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Photon-Vegetation Interactions pp 9–43Cite as

Fundamental Equations of Radiative Transfer in Leaf Canopies, and Iterative Methods for Their Solution

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Abstract

The theory of radiative transfer in turbid media was developed well enough for solving problems in astrophysics, nuclear physics and atmospheric physics (Davison 1958; Marchuk 1961 on the process of neutron transfer; Chandrasekhar 1950; Sobolev 1963 on astrophysical problems; Vladimirov 1961; Case and Zweifel 1967 for a mathematical description of transport theory). A formal way of developing radiative transfer theory in leaf canopies using the analogy of a turbid layer can be found in Shifrin (1953). Ross and colleagues further developed the theory in the mid 1960s (Ross 1962, 1964; Ross and Nilson 1963, 1965, 1967, 1968a, b: Nilson 1968a, b; Niilisk and Ross 1969).

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Abbreviations

as(z, Ω′̱ → Ω̱):

new scattering cross-section

g(z, Ω̱·Ω̱′):

any rotationally invarianted scattering phase function

G1(L, Ω′̱):

diffuse component of area scattering cross-section Ψ(L, Ω′̱)

G2(L, Ω′̱):

specular component of area scattering cross-section Ψ(L, Ω′̱)

I′0 :

constant denoting the direct radiation attenuated by the atmosphere

In(z, Ω̱):

n-th approximation to the solution of the transfer equation

Iun(L, Ω̱):

uncollided radiation

Iun 1(L, Ω̱):

incident diffuse radiation stream that has not undergone any interactions in the canopy

Iun 2(L, Ω̱):

incident direct radiation stream that has not undergone any interactions in the canopy

Ic(L, Ω̱):

radiance of photons which have been scattered one or more times in the canopy

Ic 1(L, Ω̱):

first-order scattering radiance when a monodirectional stream of photons escapes the source

Ic M(L, Ω̱):

radiance of multiply scattered photons

Jk(z, Ω̱):

radiance of the photons scattered k times

lL :

length of the mean chord of the leaf

l*L = lL/H:

parameter characterizing the leaf dimensions

rΩ0,Ω :

correlation function

R(Ω̱′, Ω̱):

canopy bidirectional reflectance factor

Ra(Ω̱′, Ω̱):

bidirectional reflectance factor of the atmosphere

Rs(Ω̱′, Ω̱):

bidirectional reflectance factor of the soil

SOSA:

successive orders of scattering approximation

t1 :

computer time needed for calculating one iteration

t2 :

computer time spent on additional operations

y(L, Ω̱):

solution of the integral transfer equation

Z:

geometrical depth of the atmosphere

α:

angle between the leaf normal and the photon incident direction

α′:

angle between the leaf normal and the photon exit direction

ΓD, (L, Ω̱′ → Ω̱):

diffuse component of the area scattering phase function

ΓS, (L, Ω̱′ → Ω̱):

specular component of the area scattering phase function

δ2 (Ω̱·Ω̱′):

surface delta function

εn :

parameter characterizing convergence of the iterative process

η1, η2 :

given accuracies

λ(r, Ω̱):

analogy of albedo for single scattering

μ*L :

cos θ*L

ρc :

rate of convergence of the iterative process

σ′(L, Ω̱0, Ω̱):

new total interaction cross-section

τ(z′, z, Ω̱):

optical depth between the points z′ and z along the direction Ω̱

T :

optical depth of the atmosphere

Ω̱*:

direction of specular reflection

Ω̱± :

part of the hemisphere in which ± (Ω̱′ Ω̱L)(Ω̱ Ω̱L) - 0

References

  • Ambarzumian VA (1942) New method of calculation of light scattering in turbid medium. Izvestiya of Acad Sci of USSR. Geogr Geophys 3:97–104 (in Russian)

    Google Scholar 

  • Bass LP, Voloschenko AM, Germogenova TA (1986) Method of discrete ordinates in radiation transport problems. Inst Appl Math Acad Sci USSR, Moscow, USSR (in Russian)

    Google Scholar 

  • Breece HT, Holmes RA (1971) Bidirectional scattering characteristics of healthy green soybean and corn leaves in vivo. Appl Opt 10:119–127

    Article  Google Scholar 

  • Bunnik NJJ (1978) The multispectral reflectance of shortwave radiation of agricultural crops in relation with their morphological and optical properties. Mededelingen Landbouwhogeschool, Wageningen, The Netherlands

    Google Scholar 

  • Case K, Zweifel P (1967) Linear transport theory. Addison-Wesley, Reading, Mass, USA

    Google Scholar 

  • Chandrasekhar S (1950) On the radiative transfer. Dover, New York, USA

    Google Scholar 

  • Chandrasekhar S (1960) Radiative transfer. Dover, New York, USA

    Google Scholar 

  • Davison B (1958) Neutron transport theory. Oxford Univ Press, Lond

    Google Scholar 

  • Germogenova TA (1986) The local properties of the solution of the transport equation. Nauka, Moscow, USSR (in Russian)

    Google Scholar 

  • Germogenova TA, Dyatlova AS (1985) Solving the transport equation with higher amount of precision. Program TEST. Inst Appl Math Acad Sci USSR, Preprint 139, Moscow (in Russian)

    Google Scholar 

  • Gerstl SAW (1986) Off-nadir optical remote sensing from satellites for vegetation identification. Proc IGARSS’86 Symp Zurich, pp 1457–1460

    Google Scholar 

  • Gerstl SAW (1988) The angular reflectance signature of the canopy hot spot in the optical regime. Proc 4th Int Coll Spec Signat Objects Rem Sens Aussois, France, pp 129–132

    Google Scholar 

  • Gerstl SAW, Zardecki A (1985) Coupled atmosphere/canopy mode for remote sensing of plant reflectance features. Appl Opt 24:94–103

    Article  PubMed  CAS  Google Scholar 

  • Goel NS, Strebel DE (1984) Simple beta distribution representation of leaf orientation in vegetation canopies. Agron J 76:800–802

    Article  Google Scholar 

  • Hall G, Watt JM (eds) (1976) Modern numerical methods for ordinary differential equations. Clarendon Press, Oxford

    Google Scholar 

  • Knyazikhin Yu (1987) Convergence of the Seidel iterative method to solve operator equation. Proc Acad Sci Estonian SSR 3:246–250 (in Russian)

    Google Scholar 

  • Knyazikhin Yu (1988a) Seidel iterations for the solution of the anisotropic transport equation in R3. USSR Computer Math and Math Phys 6:917–925 (in Russian)

    Google Scholar 

  • Knyazikhin Yu (1988b) On a class of iterative methods in linear transport theory. USSR Computer Math and Math Phys 9:1321–1331 (in Russian)

    Google Scholar 

  • Knyazikhin Yu, Marshak A (1987) The method of discrete ordinates for the solution of the transport equation (the algebraic model, the rate of convergence). Valgus, Tallinn, Estonian SSR (in Russian)

    Google Scholar 

  • Krasnoselsky MA, Vainikko GM, Zabreyko PP, Rutitsky Ya B, Stetsenko VYa (1972) Approximate solution of operator equations. Wolters-Noordhoff Publ, Groningen

    Book  Google Scholar 

  • Kuusk A (1985) The hot spot effect of uniform vegetative cover. Sov J Remote Sens 3:645–658

    Google Scholar 

  • Kuusk AE (1987) Direct sunlight scattering by the crown of a tree. USSR Earth Res Space 2:106–111 (in Russian)

    Google Scholar 

  • Larsen EW (1984) Diffusion-synthetic acceleration method for discrete ordinates problems. Transport Theory Stat Phys 1–2:107–126

    Google Scholar 

  • Marchuk GI (1961) Methods of calculation of nuclear reactors. Atomizdat, Moscow, USSR (in Russian)

    Google Scholar 

  • Marchuk GI, Lebedev VI (1971) The numerical methods in the neutron transport theory. Atomizdat, Moscow, USSR (in Russian)

    Google Scholar 

  • Marshak AL (1985) On the solution of the transport equation with periodic boundary conditions by the method of discrete ordinates. Transport Theory Stat Phys 3:323–351

    Article  Google Scholar 

  • Moldau H (1965) Vegetative course of the luminance factor of leaves of some plants. In: Questions on radiation regime of plant stand. Acad Sci ESSR (in Russian)

    Google Scholar 

  • Myneni RB, Asrar G, Kanemasu ET (1987) The theory of photon transport in leaf canopies. In: Asrar G (ed) Theory and application of optical remote sensing. Wiley & Sons, New York

    Google Scholar 

  • Myneni RB, Gutschick VP, Asrar G, Kanemasu ET (1988) Photon transport in vegetation canopies with anisotropic scattering. Part III. Scattering phase functions in two-angles. Agric For Meteorol 42:87–99

    Article  Google Scholar 

  • Niilisk H, Ross J (1969) On some features of solar radiation in vegetation. In: Basic Problems of Biol Produc. Nauka, Leningrad, USSR, pp 155–160 (in Russian)

    Google Scholar 

  • Nilson T (1968a) On the optimum geometrical arrangement of foliage in the plant cover. In: Solar Radiation Regime in Plant Stand Acad Sci ESSR, Inst Phys Astron, Tartu, USSR, pp 112–146 (in Russian)

    Google Scholar 

  • Nilson T (1968b) The calculation of spectral fluxes of shortwave radiation in plant communities. In: Solar Radiation Regime in Plant Stand Acad Sci ESSR, Inst Phys Astron, Tartu, USSR, pp 55–80 (in Russian)

    Google Scholar 

  • Nilson T, Kuusk A (1985) Approximate analytic relationships reflectance of agricultural vegetation canopies. Sov J Remote Sens 5:814–826 (in Russian)

    Google Scholar 

  • Nilson T, Kuusk A (1989) A reflectance model for the homogeneous plant canopy and its inversion. Remote Sens Environ 27:157–167

    Article  Google Scholar 

  • Reyna E, Badhwar GD (1985) Inclusion of specular reflectance in vegetative canopy models. IEEE Trans Geosc Remote Sens 23, 5:731–736

    Article  Google Scholar 

  • Riesz F Sz.-Nagy B (1972) Lecons d’analyse fonctionnele. Akademiai Kiado, Budapest, Hungary

    Google Scholar 

  • Ross J (1962) Concerning the theory of vegetable cover albedo. Sci Pap Inst Geol Geogr Acad Sci Lith SSR Vilnius 13:151–165, USSR (in Russian)

    Google Scholar 

  • Ross J (1964) Photosynthetically active radiation (PAR) in plant stands and its mathematical modeling. In: Actinometry and Atmospheric Optics Nauka Publ, Moscow, USSR pp 251–256 (in Russian)

    Google Scholar 

  • Ross J (1981) The radiation regime and architecture of plant stands. Junk Publ, The Hague

    Google Scholar 

  • Ross J, Marshak AL (1985) A Monte Carlo procedure for calculating the scattering of solar radiation by plant canopies. Sov J Remote Sens 4:783–801

    Google Scholar 

  • Ross J, Marshak AL (1988) Calculation of canopy bidirectional reflectance using the Monte Carlo method. Remote Sens Envir. 24:213–225

    Article  Google Scholar 

  • Ross J, Nilson T (1963) Concerning the theory of plant cover radiation regime. In: Investig Atm Phys Acad Sci ESSR 4:42–64, Inst Phys Astron Tartu, USSR (in Russian)

    Google Scholar 

  • Ross J, Nilson T (1965) The extinction of direct radiation in crops. In: Questions on radiation regime of plant stand. Acad Sci ESSR, Inst. Phys Astron, Tartu, USSR, pp 25–64 (in Russian)

    Google Scholar 

  • Ross J, Nilson T (1967) The radiation regime of plant communities with horizontal leaves. In: Phytoactinometric investigations of plant stand Valgus, Tallinn, USSR, pp 5–34 (in Russian)

    Google Scholar 

  • Ross J, Nilson T (1968a) The calculation of photosynthetically active radiation in plant communities. In: Solar radiation regime in plant stand Acad Sci ESSR, Inst Phys Astron, Tartu, USSR, pp 5–54 (in Russian)

    Google Scholar 

  • Ross J, Nilson T (1968b) A mathematical model of radiation regime of plant cover. In: Actinometry and atmospheric optics Valgus, Tallinn, USSR, pp 263–281 (in Russian)

    Google Scholar 

  • Shifrin KS (1953) Concerning the theory of albedo. Trans Main Geophys Obs 39, 101:244–257 (in Russian)

    Google Scholar 

  • Shultis JK, Myneni RB (1988) Radiative transfer in vegetation canopies with anisotropic scattering. J Quant Spectrosc Radiat Transfer 39, 2:115–129

    Article  Google Scholar 

  • Sobolev VV (1963) A treatise on radiative transfer. Nostrand (original publ 1956), New York

    Google Scholar 

  • Sobolev VV (1975) Light scattering in planetary atmospheres. Pergamon Press, New York, USA

    Google Scholar 

  • Vanderbilt VC (1987) Source-specular-reflector-sensor solid angles. J Opt Soc Am 4, 7:1243–1244

    Article  CAS  Google Scholar 

  • Vanderbilt VC, Grant L (1985) Plant canopy specular reflectance model. IEEE Trans Geosc Remote Sens 23, 5:722–730

    Article  Google Scholar 

  • Vladimirov VV (1963) Mathematical problems in the one-velocity theory of particle transport. AECL-1661, Chalk, River, Ontario, Canada (original publ 1961)

    Google Scholar 

  • Woolley JT (1971) Reflectance and transmittance of light by leaves. Plant Physiol 47:656–662

    Article  PubMed  CAS  Google Scholar 

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Authors
  1. Y. Knyazikhin
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  2. A. Marshak
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Editor information

Editors and Affiliations

  1. Hydrological Sciences Branch, NASA-Goddard Space Flight Center, Mail Code 974, 20771, Greenbelt, MD, USA

    Ranga B. Myneni

  2. Institute of Astrophysics and Atmospheric Physics, Estonian Academy of Sciences, Tartu, 202444, Tõravere, Estonia

    Juhan Ross

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Knyazikhin, Y., Marshak, A. (1991). Fundamental Equations of Radiative Transfer in Leaf Canopies, and Iterative Methods for Their Solution. In: Myneni, R.B., Ross, J. (eds) Photon-Vegetation Interactions. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-75389-3_2

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  • DOI: https://doi.org/10.1007/978-3-642-75389-3_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-75391-6

  • Online ISBN: 978-3-642-75389-3

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