Discrete Ordinates Method for Photon Transport in Leaf Canopies

  • R. B. Myneni
  • A. Marshak
  • Y. Knyazikhin
  • G. Asrar

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

Corn Anisotropy Attenuation Photosynthesis Expense 
A

albedo of the leaf canopy

As

albedo of the soil

c

vacuum speed of light

D

diffusion coefficient

E

energy of a photon

EQN

equal weight quadrature sets

F

radiative flux

Fd

incident flux density of diffuse sky radiation relative to a horizontal plane

Fo

incident flux density of direct solar radiation ⊥ to Ω̱0

Fr

Fresnel term

Fs

scalar flux

G

geometry factor

gL

probability density function of leaf normal orientation

h

Planck’s constant

I

specific intensity, radiance, angular flux

Id

diffuse sky intensity

Io

intensity of direct solar radiation incident in Ω̱0

Ir

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

pj

joint probability

Q

first collision source term

q

emission density

R

bidirectional reflectance factor

rLD

leaf hemispherical reflectance (for diffuse internal scattering)

rLS

leaf hemispherical reflectance (for specular reflection)

r⃗

positional variables

S

source term

T

transmissivity

t

time

tLD

leaf hemispherical transmittance

U

energy density

uL

leaf area density function

V

pressure tensor

W

quadrature weight

Xs, Ys, Zs

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

γL

leaf scattering phase function

γLD

leaf scattering phase function for diffuse internal scattering

γLS

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

θd

polar angle of incident diffuse sky radiation

θL

polar angle of the leaf normal

θ*L

polar angle of the specularly reflecting leaf normal

θO

polar angle of incident direct solar radiation

κ

mean value of the projection factors of leaf hairs

Λ

transport operator

ΛD

lower order diffusion operator

μ,

directional cosine with respect to the Z-coordinate

μ

cos θ; same as above

μ′

cos θ′

μd

cos θd

μL

cos θL

μo

cos θo

ν

photon frequency

ν′

incident photon frequency

ξ,

directional cosine with respect to the X-coordinate

ρs

bidirectional reflectance distribution function of the soil

ϱ

rebalance factors

ϱsys

global rebalance factor

σ

total interaction coefficient

σa

macroscopic absorption coefficient

σs

differential scattering coefficient

σs′,

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

φd

azimuthal angle of diffuse sky radiation

φL

azimuthal angle of the leaf normal

φ*L

azimuthal angle of the specularly reflecting leaf normal

φo

azimuthal angle of direct solar radiation

φt

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

Ω̱d

solid angle containing the incident diffuse sky radiation

Ω̱L

solid angle containing the leaf normal

Ω̱*L

normal of specularly reflecting leaves

Ω̱o

solid angle containing the incident direct solar radiation

Ω•∇⃗

streaming operator

ω

single scattering albedo

ωL

leaf albedo

index of heliotropism

iteration index

renormalization factors

azimuthally symmetric form of σs

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References

  1. Allen LH Jr (1974) Model of light penetration into a wide row crop. Agron J 66:41–47CrossRefGoogle Scholar
  2. Anselone PM (1961) Convergence of Chandrasekhar’s method for inhomogeneous transfer problems. J Math Meth 10:537–541Google Scholar
  3. Bass LP, Voloschenko AM, Germogenova TA (1986) Methods of discrete ordinates in radiation transport problems. Inst Appl Math, The USSR Acad Sci, Moscow (in Russian)Google Scholar
  4. Bell GI, Glasstone S (1970) Nuclear reactor theory. Van Nostrand Reinhold, New YorkGoogle Scholar
  5. Blomquist RN, Miller WF Jr (1980) A rigorous treatment of transverse buckling effects in two-dimensional neutron transport computations. Nucl Sci Eng 73:125–131Google Scholar
  6. Borel-Donohue CC (1988) Models for backscattering of millimeter waves from vegetation canopies. PhD Thesis, Dept Electrical and Computer Eng, Univ Massachusetts at Amherst, Mass 01003Google Scholar
  7. Breece HT, Holmes RA (1971) Bidirectional scattering characteristics of healthy green soybean and corn leaves in vivo. Appl Opt 10:119–127CrossRefGoogle Scholar
  8. Briggs LL, Miller WF Jr, Lewis EE (1975) Ray effect mitigation in discrete ordinates like angular finite element approximations in neutron transport. Nucl Sci Eng 57:205–212Google Scholar
  9. Bunnik NJJ (1978) The multispectral reflectance of shortwave radiation by agricultural crops in relation with their morphological and optical properties. Pudoc Publ, Wageningen, The NetherlandsGoogle Scholar
  10. Carbone RY, Lathrop KD (1969) SN-calculation of highly forward peaked neutron angular fluxes using assymetrical quadrature sets. Nucl Sci Eng 35:139–147Google Scholar
  11. Carlson BG (1970) Transport theory: discrete ordinates quadrature over the unit sphere. LANL Rept LA-4554, Los Alamos Natl LabGoogle Scholar
  12. Carlson BG (1971) Tables of equal weight quadrature EQN over the unit sphere. LANL Rept LA-4734, Los Alamos Natl LabGoogle Scholar
  13. Carlson BG, Lathrop KD (1968) Transport theory: the method of discrete ordinates. In: Greenspan H, Kelber CN, Okrent D (eds) Computing methods in reactor physics. Gordon and Breech, New York, pp 167–265Google Scholar
  14. Carlson BG, Lee CE (1961) Mechanical quadrature and the transport equation. LANL Rept LA-2583, Los Alamos Natl LabGoogle Scholar
  15. Chandrasekhar S (1960) Radiative transfer. Dover, New YorkGoogle Scholar
  16. de Wit CT (1965) Photosynthesis of leaf canopies. Pudoc Publ, Wageningen, The NetherlandsGoogle Scholar
  17. Duderstadt JJ, Martin WR (1979) Transport theory. Wiley, New YorkGoogle Scholar
  18. Feynman RP, Leighton RB, Sands M (1963) The Feynman lectures on physics, Vol. 1. Addison-Wesley, Reading, Massachusetts, Chap 33Google Scholar
  19. Froehlich R (1967) A theoretical foundation for coarse mesh variational techniques. USAEC Rept GA-7870, Gulf General AtomicGoogle Scholar
  20. Germogenova TA (1987) The local properties of the solutions of the transport equation. Nauka Publ Moscow (in Russian)Google Scholar
  21. Gerstl SAW, Simmer C (1986) Radiation physics and modeling for off-nadir satellite-sensing of non-Lambertian surfaces. Remote Sens Environ 20:1–29CrossRefGoogle Scholar
  22. Gerstl SAW, Zardecki A (1985a) Discrete ordinates finite element method for atmospheric radiative transfer and remote sensing. Appl Opt 24:81–93PubMedCrossRefGoogle Scholar
  23. Gerstl SAW, Zardecki A (1985b) Coupled atmosphere/canopy model for remote sensing of plant reflectance features. Appl Opt 24:94–103PubMedCrossRefGoogle Scholar
  24. Goel NS (1988) Models of vegetation canopy reflectance and their use in estimation of biophysical parameters from reflectance data. Remote Sens Rev 4:1–222CrossRefGoogle Scholar
  25. Hill TR (1975) ONETRAN: A discrete ordinates finite element code for the solution of the one dimensional multigroup transport equation. LANL Rept LA-5990-MS, Los Alamos Natl LabCrossRefGoogle Scholar
  26. Kimes DS, Kirchner JA (1982) Radiative transfer model for heterogeneous 3D scenes. Appl Opt 21:4119–4129PubMedCrossRefGoogle Scholar
  27. Knyazikhin Yu, Marshak AL (1987) The method of discrete ordinates for the solution of the transport equation (The Algebraical Model and the Rate of Convergence). Valgus Publ Tallinn (in Russian)Google Scholar
  28. Kuusk A (1985) The hot spot effect of a uniform vegetative cover. Sov J Remote Sens 3:645–658Google Scholar
  29. Kuusk A, Nilson T (1989) A reflectance model for the homogeneous plant canopy and its inversion. Remote Sens Environ 27:157–167CrossRefGoogle Scholar
  30. Lathrop KD (1968) Ray effects in discrete ordinates equations. Nucl Sci Eng 32:357–364Google Scholar
  31. Lathrop KD (1971) Remedies for ray effects. Nucl Sci Eng 45:255–261Google Scholar
  32. Lathrop KD (1972) Discrete-ordinates methods for the numerical solution of the transport equation. Reactor Tech 15:107–135Google Scholar
  33. Lathrop KD (1976) THREETRAN: A program to solve the multigroup discrete ordinates transport equation in (x, y, z) Geometry. USAEC Rept LA-6333-MS, Los Alamos Natl LabCrossRefGoogle Scholar
  34. Lathrop KD, Brinkley FW (1970) Theory and use of the gereral-geometry TWOTRAN program. USAEC Rept LA-4432, Los Alamos Natl LabCrossRefGoogle Scholar
  35. Lathrop KD, Carlson BG (1965) Discrete ordinates angular quadrature of the neutron transport equation. LANL Rept LA-3186, Los Alamos Natl LabGoogle Scholar
  36. Lathrop KD, Carlson BG (1971) Properties of new numerical approximations to the transport equation. J Quant Spectroscp Radiat Transfer 11:921–948CrossRefGoogle Scholar
  37. Lee CE (1962) Discrete SN approximation to transport theory. LANL Rept LA-2595, Los Alamos Natl LabGoogle Scholar
  38. Lewis EE, Miller WF Jr (1984) Computational methods of neutron transport. Wiley-Intersci, New YorkGoogle Scholar
  39. Li X, Strahler AH (1986) Geometric-optical bidirectional reflectance modeling of a coniferous forest canopy. IEEE Trans Geosci Remote Sens GE-24:906–919Google Scholar
  40. Marchuk GI, Lebedev VI (1971) Numerical methods in neutron transport theory. Atomizdat Publ, Moscow (in Russian)Google Scholar
  41. Marshak AL (1989) Consideration of the effect of hot spot for the transport equation in plant canopies. J Quant Spectroscp Radiat Transfer 42:615–630CrossRefGoogle Scholar
  42. Miller WF Jr, Reed WH (1977) Ray effect mitigation methods for two-dimensional neutron transport theory. Nucl Sci Eng 62:391–399Google Scholar
  43. 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
  44. Myneni RB, Kanemasu ET (1988) The hot spot of vegetation canopies. J Quant Spectroscp Radiat Transfer 40:165–168CrossRefGoogle Scholar
  45. Myneni RB, Asrar G, Kanemasu ET (1988a) Finite element discrete ordinates method for radiative transfer in non-rotationally invariant scattering media: Application to the leaf canopy problem. J Quant Spectroscp Radiat Transfer 40:147–155CrossRefGoogle Scholar
  46. Myneni RB, Gutschick VP, Asrar G, Kanemasu ET (1988b) Photon transport in vegetation canopies with anisotropic scattering: Part I. The scattering phase functions in the one-angle problem. Agric For Meteorol 42:1–16CrossRefGoogle Scholar
  47. Myneni RB, Gutschick VP, Asrar G, Kanemasu ET (1988c) Photon transport in vegetation canopies with anisotropic scattering: Part II. Discrete-ordinates finite-difference exact-kernel technique for photon transport in slab geometry for the one-angle problem. Agric For Meteorol 42:17–40CrossRefGoogle Scholar
  48. Myneni RB, Gutschick VP, Asrar G, Kanemasu ET (1988d) Photon transport in vegetation canopies with anisotropic scattering: Part III. The scattering phase functions in the two-angle problem. Agric For Meteorol 42:87–99CrossRefGoogle Scholar
  49. Myneni RB, Gutschick VP, Asrar G, Kanemasu ET (1988e) Photon transport in vegetation canopies with anisotropic scattering: Part IV. Discrete-ordinates finite-difference exact-kernel technique for photon transport in slab geometry for the two-angle problem. Agric For Meteorol 42:101–120CrossRefGoogle Scholar
  50. Myneni RB, Ross J, Asrar G (1989a) A review on the theory of photon transport in leaf canopies in slab geometry. Agric For Meteorol 45:1–165, (special issue)Google Scholar
  51. Myneni RB, Asrar G, Kanemasu ET (1989b) The theory of photon transport in leaf canopies. In: Asrar G (ed) Theory and applications of optical remote sensing. Wiley, New York, pp 167–265Google Scholar
  52. Myneni RB, Asrar G, Gerstl SAW (1990) Radiative transfer in three dimensional leaf canopies. Trans Theory Stat Phys 19: 1–54CrossRefGoogle Scholar
  53. Nakamura S (1970) A variational rebalancing method for linear iterative convergence schemes of neutron diffusion and transport equation. Nucl Sci Eng 39:278–284Google Scholar
  54. Nelson P (1973) Convergence of the discrete ordinates method for anisotropically scattering multiplying particles in a subcritical slab. SIAM J Numer Anal 10:175–181CrossRefGoogle Scholar
  55. Nelson P, Victory HD (1979) Theoretical properties of one-dimensional discrete ordinates. SIAM J Numer Anal 16:270–283CrossRefGoogle Scholar
  56. Nilson T (1971) A theoretical analysis of the frequency of gaps in plant stands. Agric Meteorol 8:25–38CrossRefGoogle Scholar
  57. Norman JM, Wells JM (1983) Radiative transfer in an array of canopies. Agron J 75:481–488CrossRefGoogle Scholar
  58. Odom JP (1975) Neutron transport with highly anisotropic scattering. PhD Diss, Kansas State Univ, Manhattan, KS 66506, USAGoogle Scholar
  59. Pomraning GC (1973) The equations of radiation hydrodynamics. Pergamon Press, OxfordGoogle Scholar
  60. Ranson KJ, Biehl LL (1983) Corn canopy reflectance modeling data set. LARS Tech Rept 071584, Purdue Univ, Indiana 47907Google Scholar
  61. Reed WH (1971) The effectiveness of acceleration techniques for iterative methods in transport theory. Nucl Sci Eng 45:245–249Google Scholar
  62. Reyna E, Badhwar GD (1985) Inclusion of specular reflectance in vegetation canopy models. IEEE Trans Geosc Remote Sens 23:731–736CrossRefGoogle Scholar
  63. Risner JM (1985) Semi-analytical evaluation of the scattering source term in discrete ordinates transport calculations. MS Thesis, Kansas State Univ, Manhattan, KS 66506, USAGoogle Scholar
  64. Rogers DF (1986) Procedural elements for computer graphics. McGraw-Hill, New YorkGoogle Scholar
  65. Ross J (1981) The radiation regime and architecture of plant stands. Junk Publ, Den Hague, The NetherlandsGoogle Scholar
  66. Ross J, Marshak AL (1988a) Calculation of canopy bidirectional reflectance using the Monte-Carlo method. Remote Sens Environ 24:213–225CrossRefGoogle Scholar
  67. Ross J, Marshak AL (1988b) Estimation of the influence of the leaf normal orientation and reflectance specular component on the canopy phase function. Atmos Opt 1:76–85 (in Russian)Google Scholar
  68. Ross J, Nilson T (1966) A mathematical model of the radiation regime of vegetation. In: Pyldmaa VK (ed) Actinometry and atmospheric optics. Israel Prog Sci Transl Jerusalem, pp 253–270Google Scholar
  69. Ross J, Nilson T (1967) The spatial orientation of leaves in crop stands. In: Nichiprovich AA (ed) Photosynthesis of productive systems. Israel Prog Sci Transl, Jerusalem, pp 86–99Google Scholar
  70. Ross J, Nilson T (1968) A mathematical model of radiation regime of plant cover. In: Actinometry and atmospheric optics. Valgus Publ, Tallinn, pp 263–281 (in Russian)Google Scholar
  71. Roth SD (1982) Raycasting for modeling solids. Compt Grap Image Proc 18:109–144CrossRefGoogle Scholar
  72. Shultis JK, Myneni RB 1988 Radiative transfer in vegetation canopies with anisotropic scattering. J Quant Spectroscp Radiat Transfer 39:115–129CrossRefGoogle Scholar
  73. Simmer C (1987) Modelled angular reflectance of plant canopies involving canopy phase function and shadowing. LANL Rept LA-UR-87–2638, Los Alamos Natl LabGoogle Scholar
  74. Tammet HF (1975) Introduction to the theory of linear finite spectrometry. Valgus Publ Tallinn (in Russian)Google Scholar
  75. Vanderbilt VC, Grant L (1985) Plant canopy specular reflectance model. IEEE Trans Geosc Remote Sens 23:722–730CrossRefGoogle Scholar
  76. Verstraete MM (1987) Radiation transfer in plant canopies: transmission of direct solar radiation and the role of leaf orientation. J Geophys Res 92:10985–10995CrossRefGoogle Scholar
  77. Voevodin VV, Tartyshinkov EE (1987) Computational processes with the Tioplit’s matrix. Nauka Publ, Moscow (in Russian)Google Scholar
  78. Wick CG (1943) Über ebene Diffusions probleme. Z Phys 120:702–718Google Scholar
  79. Wilson BF (1966) Development of the shoot system of Acer rurum L. Harvard For Pap 14:21–30Google Scholar
  80. Woolley JT (1971) Reflectance and transmittance of light by leaves. Plant Physiol 47:656–662PubMedCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • R. B. Myneni
  • A. Marshak
  • Y. Knyazikhin
  • G. Asrar

There are no affiliations available

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