Abstract

The plate description of a typical, thin, compact plant leaf, introduced previously, has been generalized to the noncompact case and applied to experimental data including average reflectance and transmittance measurements on 200 mature, field-cotton leaves. A compact leaf has few and a noncompact leaf has many intercellular air spaces in the mesophyll. No statistically significant difference was found between the average leaf thickness and the mean effective water thickness of the leaves. The Kubelka–Munk scattering coefficient s for a typical leaf, measured at the 1-μ spectral region, is approximated by the relation s = r/t, where r and t, respectively, are the reflectance and transmittance of the leaf. The approximate equality of r and t, noted by previous investigators, is explained on the basis of the scattering of diffuse light within the leaf by critical internal reflections. Predictions from the plate (P) model for cotton leaves compare favorably with those of the Kubelka–Munk (K–M) and Melamed (M) theories. Applied to vegetation, all three theories predict a characteristic linear dimension related to the cellular structure of the leaf.

© 1970 Optical Society of America

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  1. W. A. Allen and A. J. Richardson, J. Opt. Soc. Am. 58, 1023 (1968).
    [Crossref]
  2. P. Kubelka and F. Munk, Z. Tech. Physik 12, 593 (1931).
  3. W. A. Allen, H. W. Gausman, A. J. Richardson, and J. R. Thomas, J. Opt. Soc. Am. 59, 1376 (1969).
    [Crossref]
  4. N. T. Melamed, J. Appl. Phys. 34, 560 (1963).
    [Crossref]
  5. W. W. Wendlandt and H. G. Hecht, Reflectance Spectroscopy (Interscience Publ., John Wiley & Sons, Inc., New York, 1966), p. 62.
  6. A. C. Hardy and F. H. Perrin, The Principles of Optics (McGraw–Hill Book Co., New York, 1932), p. 24.
  7. Seishi Isobe, Bull. Natl. Inst. Agricultural Sci. (Japan) Ser. A, No.  16 (March1969).
  8. D. M. Gates, H. J. Keegan, J. C. Schleter, and V. R. Weidner, Appl. Opt. 4, 11 (1965).
    [Crossref]
  9. E. I. Rabinowitch, Photosynthesis and Related Processes (Interscience Publ., John Wiley & Sons, Inc., New York, 1961), Vol. II, Part 1.
  10. R. A. Moss and W. E. Loomis, Plant Physiol. 27, 370 (1952).
  11. H. W. Gausman, W. A. Allen, V. I. Myers, and R. Cardenas, Agron. J. 61, 374 (1969).
    [Crossref]
  12. V. I. Myers and W. A. Allen, Appl. Opt. 7, 1819 (1968).
    [Crossref] [PubMed]
  13. Trade names are included for information only and do not constitute endorsement by the U. S. Department of Agriculture.
  14. C. T. de Wit, Agr. Res. Rep. 663. Inst. Biol. Chem. Res. on Field Crops and Herbage, Wageningen, Netherlands (1965).
  15. G. Kortüm, W. Braun, and G. Herzog, Angew. Chem. 2, 333 (1963).
    [Crossref]
  16. G. G. Stokes, Proc. Roy. Soc. (London) 11, 545 (1862).
  17. G. W. Ingle. Note on coincidence of equations for the reflectance of translucent films. American Society for Testing and Materials Bulletin 116, 32 (1942).
  18. F. Stern, Appl. Opt. 3, 111 (1964).
    [Crossref]
  19. P. Kubelka, J. Opt. Soc. Am. 38, 448 (1948).
    [Crossref] [PubMed]
  20. M. Centeno, J. Opt. Soc. Am. 31, 245 (1941).
    [Crossref]
  21. R. Willstätter and A. Stoll, Untersuchungen über die Assimilation der Kohlensäure (Springer-Verlag, Berlin, 1913).
  22. M. D. Heilman, C. L. Gonzalez, W. A. Swanson, and W. J. Rippert, Agron. J. 60, 578 (1968).
    [Crossref]

1969 (3)

W. A. Allen, H. W. Gausman, A. J. Richardson, and J. R. Thomas, J. Opt. Soc. Am. 59, 1376 (1969).
[Crossref]

H. W. Gausman, W. A. Allen, V. I. Myers, and R. Cardenas, Agron. J. 61, 374 (1969).
[Crossref]

Seishi Isobe, Bull. Natl. Inst. Agricultural Sci. (Japan) Ser. A, No.  16 (March1969).

1968 (3)

1965 (1)

1964 (1)

1963 (2)

N. T. Melamed, J. Appl. Phys. 34, 560 (1963).
[Crossref]

G. Kortüm, W. Braun, and G. Herzog, Angew. Chem. 2, 333 (1963).
[Crossref]

1952 (1)

R. A. Moss and W. E. Loomis, Plant Physiol. 27, 370 (1952).

1948 (1)

1942 (1)

G. W. Ingle. Note on coincidence of equations for the reflectance of translucent films. American Society for Testing and Materials Bulletin 116, 32 (1942).

1941 (1)

M. Centeno, J. Opt. Soc. Am. 31, 245 (1941).
[Crossref]

1931 (1)

P. Kubelka and F. Munk, Z. Tech. Physik 12, 593 (1931).

1862 (1)

G. G. Stokes, Proc. Roy. Soc. (London) 11, 545 (1862).

Allen, W. A.

Braun, W.

G. Kortüm, W. Braun, and G. Herzog, Angew. Chem. 2, 333 (1963).
[Crossref]

Cardenas, R.

H. W. Gausman, W. A. Allen, V. I. Myers, and R. Cardenas, Agron. J. 61, 374 (1969).
[Crossref]

Centeno, M.

M. Centeno, J. Opt. Soc. Am. 31, 245 (1941).
[Crossref]

de Wit, C. T.

C. T. de Wit, Agr. Res. Rep. 663. Inst. Biol. Chem. Res. on Field Crops and Herbage, Wageningen, Netherlands (1965).

Gates, D. M.

Gausman, H. W.

H. W. Gausman, W. A. Allen, V. I. Myers, and R. Cardenas, Agron. J. 61, 374 (1969).
[Crossref]

W. A. Allen, H. W. Gausman, A. J. Richardson, and J. R. Thomas, J. Opt. Soc. Am. 59, 1376 (1969).
[Crossref]

Gonzalez, C. L.

M. D. Heilman, C. L. Gonzalez, W. A. Swanson, and W. J. Rippert, Agron. J. 60, 578 (1968).
[Crossref]

Hardy, A. C.

A. C. Hardy and F. H. Perrin, The Principles of Optics (McGraw–Hill Book Co., New York, 1932), p. 24.

Hecht, H. G.

W. W. Wendlandt and H. G. Hecht, Reflectance Spectroscopy (Interscience Publ., John Wiley & Sons, Inc., New York, 1966), p. 62.

Heilman, M. D.

M. D. Heilman, C. L. Gonzalez, W. A. Swanson, and W. J. Rippert, Agron. J. 60, 578 (1968).
[Crossref]

Herzog, G.

G. Kortüm, W. Braun, and G. Herzog, Angew. Chem. 2, 333 (1963).
[Crossref]

Ingle, G. W.

G. W. Ingle. Note on coincidence of equations for the reflectance of translucent films. American Society for Testing and Materials Bulletin 116, 32 (1942).

Isobe, Seishi

Seishi Isobe, Bull. Natl. Inst. Agricultural Sci. (Japan) Ser. A, No.  16 (March1969).

Keegan, H. J.

Kortüm, G.

G. Kortüm, W. Braun, and G. Herzog, Angew. Chem. 2, 333 (1963).
[Crossref]

Kubelka, P.

P. Kubelka, J. Opt. Soc. Am. 38, 448 (1948).
[Crossref] [PubMed]

P. Kubelka and F. Munk, Z. Tech. Physik 12, 593 (1931).

Loomis, W. E.

R. A. Moss and W. E. Loomis, Plant Physiol. 27, 370 (1952).

Melamed, N. T.

N. T. Melamed, J. Appl. Phys. 34, 560 (1963).
[Crossref]

Moss, R. A.

R. A. Moss and W. E. Loomis, Plant Physiol. 27, 370 (1952).

Munk, F.

P. Kubelka and F. Munk, Z. Tech. Physik 12, 593 (1931).

Myers, V. I.

H. W. Gausman, W. A. Allen, V. I. Myers, and R. Cardenas, Agron. J. 61, 374 (1969).
[Crossref]

V. I. Myers and W. A. Allen, Appl. Opt. 7, 1819 (1968).
[Crossref] [PubMed]

Perrin, F. H.

A. C. Hardy and F. H. Perrin, The Principles of Optics (McGraw–Hill Book Co., New York, 1932), p. 24.

Rabinowitch, E. I.

E. I. Rabinowitch, Photosynthesis and Related Processes (Interscience Publ., John Wiley & Sons, Inc., New York, 1961), Vol. II, Part 1.

Richardson, A. J.

Rippert, W. J.

M. D. Heilman, C. L. Gonzalez, W. A. Swanson, and W. J. Rippert, Agron. J. 60, 578 (1968).
[Crossref]

Schleter, J. C.

Stern, F.

Stokes, G. G.

G. G. Stokes, Proc. Roy. Soc. (London) 11, 545 (1862).

Stoll, A.

R. Willstätter and A. Stoll, Untersuchungen über die Assimilation der Kohlensäure (Springer-Verlag, Berlin, 1913).

Swanson, W. A.

M. D. Heilman, C. L. Gonzalez, W. A. Swanson, and W. J. Rippert, Agron. J. 60, 578 (1968).
[Crossref]

Thomas, J. R.

Weidner, V. R.

Wendlandt, W. W.

W. W. Wendlandt and H. G. Hecht, Reflectance Spectroscopy (Interscience Publ., John Wiley & Sons, Inc., New York, 1966), p. 62.

Willstätter, R.

R. Willstätter and A. Stoll, Untersuchungen über die Assimilation der Kohlensäure (Springer-Verlag, Berlin, 1913).

Agron. J. (2)

H. W. Gausman, W. A. Allen, V. I. Myers, and R. Cardenas, Agron. J. 61, 374 (1969).
[Crossref]

M. D. Heilman, C. L. Gonzalez, W. A. Swanson, and W. J. Rippert, Agron. J. 60, 578 (1968).
[Crossref]

American Society for Testing and Materials Bulletin (1)

G. W. Ingle. Note on coincidence of equations for the reflectance of translucent films. American Society for Testing and Materials Bulletin 116, 32 (1942).

Angew. Chem. (1)

G. Kortüm, W. Braun, and G. Herzog, Angew. Chem. 2, 333 (1963).
[Crossref]

Appl. Opt. (3)

Bull. Natl. Inst. Agricultural Sci. (Japan) Ser. A (1)

Seishi Isobe, Bull. Natl. Inst. Agricultural Sci. (Japan) Ser. A, No.  16 (March1969).

J. Appl. Phys. (1)

N. T. Melamed, J. Appl. Phys. 34, 560 (1963).
[Crossref]

J. Opt. Soc. Am. (4)

Plant Physiol. (1)

R. A. Moss and W. E. Loomis, Plant Physiol. 27, 370 (1952).

Proc. Roy. Soc. (London) (1)

G. G. Stokes, Proc. Roy. Soc. (London) 11, 545 (1862).

Z. Tech. Physik (1)

P. Kubelka and F. Munk, Z. Tech. Physik 12, 593 (1931).

Other (6)

W. W. Wendlandt and H. G. Hecht, Reflectance Spectroscopy (Interscience Publ., John Wiley & Sons, Inc., New York, 1966), p. 62.

A. C. Hardy and F. H. Perrin, The Principles of Optics (McGraw–Hill Book Co., New York, 1932), p. 24.

E. I. Rabinowitch, Photosynthesis and Related Processes (Interscience Publ., John Wiley & Sons, Inc., New York, 1961), Vol. II, Part 1.

R. Willstätter and A. Stoll, Untersuchungen über die Assimilation der Kohlensäure (Springer-Verlag, Berlin, 1913).

Trade names are included for information only and do not constitute endorsement by the U. S. Department of Agriculture.

C. T. de Wit, Agr. Res. Rep. 663. Inst. Biol. Chem. Res. on Field Crops and Herbage, Wageningen, Netherlands (1965).

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Figures (4)

Fig. 1
Fig. 1

Function Φ(b) = (bb−1)−1 logb2 for use in Eq. (17). The arrows and their widths designate the appropriate values and ranges of b for 200 mature field-cotton leaves at the indicated wavelengths.

Fig. 2
Fig. 2

Mean Kubelka–Munk scattering coefficient s for 200 mature field-cotton leaves.

Fig. 3
Fig. 3

Mean dispersion curve for 200 mature field-cotton leaves.

Fig. 4
Fig. 4

Computer plot of average diffuse reflectance for 200 mature field-cotton leaves. Four points are printed at each wavelength but some are superimposed. The points represent ± one standard deviation about two different curves. The lower curve applies to single leaves and the upper curve is the calculated infinite reflectance R for these leaves.

Tables (4)

Tables Icon

Table I Scattering parameters with standard errors for leaves over the range 0.75–1.05 μ. The quantity s is the K–M scattering coefficient and n is the effective index of refraction.

Tables Icon

Table II Mean absorption coefficients of 200 mature field-cotton leaves k(cm−1) for Kubelka–Munk (K–M), Plate (P), and Melamed (M) reflectance theories compared with values for pure water.a

Tables Icon

Table III Mean absorption coefficients k(cm−1) of 200 mature field-cotton leaves for K–M, P, and M reflectance theories.

Tables Icon

Table IV Mean characteristic dimensionsa in microns from the K–M, P, and M reflectance theories for 200 mature field-cotton leaves compared with measured leaf thickness.

Equations (21)

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t - 1 ( a - a - 1 ) = r - 1 ( b - b - 1 ) = a b - a - 1 b - 1 ,
t - 1 ( a 0 - a 0 - 1 ) = r - 1 ( b 0 N - b 0 - N ) = a 0 b 0 N - a 0 - 1 b 0 - N ,
a = a 0 , b = b 0 N .
t 0 - 1 ( a 0 - a 0 - 1 ) = r 0 - 1 ( b 0 - b 0 - 1 ) = a 0 b 0 - a 0 - 1 b 0 - 1 .
t 0 - 1 ( a - a - 1 ) = r 0 - 1 ( b 1 / N - b - 1 / N ) = a b 1 / N - a - 1 b - 1 / N .
s = 2 a ( a 2 - 1 ) - 1 log b , s 0 = 2 a 0 ( a 0 2 - 1 ) - 1 log b 0 .
N = s / s 0 .
τ = ( 1 - z ) e - z + z 2 z x - 1 e - x d x ,
r = ( 1 - T 12 ) + τ 2 T 12 2 ( n 2 - T 12 ) n 4 - τ 2 ( n 2 - T 12 ) 2 ,
t = τ n 2 T 12 2 / [ n 4 - τ 2 ( n 2 - T 12 ) 2 ] ,
a = ( 1 + r 2 - t 2 + Δ ) / 2 r ,
b = ( 1 - r 2 + t 2 + Δ ) / 2 t ,
Δ 2 = ( 1 + r + t ) ( 1 + r - t ) ( 1 - r + t ) ( 1 - r - t ) ,
s = ( a - a - 1 ) - 1 log b 2 ,
τ = exp ( - 2 k D ) .
s = ( b - b - 1 ) / ( a b - a - 1 b - 1 ) log b 2 ( a - a - 1 ) / ( a b - a - 1 b - 1 ) b - b - 1 .
s = ( r / t ) Φ ( b ) ,
Φ ( b ) ( b - b - 1 ) - 1 log b 2
r t = ( 1 - T 12 ) [ n 4 - τ 2 ( n 2 - T 12 ) 2 ] + τ 2 T 12 2 ( n 2 - T 12 ) τ n 2 T 12 2 .
r t = ( 1 - T 12 ) ( 2 n 2 - T 12 ) + T 12 ( n 2 - T 12 ) n 2 T 12 .
r / t = 1 - n - 2 + 2 R 12 .