–72 (in Russian)Google Scholar
- Nilson T, Kuusk A (1984) Approximate analytic relationships for the reflectance of agricultural vegetation canopies. Earth Res Space 5:76–83.Google Scholar
- Nilson T, Kuusk A (1984) Approximate analytic relationships for the reflectance of agricultural vegetation canopies. (in Russian, English translation in Sov J Remote Sens 1985, 4(5):814–826)Google Scholar
- Nilson T, Kuusk A (1989) A reflectance model for the homogeneous plant canopy and its inversion. Remote Sens Environ 27:157–167CrossRefGoogle Scholar
- Norman JM, Welles JM (1983) Radiative transfer in an array of canopies. Agron J 75:481–488CrossRefGoogle Scholar
- Oker-Blom P (1984) Penumbral effects of within-plant shading on radiation distribution and leaf photosynthesis: A Monte Carlo simulation. Photosynthetica 18:522–528Google Scholar
- Oker-Blom P, Smolander H (1988) The ratio of shoot silhouette area to total needle area in Scots pine. For Sci 34:894–905Google Scholar
- Ross J (1975) The radiation regime and architecture of plant stands. Gidrometeoizdat. Leningrad, 342 pp (in Russian, the English translation 1981, Junk Publ, The Hague)Google Scholar
- Ross J, Nilson T (1975) Radiation exchange in plant canopies. In: de Vries DA, Afgan NH (eds) Heat and mass transfer in the biosphere. Scripta Book Co, Washington DC, pp 327–336Google Scholar
- Sobolev VV (1972) Scattering of radiation in the atmospheres of planets. Nauka, Moscow, 335 pp (in Russian)Google Scholar
- Vanderbilt VC (1980) A model of plant canopy polarization response. 6th Annu Symp Mach Process Remotely Sensed Data Soil Inf Syst Remote Sens Soil Surv, West Lafayette, Ind 1980, New York, pp 98–108Google Scholar
- Vanderbilt VC, Grant L (1985) Plant canopy specular reflectance model. IEEE Trans Geosci Remote Sens GE-23:722–730CrossRefGoogle Scholar
- Walthall CL, Norman JM, Welles JM, Campbell G, Blad BL (1985) Simple equation to approximate the bidirectional reflectance from vegetative canopies and bare soil surfaces. Appl Opt 24:383–387PubMedCrossRefGoogle Scholar