Abstract

We propose a nondestructive or optical method of measuring the chlorophyll content in a leaf after constructing a mathematical model of reflectance and transmittance of plant leaves as a function of their chlorophyll pigment content. The model is based on the Kubelka–Munk theory and involves the modeling of the multiple reflection of light in a leaf that is assumed to be composed of a stack of four layers. It also includes the assumption that the scattering coefficient and the absorption coefficient of the Kubelka–Munk theory can be expressed as a linear function of the pigment content of a plant leaf. In the proposed method, the chlorophyll content is calculated from reflectances and transmittances at three bands whose center wavelengths are 880,720, and 700 nm. Experiments were performed to confirm the applicability of the model and the method. Reflectance and transmittance calculated with the model showed good agreement with measured values. Furthermore, several unmeasurable constants necessary in the calculation were determined by a least-squares fit. We also confirmed that these results were consistent with several well-known facts in the botanical field. The method proposed here showed a small estimation error of 6.6 μg/cm2 over the 0–80 μg/ cm2 chlorophyll content range for all kinds of plant tested.

© 1991 Optical Society of America

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References

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  1. M. Aoki, K. Yabuki, T. Totsuka, “An evaluation of chlorophyll content of leaves based on the spectral reflectivity in several plants,” Res. Rep. Natl. Inst. Environ. Stud. Jpn. 66, 125–130 (1981).
  2. Minolta Camera Company, Ltd., Model SPAD-501 (1984).
  3. C. J. Tucker, L. D. Miller, R. L. Pearson, in Proceedings Second Annual Remote Sensing of Earth Resources Conference, F. Shahroki, ed. (U. Tennessee Press, Tullahoma, Tenn., 1973) p. 601.
  4. R. L. Pearson, L. D. Miller, C. J. Tucker, “Hand-held spectral radiometer to estimate gramineous biomass,” Appl. Opt. 15, 416–418 (1976).
    [CrossRef] [PubMed]
  5. F. E. Hoge, R. N. Swift, “Chlorophyll pigment concentration using spectral curvature algorithms: an evaluation of present and proposed satellite ocean color sensor bands,” Appl. Opt. 25, 3677–3682 (1986).
    [CrossRef] [PubMed]
  6. J. W. Campbell, W. E. Esaias, “Basis for spectral curvature algorithms in remote sensing of chlorophyll,” Appl. Opt. 22, 1084–1093 (1983).
    [CrossRef] [PubMed]
  7. H. R. Gordon, D. K. Clark, J. W. Brown, R. H. Evans, W. W. Broenkow, “Phytoplankton pigment concentrations in the Middle Atlantic Bight: comparison of ship determinations and CZCS estimates,” Appl. Opt. 22, 20–36 (1983).
    [CrossRef] [PubMed]
  8. R. Willstatter, A. Stoll, Untersuchungen über die Assimilation der Kohlensäure (Springer-Verlag, Berlin, 1918).
  9. R. Kumar, L. Silva, “Light ray tracing through a leaf cross section,” Appl. Opt. 12, 2950–2954 (1973).
    [CrossRef] [PubMed]
  10. C. J. Tucker, M. W. Garratt, “Leaf optical system modeled as a stochastic process,” Appl. Opt. 16, 635–642 (1977).
    [CrossRef] [PubMed]
  11. D. M. Gates, H. J. Keegan, J. C. Schleter, V. R. Weidner, “Spectral properties of plants,” Appl. Opt. 4, 11–20 (1965).
    [CrossRef]
  12. W. A. Allen, H. W. Gausman, A. J. Richardson, “Willstatter-Stoll theory of leaf reflectance evaluated by ray tracing,” Appl. Opt. 12, 2448–2453 (1973).
    [CrossRef] [PubMed]
  13. J. W. Ryde, B. S. Cooper, “The scattering of light by turbid media,” Proc. R. Soc. London Ser. A 131, 451–475 (1931).
    [CrossRef]
  14. S. Q. Duntley, “The optical properties of diffusing materials,” J. Opt. Soc. Am. 32, 61–70 (1942).
    [CrossRef]
  15. E. O. Hulburt, “Propagation of radiation in a scattering and absorbing medium,” J. Opt. Soc. Am. 33, 42–45 (1943).
    [CrossRef]
  16. A. Schuster, “Radiation through a foggy atmosphere,” Astrophys. J. 21, 1–22 (1905).
    [CrossRef]
  17. P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).
  18. L. Fukshansky, N. Kazarinova, “Extension of the Kubelka–Munk theory of light propagation in intensely scattering materials to fluorescent media,” J. Opt. Soc. Am. 70, 1101–1111 (1980).
    [CrossRef]
  19. P. Kubelka, “New contributions to the optics of intensely light scattering materials,” J. Opt. Soc. Am. 38, 448–457 (1948).
    [CrossRef] [PubMed]
  20. P. S. Mudgett, L. W. Richards, “Multiple scattering calculations for technology,” Appl. Opt. 10, 1485–1502 (1971).
    [CrossRef] [PubMed]
  21. N. Fukshansky-Kazarinova, W. Lork, E. Schafer, L. Fukshansky, “Photon flux gradiants in layered turbid media: applications to biological tissues,” Appl. Opt. 25, 780–788 (1986).
    [CrossRef] [PubMed]
  22. A. L. Lathrop, “Diffuse scattered radiation theories of Duntley and of Kubelka–Munk,” J. Opt. Soc. Am. 55, 1097–1104 (1965).
    [CrossRef]
  23. M. Seyfried, L. Fukshansky, “Light gradients in plant tissue,” Appl. Opt. 22, 1402–1408 (1983).
    [CrossRef] [PubMed]
  24. W. A. Allen, A. J. Richardson, “Interaction of light with a plant canopy,” J. Opt. Soc. Am. 58, 1023–1028 (1968).
    [CrossRef]
  25. V. I. Myers, W. A. Allen, “Electrooptical remote sensing methods as nondestructive testing and measuring techniques in agriculture,” Appl. Opt. 7, 1819–1838 (1968).
    [CrossRef] [PubMed]
  26. N. Yamada, S. Fujimura, “A mathematical model of reflectance and transmittance of plant leaves as a function of chlorophyll pigment content,” in Proceedings of the International Geoscience and Remote Sensing Symposium, T. D. Guyenne, J. J. Hunt, eds. (European Space Agency, Noodwijk, The Netherlands, 1988), pp. 833–834.
  27. E. Charney, F. S. Brackett, “The spectral dependence of scattering from a spherical alga and its implications for the state of organization of the light-accepting pigments,” Arch. Biochem. Biophys. 92, 1–12 (1961).
    [CrossRef] [PubMed]
  28. M. Seyfried, L. Fukshansky, E. Schafer, “Correcting remission and transmission spectra of plant tissue measured in glass cuvettes: a technique,” Appl. Opt. 22, 492–496 (1983).
    [CrossRef] [PubMed]
  29. C. L. Comar, F. P. Zscheile, “Analysis of plant extracts for chlorophylls a and b by a photoelectric spectrophotometric method,” Plant Physiol. 17, 198–209 (1942).
    [CrossRef] [PubMed]
  30. L. P. Vernon, G. R. Seely, eds., The Chlorophylls (Academic, New York, 1966).
  31. T. W. Goodwin, ed., Chemistry and Biochemistry of Plant Pigments (Academic, New York, 1976).

1986 (2)

1983 (4)

1981 (1)

M. Aoki, K. Yabuki, T. Totsuka, “An evaluation of chlorophyll content of leaves based on the spectral reflectivity in several plants,” Res. Rep. Natl. Inst. Environ. Stud. Jpn. 66, 125–130 (1981).

1980 (1)

1977 (1)

1976 (1)

1973 (2)

1971 (1)

1968 (2)

1965 (2)

1961 (1)

E. Charney, F. S. Brackett, “The spectral dependence of scattering from a spherical alga and its implications for the state of organization of the light-accepting pigments,” Arch. Biochem. Biophys. 92, 1–12 (1961).
[CrossRef] [PubMed]

1948 (1)

1943 (1)

1942 (2)

S. Q. Duntley, “The optical properties of diffusing materials,” J. Opt. Soc. Am. 32, 61–70 (1942).
[CrossRef]

C. L. Comar, F. P. Zscheile, “Analysis of plant extracts for chlorophylls a and b by a photoelectric spectrophotometric method,” Plant Physiol. 17, 198–209 (1942).
[CrossRef] [PubMed]

1931 (2)

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).

J. W. Ryde, B. S. Cooper, “The scattering of light by turbid media,” Proc. R. Soc. London Ser. A 131, 451–475 (1931).
[CrossRef]

1905 (1)

A. Schuster, “Radiation through a foggy atmosphere,” Astrophys. J. 21, 1–22 (1905).
[CrossRef]

Allen, W. A.

Aoki, M.

M. Aoki, K. Yabuki, T. Totsuka, “An evaluation of chlorophyll content of leaves based on the spectral reflectivity in several plants,” Res. Rep. Natl. Inst. Environ. Stud. Jpn. 66, 125–130 (1981).

Brackett, F. S.

E. Charney, F. S. Brackett, “The spectral dependence of scattering from a spherical alga and its implications for the state of organization of the light-accepting pigments,” Arch. Biochem. Biophys. 92, 1–12 (1961).
[CrossRef] [PubMed]

Broenkow, W. W.

Brown, J. W.

Campbell, J. W.

Charney, E.

E. Charney, F. S. Brackett, “The spectral dependence of scattering from a spherical alga and its implications for the state of organization of the light-accepting pigments,” Arch. Biochem. Biophys. 92, 1–12 (1961).
[CrossRef] [PubMed]

Clark, D. K.

Comar, C. L.

C. L. Comar, F. P. Zscheile, “Analysis of plant extracts for chlorophylls a and b by a photoelectric spectrophotometric method,” Plant Physiol. 17, 198–209 (1942).
[CrossRef] [PubMed]

Cooper, B. S.

J. W. Ryde, B. S. Cooper, “The scattering of light by turbid media,” Proc. R. Soc. London Ser. A 131, 451–475 (1931).
[CrossRef]

Duntley, S. Q.

Esaias, W. E.

Evans, R. H.

Fujimura, S.

N. Yamada, S. Fujimura, “A mathematical model of reflectance and transmittance of plant leaves as a function of chlorophyll pigment content,” in Proceedings of the International Geoscience and Remote Sensing Symposium, T. D. Guyenne, J. J. Hunt, eds. (European Space Agency, Noodwijk, The Netherlands, 1988), pp. 833–834.

Fukshansky, L.

Fukshansky-Kazarinova, N.

Garratt, M. W.

Gates, D. M.

Gausman, H. W.

Gordon, H. R.

Hoge, F. E.

Hulburt, E. O.

Kazarinova, N.

Keegan, H. J.

Kubelka, P.

P. Kubelka, “New contributions to the optics of intensely light scattering materials,” J. Opt. Soc. Am. 38, 448–457 (1948).
[CrossRef] [PubMed]

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).

Kumar, R.

Lathrop, A. L.

Lork, W.

Miller, L. D.

R. L. Pearson, L. D. Miller, C. J. Tucker, “Hand-held spectral radiometer to estimate gramineous biomass,” Appl. Opt. 15, 416–418 (1976).
[CrossRef] [PubMed]

C. J. Tucker, L. D. Miller, R. L. Pearson, in Proceedings Second Annual Remote Sensing of Earth Resources Conference, F. Shahroki, ed. (U. Tennessee Press, Tullahoma, Tenn., 1973) p. 601.

Mudgett, P. S.

Munk, F.

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).

Myers, V. I.

Pearson, R. L.

R. L. Pearson, L. D. Miller, C. J. Tucker, “Hand-held spectral radiometer to estimate gramineous biomass,” Appl. Opt. 15, 416–418 (1976).
[CrossRef] [PubMed]

C. J. Tucker, L. D. Miller, R. L. Pearson, in Proceedings Second Annual Remote Sensing of Earth Resources Conference, F. Shahroki, ed. (U. Tennessee Press, Tullahoma, Tenn., 1973) p. 601.

Richards, L. W.

Richardson, A. J.

Ryde, J. W.

J. W. Ryde, B. S. Cooper, “The scattering of light by turbid media,” Proc. R. Soc. London Ser. A 131, 451–475 (1931).
[CrossRef]

Schafer, E.

Schleter, J. C.

Schuster, A.

A. Schuster, “Radiation through a foggy atmosphere,” Astrophys. J. 21, 1–22 (1905).
[CrossRef]

Seyfried, M.

Silva, L.

Stoll, A.

R. Willstatter, A. Stoll, Untersuchungen über die Assimilation der Kohlensäure (Springer-Verlag, Berlin, 1918).

Swift, R. N.

Totsuka, T.

M. Aoki, K. Yabuki, T. Totsuka, “An evaluation of chlorophyll content of leaves based on the spectral reflectivity in several plants,” Res. Rep. Natl. Inst. Environ. Stud. Jpn. 66, 125–130 (1981).

Tucker, C. J.

Weidner, V. R.

Willstatter, R.

R. Willstatter, A. Stoll, Untersuchungen über die Assimilation der Kohlensäure (Springer-Verlag, Berlin, 1918).

Yabuki, K.

M. Aoki, K. Yabuki, T. Totsuka, “An evaluation of chlorophyll content of leaves based on the spectral reflectivity in several plants,” Res. Rep. Natl. Inst. Environ. Stud. Jpn. 66, 125–130 (1981).

Yamada, N.

N. Yamada, S. Fujimura, “A mathematical model of reflectance and transmittance of plant leaves as a function of chlorophyll pigment content,” in Proceedings of the International Geoscience and Remote Sensing Symposium, T. D. Guyenne, J. J. Hunt, eds. (European Space Agency, Noodwijk, The Netherlands, 1988), pp. 833–834.

Zscheile, F. P.

C. L. Comar, F. P. Zscheile, “Analysis of plant extracts for chlorophylls a and b by a photoelectric spectrophotometric method,” Plant Physiol. 17, 198–209 (1942).
[CrossRef] [PubMed]

Appl. Opt. (13)

R. L. Pearson, L. D. Miller, C. J. Tucker, “Hand-held spectral radiometer to estimate gramineous biomass,” Appl. Opt. 15, 416–418 (1976).
[CrossRef] [PubMed]

F. E. Hoge, R. N. Swift, “Chlorophyll pigment concentration using spectral curvature algorithms: an evaluation of present and proposed satellite ocean color sensor bands,” Appl. Opt. 25, 3677–3682 (1986).
[CrossRef] [PubMed]

J. W. Campbell, W. E. Esaias, “Basis for spectral curvature algorithms in remote sensing of chlorophyll,” Appl. Opt. 22, 1084–1093 (1983).
[CrossRef] [PubMed]

H. R. Gordon, D. K. Clark, J. W. Brown, R. H. Evans, W. W. Broenkow, “Phytoplankton pigment concentrations in the Middle Atlantic Bight: comparison of ship determinations and CZCS estimates,” Appl. Opt. 22, 20–36 (1983).
[CrossRef] [PubMed]

R. Kumar, L. Silva, “Light ray tracing through a leaf cross section,” Appl. Opt. 12, 2950–2954 (1973).
[CrossRef] [PubMed]

C. J. Tucker, M. W. Garratt, “Leaf optical system modeled as a stochastic process,” Appl. Opt. 16, 635–642 (1977).
[CrossRef] [PubMed]

D. M. Gates, H. J. Keegan, J. C. Schleter, V. R. Weidner, “Spectral properties of plants,” Appl. Opt. 4, 11–20 (1965).
[CrossRef]

W. A. Allen, H. W. Gausman, A. J. Richardson, “Willstatter-Stoll theory of leaf reflectance evaluated by ray tracing,” Appl. Opt. 12, 2448–2453 (1973).
[CrossRef] [PubMed]

P. S. Mudgett, L. W. Richards, “Multiple scattering calculations for technology,” Appl. Opt. 10, 1485–1502 (1971).
[CrossRef] [PubMed]

N. Fukshansky-Kazarinova, W. Lork, E. Schafer, L. Fukshansky, “Photon flux gradiants in layered turbid media: applications to biological tissues,” Appl. Opt. 25, 780–788 (1986).
[CrossRef] [PubMed]

M. Seyfried, L. Fukshansky, “Light gradients in plant tissue,” Appl. Opt. 22, 1402–1408 (1983).
[CrossRef] [PubMed]

M. Seyfried, L. Fukshansky, E. Schafer, “Correcting remission and transmission spectra of plant tissue measured in glass cuvettes: a technique,” Appl. Opt. 22, 492–496 (1983).
[CrossRef] [PubMed]

V. I. Myers, W. A. Allen, “Electrooptical remote sensing methods as nondestructive testing and measuring techniques in agriculture,” Appl. Opt. 7, 1819–1838 (1968).
[CrossRef] [PubMed]

Arch. Biochem. Biophys. (1)

E. Charney, F. S. Brackett, “The spectral dependence of scattering from a spherical alga and its implications for the state of organization of the light-accepting pigments,” Arch. Biochem. Biophys. 92, 1–12 (1961).
[CrossRef] [PubMed]

Astrophys. J. (1)

A. Schuster, “Radiation through a foggy atmosphere,” Astrophys. J. 21, 1–22 (1905).
[CrossRef]

J. Opt. Soc. Am. (6)

Plant Physiol. (1)

C. L. Comar, F. P. Zscheile, “Analysis of plant extracts for chlorophylls a and b by a photoelectric spectrophotometric method,” Plant Physiol. 17, 198–209 (1942).
[CrossRef] [PubMed]

Proc. R. Soc. London Ser. A (1)

J. W. Ryde, B. S. Cooper, “The scattering of light by turbid media,” Proc. R. Soc. London Ser. A 131, 451–475 (1931).
[CrossRef]

Res. Rep. Natl. Inst. Environ. Stud. Jpn. (1)

M. Aoki, K. Yabuki, T. Totsuka, “An evaluation of chlorophyll content of leaves based on the spectral reflectivity in several plants,” Res. Rep. Natl. Inst. Environ. Stud. Jpn. 66, 125–130 (1981).

Z. Tech. Phys. (1)

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).

Other (6)

Minolta Camera Company, Ltd., Model SPAD-501 (1984).

C. J. Tucker, L. D. Miller, R. L. Pearson, in Proceedings Second Annual Remote Sensing of Earth Resources Conference, F. Shahroki, ed. (U. Tennessee Press, Tullahoma, Tenn., 1973) p. 601.

R. Willstatter, A. Stoll, Untersuchungen über die Assimilation der Kohlensäure (Springer-Verlag, Berlin, 1918).

L. P. Vernon, G. R. Seely, eds., The Chlorophylls (Academic, New York, 1966).

T. W. Goodwin, ed., Chemistry and Biochemistry of Plant Pigments (Academic, New York, 1976).

N. Yamada, S. Fujimura, “A mathematical model of reflectance and transmittance of plant leaves as a function of chlorophyll pigment content,” in Proceedings of the International Geoscience and Remote Sensing Symposium, T. D. Guyenne, J. J. Hunt, eds. (European Space Agency, Noodwijk, The Netherlands, 1988), pp. 833–834.

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Figures (10)

Fig. 1
Fig. 1

Actual structure of a dicotyledonous leaf.

Fig. 2
Fig. 2

Simplified sketch of the radiant energy reflected and absorbed in a macrohomogenous layer.

Fig. 3
Fig. 3

Radiant-energy flow in a four-layered leaf.

Fig. 4
Fig. 4

Model for the structure of a dicotyledonous leaf.

Fig. 5
Fig. 5

Model shown in Fig. 4 simplified by the inclusion of the assumptions.

Fig. 6
Fig. 6

Reflectance and transmittance of hydrangea leaves at wavelengths of (a) 880 nm and (b) 700 nm. The symbols show the experimental data; the curves, the model predictions. Reflectance from above is denoted by ■ and the lower solid curve, reflectance from below by □ and the dashed curve [in (b) only], and transmittance by ● and the upper solid curve.

Fig. 7
Fig. 7

Spectral characteristics of (a) α, (b) V01 (solid curve) and V02 (dashed curve), and (c) U01 (solid curve) and U02 (dashed curve) determined by a least-squares fit for hydrangea leaves.

Fig. 8
Fig. 8

Results of the nondestructive measurement by the method proposed in this paper. ■, ●, □, and ○ denote results for hydrangea, chinquapin, rice, and dragon tree, respectively.

Fig. 9
Fig. 9

Simulated results for the method proposed in this paper when the true r0 varies from 0.03 to 0.07.

Fig. 10
Fig. 10

Reflectance from above ra as a function of chlorophyll content M at wavelengths of 740, 720, 700, and 680 nm. Curves are model predictions obtained in Section IV.

Tables (1)

Tables Icon

Table I Values of β and ra (360 nm) Determined for Each Type of Plant

Equations (40)

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d I ( x ) d x = υ I ( x ) u I ( x ) + u J ( x ) , d J ( x ) d x = υ J ( x ) u J ( x ) + u I ( x ) .
u = m σ , υ = m α ;
r = U / ( A + B coth B ) , t = B / ( A sinh B + B cosh B ) , if V 0 ,
r = U / ( 1 + U ) , t = 1 / ( 1 + U ) if V = 0 ,
U = u D , V = υ D , A = U + V , B = A 2 U 2 ,
r = 0 , t = exp V if U = 0 .
U = M σ + U 0 , V = M α + V 0 ,
U = η M σ + U 0 , V = η M α + V 0 .
exp B = ( U r A + r B ) / U t , exp B = ( U r A r B ) / U t .
A U = ( 1 t 2 + r 2 ) 2 r = def Ψ A ( r , t ) ,
V U = A U 1 = ( 1 r ) 2 t 2 2 r = def Ψ V ( r , t ) ,
B U = ( A 2 U 2 1 ) 1 / 2 = { [ ( 1 r ) 2 t 2 ] [ ( 1 + r ) 2 t 2 ] } 1 / 2 2 r = def Ψ B ( r , t ) .
B = ln 1 r Ψ A ( r , t ) + r Ψ B ( r , t ) t .
U = 1 Ψ B ( r , t ) ln 1 r Ψ A ( r , t ) + r Ψ B ( r , t ) t , V = Ψ V ( r , t ) Ψ B ( r , t ) ln 1 r Ψ A ( r , t ) + r Ψ B ( r , t ) t .
U = r / t , V = 0 if r + t = 1 ,
U = 0 , V = ln t if r = 0 .
J k = r k I k + t k J k + 1 , I k + 1 = t k I k + r k J k + 1 .
G k = def 1 t k [ t k 2 r k 2 r k r k 1 ] ,
[ I k + 1 J k + 1 ] = G k [ I k J k ] ,
[ I 4 J 4 ] = G [ I 0 J 0 ] .
G = def G 3 G 2 G 1 G 0 .
r a = g 21 / g 22 , r b = g 12 / g 22 , t = t a = t b = 1 / g 22 .
g 11 g 22 g 12 g 21 = 1
J 0 = r a I 0 + t b J 4 , I 4 = t a I 0 + r b J 4 ,
G = 1 t b ( t a t b r a r b r b r a 1 ) .
G I = G 3 1 G G 0 1 ,
G 3 = G 0 = 1 1 r 0 [ 1 2 r 0 r 0 r 0 1 ] ,
r I a = g I 21 / g I 22 , r I b = g I 12 / g I 22 , t I a = ( g I 11 g I 22 g I 12 g I 21 ) / g I 22 , t I b = 1 / g I 22 .
U 1 = η 1 P M σ + U 01 , U 2 = η 2 ( 1 P ) M σ + U 02 , V 1 = η 1 P M α + V 01 , V 2 = η 2 ( 1 P ) M α + V 02 ,
r a ( λ 0 ) + r b ( λ 0 ) 2 r a ( λ 0 ) r a ( λ l ) , r a ( λ 0 ) + r b ( λ 0 ) 2 r b ( λ 0 ) r b ( λ l ) ,
U 2 ( λ 0 ) = U 2 ( λ 0 ) = r I ( λ 0 ) / t I ( λ 0 ) = g I 12 ( λ 0 ) .
t 1 ( λ k ) = [ g I 21 ( λ k ) / g I 12 ( λ k ) ] 1 / 2 , r 2 ( λ k ) = g I 12 ( λ k ) / g I 22 ( λ k ) , t 2 ( λ k ) = [ g I 12 ( λ k ) / g I 21 ( λ k ) ] 1 / 2 / g I 22 ( λ k ) .
G I = G 2 G 1 = 1 t 1 t 2 [ t 1 2 ( t 2 2 r 2 2 ) r 2 t 1 2 r 2 1 ]
V 1 ( λ k ) = ln t 1 ( λ k ) η 1 ( λ k ) = 2 ln t 1 ( λ k ) ,
V 2 ( λ k ) = V 2 ( λ k ) U 2 ( λ 0 ) U 2 ( λ k ) = Ψ V [ r 2 ( λ k ) , t 2 ( λ k ) ] U 2 ( λ 0 ) = [ 1 r 2 ( λ k ) ] 2 t 2 ( λ k ) 2 2 r 2 ( λ k ) U 2 ( λ 0 ) .
Δ V k = def V k ( λ 2 ) V k ( λ 1 ) .
Δ V 1 = V 1 ( λ 2 ) V 1 ( λ 1 ) = [ α ( λ 2 ) α ( λ 1 ) ] P M + [ V 01 ( λ 2 ) V 01 ( λ 1 ) ] [ α ( λ 2 ) α ( λ 1 ) ] P M ,
Δ V 2 = V 2 ( λ 2 ) V 2 ( λ 1 ) = [ α ( λ 2 ) α ( λ 1 ) ] ( 1 P ) M + [ V 02 ( λ 2 ) V 02 ( λ 1 ) ] [ α ( λ 2 ) α ( λ 1 ) ] ( 1 P ) M .
M = β ( Δ V 1 + Δ V 2 ) .
β = def 1 / [ α ( λ 2 ) α ( λ 1 ) ] .

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