Abstract

A theoretical modeling is proposed to predict the efficiency factors for attenuation, total scattering, and backscattering for spherical and homogeneous phytoplanktonic cells in suspension. The input parameters of this modeling are the actual size distribution, the spectral values of absorption by the living cells, and an adjustable value of the real part of the refractive index. The variations in these parameters lead to very diverse spectral behavior of the efficiency factors. Theoretical predictions are compared to experimental results for some species to evaluate the reliability of the model for algal cells of various indices and morphologies.

© 1986 Optical Society of America

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References

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  1. P. Latimer, “Influence of Selective Light Scattering on Measurements of Absorption Spectra of Chlorella,” Plant Physiol. 34, 193 (1959).
    [CrossRef] [PubMed]
  2. K. G. Privoznik, K. J. Daniel, F. P. Incropera, “Absorption, Extinction and Phase Function Measurements for Algal Suspensions of Chlorella pyrenoidosa,” J. Quant. Spectrosc. Radiat. Transfer 20, 345 (1978).
    [CrossRef]
  3. D. A. Kiefer, R. J. Olson, W. H. Wilson, “Reflectance Spectroscopy of Marine Phytoplankton. Part 1. Optical Properties as Related to Age and Growth Rate,” Limnol. Oceanogr. 24, 664 (1979).
    [CrossRef]
  4. A. Morel, A. Bricaud, “Theoretical Results Concerning Light Absorption in a Discrete Medium, and Application to Specific Absorption of Phytoplankton,” Deep-Sea Res. 28A, 1375 (1981).
    [CrossRef]
  5. A. Bricaud, A. Morel, L. Prieur, “Optical Efficiency Factors of Some Phytoplankters,” Limnol. Oceanogr. 28, 5, 816 (1983).
    [CrossRef]
  6. A. Morel, R. C. Smith, “Terminology and Units in Optical Oceanography,” Mar. Geod. 5, 335 (1982).
    [CrossRef]
  7. R. W. Preisendorfer, “Application of Radiative Transfer Theory to Light Measurements in the Sea,” Int. Geophys. Geod. Monogr. 10, 11 (1961).
  8. G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidalen Metall-lösungen,” Ann. Phys. 25, 377 (1908).
    [CrossRef]
  9. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  10. F. D. Bryant, B. A. Seiber, P. Latimer, “Absolute Optical Cross Sections of Cells and Chloroplasts,” Arch. Biochem. Biophys. 135, 97 (1969).
    [CrossRef] [PubMed]
  11. J. L. Mueller, “The Influence of Phytoplankton on Ocean Color Spectra,” Ph.D. Thesis, Oregon State U. (1973).
  12. A. Morel, A. Bricaud, “Theoretical Results Concerning the Optics of Phytoplankton, with Special Reference to Remote Sensing Applications,” in Oceanography from Space, J. F. R. Gower, Ed. (Plenum, New York, 1981), p. 313.
    [CrossRef]
  13. A. S. Davydov, Izv. Akad. Nauk. SSSR Ser. Fiz. 5, 523 (1953), in Russian.
  14. R. E. Norris, “Prasinophytes,” in Phytoflagellates, E. Cox, Ed. (Elsevier North-Holland, Amsterdam, 1980), p. 85.
  15. H. H. Gran, “Pelagic Plant Life,” in Depths of the Ocean, X. Murray, X. Hjort, Eds. (Macmillan, London, 1912), p. 307.
  16. P. M. Holligan, M. Viollier, D. S. Harbour, P. Camus, M. Champagne-Philippe, “Satellite and Ship Studies of Coccolithophore Production Along a Continental Shelf Edge,” Nature London 304, 339 (1983).
    [CrossRef]
  17. A. Morel, L. Prieur, “Analysis of Variations in Ocean Color,” Limnol. Oceanogr. 22, 709 (1977).
    [CrossRef]
  18. R. C. Weast, Ed., Handbook of Chemistry and Physics (CRC Press, Cleveland, 1977).
  19. E. Aas, “The Refractive Index of Phytoplankton,” Inst. Rep. Ser., U. Oslo, 46 (1981), 61 pp.
  20. E. Aas, “Influence of Shape and Structure on Light Scattering by Marine Particles,” Inst. Rep. Ser., U. Oslo, 53 (1984), 112 pp.
  21. B. E. Pyle, A. Brunsting, P. Latimer, “Detection of the Vacuole of Yeast Cells in Suspension by Transmittance Radiometry,” Appl. Opt. 18, 3615 (1979).
    [CrossRef] [PubMed]
  22. W. Wiscombe, A. Mugnai, “Exact Calculations of Scattering from Moderately Non-Spherical Tn-Particles: Comparison with Equivalent Spheres,” in Light Scattering by Irregularly Shaped Particles, D. W. Schuerman, Ed. (Plenum, New York, 1980), p. 141.
    [CrossRef]
  23. P. Latimer, “Light Scattering by a Homogeneous Sphere with Radial Projections,” Appl. Opt. 23, 442 (1984).
    [CrossRef] [PubMed]
  24. R. A. Meyer, “Light Scattering from Biological Cells: Dependence of Backscatter Radiation on Membrane Thickness and Refractive Index,” Appl. Opt. 18, 585 (1979).
    [CrossRef] [PubMed]

1984

1983

P. M. Holligan, M. Viollier, D. S. Harbour, P. Camus, M. Champagne-Philippe, “Satellite and Ship Studies of Coccolithophore Production Along a Continental Shelf Edge,” Nature London 304, 339 (1983).
[CrossRef]

A. Bricaud, A. Morel, L. Prieur, “Optical Efficiency Factors of Some Phytoplankters,” Limnol. Oceanogr. 28, 5, 816 (1983).
[CrossRef]

1982

A. Morel, R. C. Smith, “Terminology and Units in Optical Oceanography,” Mar. Geod. 5, 335 (1982).
[CrossRef]

1981

A. Morel, A. Bricaud, “Theoretical Results Concerning Light Absorption in a Discrete Medium, and Application to Specific Absorption of Phytoplankton,” Deep-Sea Res. 28A, 1375 (1981).
[CrossRef]

1979

1978

K. G. Privoznik, K. J. Daniel, F. P. Incropera, “Absorption, Extinction and Phase Function Measurements for Algal Suspensions of Chlorella pyrenoidosa,” J. Quant. Spectrosc. Radiat. Transfer 20, 345 (1978).
[CrossRef]

1977

A. Morel, L. Prieur, “Analysis of Variations in Ocean Color,” Limnol. Oceanogr. 22, 709 (1977).
[CrossRef]

1969

F. D. Bryant, B. A. Seiber, P. Latimer, “Absolute Optical Cross Sections of Cells and Chloroplasts,” Arch. Biochem. Biophys. 135, 97 (1969).
[CrossRef] [PubMed]

1961

R. W. Preisendorfer, “Application of Radiative Transfer Theory to Light Measurements in the Sea,” Int. Geophys. Geod. Monogr. 10, 11 (1961).

1959

P. Latimer, “Influence of Selective Light Scattering on Measurements of Absorption Spectra of Chlorella,” Plant Physiol. 34, 193 (1959).
[CrossRef] [PubMed]

1953

A. S. Davydov, Izv. Akad. Nauk. SSSR Ser. Fiz. 5, 523 (1953), in Russian.

1908

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidalen Metall-lösungen,” Ann. Phys. 25, 377 (1908).
[CrossRef]

Aas, E.

E. Aas, “The Refractive Index of Phytoplankton,” Inst. Rep. Ser., U. Oslo, 46 (1981), 61 pp.

E. Aas, “Influence of Shape and Structure on Light Scattering by Marine Particles,” Inst. Rep. Ser., U. Oslo, 53 (1984), 112 pp.

Bricaud, A.

A. Bricaud, A. Morel, L. Prieur, “Optical Efficiency Factors of Some Phytoplankters,” Limnol. Oceanogr. 28, 5, 816 (1983).
[CrossRef]

A. Morel, A. Bricaud, “Theoretical Results Concerning Light Absorption in a Discrete Medium, and Application to Specific Absorption of Phytoplankton,” Deep-Sea Res. 28A, 1375 (1981).
[CrossRef]

A. Morel, A. Bricaud, “Theoretical Results Concerning the Optics of Phytoplankton, with Special Reference to Remote Sensing Applications,” in Oceanography from Space, J. F. R. Gower, Ed. (Plenum, New York, 1981), p. 313.
[CrossRef]

Brunsting, A.

Bryant, F. D.

F. D. Bryant, B. A. Seiber, P. Latimer, “Absolute Optical Cross Sections of Cells and Chloroplasts,” Arch. Biochem. Biophys. 135, 97 (1969).
[CrossRef] [PubMed]

Camus, P.

P. M. Holligan, M. Viollier, D. S. Harbour, P. Camus, M. Champagne-Philippe, “Satellite and Ship Studies of Coccolithophore Production Along a Continental Shelf Edge,” Nature London 304, 339 (1983).
[CrossRef]

Champagne-Philippe, M.

P. M. Holligan, M. Viollier, D. S. Harbour, P. Camus, M. Champagne-Philippe, “Satellite and Ship Studies of Coccolithophore Production Along a Continental Shelf Edge,” Nature London 304, 339 (1983).
[CrossRef]

Daniel, K. J.

K. G. Privoznik, K. J. Daniel, F. P. Incropera, “Absorption, Extinction and Phase Function Measurements for Algal Suspensions of Chlorella pyrenoidosa,” J. Quant. Spectrosc. Radiat. Transfer 20, 345 (1978).
[CrossRef]

Davydov, A. S.

A. S. Davydov, Izv. Akad. Nauk. SSSR Ser. Fiz. 5, 523 (1953), in Russian.

Gran, H. H.

H. H. Gran, “Pelagic Plant Life,” in Depths of the Ocean, X. Murray, X. Hjort, Eds. (Macmillan, London, 1912), p. 307.

Harbour, D. S.

P. M. Holligan, M. Viollier, D. S. Harbour, P. Camus, M. Champagne-Philippe, “Satellite and Ship Studies of Coccolithophore Production Along a Continental Shelf Edge,” Nature London 304, 339 (1983).
[CrossRef]

Holligan, P. M.

P. M. Holligan, M. Viollier, D. S. Harbour, P. Camus, M. Champagne-Philippe, “Satellite and Ship Studies of Coccolithophore Production Along a Continental Shelf Edge,” Nature London 304, 339 (1983).
[CrossRef]

Incropera, F. P.

K. G. Privoznik, K. J. Daniel, F. P. Incropera, “Absorption, Extinction and Phase Function Measurements for Algal Suspensions of Chlorella pyrenoidosa,” J. Quant. Spectrosc. Radiat. Transfer 20, 345 (1978).
[CrossRef]

Kiefer, D. A.

D. A. Kiefer, R. J. Olson, W. H. Wilson, “Reflectance Spectroscopy of Marine Phytoplankton. Part 1. Optical Properties as Related to Age and Growth Rate,” Limnol. Oceanogr. 24, 664 (1979).
[CrossRef]

Latimer, P.

P. Latimer, “Light Scattering by a Homogeneous Sphere with Radial Projections,” Appl. Opt. 23, 442 (1984).
[CrossRef] [PubMed]

B. E. Pyle, A. Brunsting, P. Latimer, “Detection of the Vacuole of Yeast Cells in Suspension by Transmittance Radiometry,” Appl. Opt. 18, 3615 (1979).
[CrossRef] [PubMed]

F. D. Bryant, B. A. Seiber, P. Latimer, “Absolute Optical Cross Sections of Cells and Chloroplasts,” Arch. Biochem. Biophys. 135, 97 (1969).
[CrossRef] [PubMed]

P. Latimer, “Influence of Selective Light Scattering on Measurements of Absorption Spectra of Chlorella,” Plant Physiol. 34, 193 (1959).
[CrossRef] [PubMed]

Meyer, R. A.

Mie, G.

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidalen Metall-lösungen,” Ann. Phys. 25, 377 (1908).
[CrossRef]

Morel, A.

A. Bricaud, A. Morel, L. Prieur, “Optical Efficiency Factors of Some Phytoplankters,” Limnol. Oceanogr. 28, 5, 816 (1983).
[CrossRef]

A. Morel, R. C. Smith, “Terminology and Units in Optical Oceanography,” Mar. Geod. 5, 335 (1982).
[CrossRef]

A. Morel, A. Bricaud, “Theoretical Results Concerning Light Absorption in a Discrete Medium, and Application to Specific Absorption of Phytoplankton,” Deep-Sea Res. 28A, 1375 (1981).
[CrossRef]

A. Morel, L. Prieur, “Analysis of Variations in Ocean Color,” Limnol. Oceanogr. 22, 709 (1977).
[CrossRef]

A. Morel, A. Bricaud, “Theoretical Results Concerning the Optics of Phytoplankton, with Special Reference to Remote Sensing Applications,” in Oceanography from Space, J. F. R. Gower, Ed. (Plenum, New York, 1981), p. 313.
[CrossRef]

Mueller, J. L.

J. L. Mueller, “The Influence of Phytoplankton on Ocean Color Spectra,” Ph.D. Thesis, Oregon State U. (1973).

Mugnai, A.

W. Wiscombe, A. Mugnai, “Exact Calculations of Scattering from Moderately Non-Spherical Tn-Particles: Comparison with Equivalent Spheres,” in Light Scattering by Irregularly Shaped Particles, D. W. Schuerman, Ed. (Plenum, New York, 1980), p. 141.
[CrossRef]

Norris, R. E.

R. E. Norris, “Prasinophytes,” in Phytoflagellates, E. Cox, Ed. (Elsevier North-Holland, Amsterdam, 1980), p. 85.

Olson, R. J.

D. A. Kiefer, R. J. Olson, W. H. Wilson, “Reflectance Spectroscopy of Marine Phytoplankton. Part 1. Optical Properties as Related to Age and Growth Rate,” Limnol. Oceanogr. 24, 664 (1979).
[CrossRef]

Preisendorfer, R. W.

R. W. Preisendorfer, “Application of Radiative Transfer Theory to Light Measurements in the Sea,” Int. Geophys. Geod. Monogr. 10, 11 (1961).

Prieur, L.

A. Bricaud, A. Morel, L. Prieur, “Optical Efficiency Factors of Some Phytoplankters,” Limnol. Oceanogr. 28, 5, 816 (1983).
[CrossRef]

A. Morel, L. Prieur, “Analysis of Variations in Ocean Color,” Limnol. Oceanogr. 22, 709 (1977).
[CrossRef]

Privoznik, K. G.

K. G. Privoznik, K. J. Daniel, F. P. Incropera, “Absorption, Extinction and Phase Function Measurements for Algal Suspensions of Chlorella pyrenoidosa,” J. Quant. Spectrosc. Radiat. Transfer 20, 345 (1978).
[CrossRef]

Pyle, B. E.

Seiber, B. A.

F. D. Bryant, B. A. Seiber, P. Latimer, “Absolute Optical Cross Sections of Cells and Chloroplasts,” Arch. Biochem. Biophys. 135, 97 (1969).
[CrossRef] [PubMed]

Smith, R. C.

A. Morel, R. C. Smith, “Terminology and Units in Optical Oceanography,” Mar. Geod. 5, 335 (1982).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Viollier, M.

P. M. Holligan, M. Viollier, D. S. Harbour, P. Camus, M. Champagne-Philippe, “Satellite and Ship Studies of Coccolithophore Production Along a Continental Shelf Edge,” Nature London 304, 339 (1983).
[CrossRef]

Wilson, W. H.

D. A. Kiefer, R. J. Olson, W. H. Wilson, “Reflectance Spectroscopy of Marine Phytoplankton. Part 1. Optical Properties as Related to Age and Growth Rate,” Limnol. Oceanogr. 24, 664 (1979).
[CrossRef]

Wiscombe, W.

W. Wiscombe, A. Mugnai, “Exact Calculations of Scattering from Moderately Non-Spherical Tn-Particles: Comparison with Equivalent Spheres,” in Light Scattering by Irregularly Shaped Particles, D. W. Schuerman, Ed. (Plenum, New York, 1980), p. 141.
[CrossRef]

Ann. Phys.

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidalen Metall-lösungen,” Ann. Phys. 25, 377 (1908).
[CrossRef]

Appl. Opt.

Arch. Biochem. Biophys.

F. D. Bryant, B. A. Seiber, P. Latimer, “Absolute Optical Cross Sections of Cells and Chloroplasts,” Arch. Biochem. Biophys. 135, 97 (1969).
[CrossRef] [PubMed]

Deep-Sea Res.

A. Morel, A. Bricaud, “Theoretical Results Concerning Light Absorption in a Discrete Medium, and Application to Specific Absorption of Phytoplankton,” Deep-Sea Res. 28A, 1375 (1981).
[CrossRef]

Int. Geophys. Geod. Monogr.

R. W. Preisendorfer, “Application of Radiative Transfer Theory to Light Measurements in the Sea,” Int. Geophys. Geod. Monogr. 10, 11 (1961).

Izv. Akad. Nauk. SSSR Ser. Fiz.

A. S. Davydov, Izv. Akad. Nauk. SSSR Ser. Fiz. 5, 523 (1953), in Russian.

J. Quant. Spectrosc. Radiat. Transfer

K. G. Privoznik, K. J. Daniel, F. P. Incropera, “Absorption, Extinction and Phase Function Measurements for Algal Suspensions of Chlorella pyrenoidosa,” J. Quant. Spectrosc. Radiat. Transfer 20, 345 (1978).
[CrossRef]

Limnol. Oceanogr.

D. A. Kiefer, R. J. Olson, W. H. Wilson, “Reflectance Spectroscopy of Marine Phytoplankton. Part 1. Optical Properties as Related to Age and Growth Rate,” Limnol. Oceanogr. 24, 664 (1979).
[CrossRef]

A. Bricaud, A. Morel, L. Prieur, “Optical Efficiency Factors of Some Phytoplankters,” Limnol. Oceanogr. 28, 5, 816 (1983).
[CrossRef]

A. Morel, L. Prieur, “Analysis of Variations in Ocean Color,” Limnol. Oceanogr. 22, 709 (1977).
[CrossRef]

Mar. Geod.

A. Morel, R. C. Smith, “Terminology and Units in Optical Oceanography,” Mar. Geod. 5, 335 (1982).
[CrossRef]

Nature London

P. M. Holligan, M. Viollier, D. S. Harbour, P. Camus, M. Champagne-Philippe, “Satellite and Ship Studies of Coccolithophore Production Along a Continental Shelf Edge,” Nature London 304, 339 (1983).
[CrossRef]

Plant Physiol.

P. Latimer, “Influence of Selective Light Scattering on Measurements of Absorption Spectra of Chlorella,” Plant Physiol. 34, 193 (1959).
[CrossRef] [PubMed]

Other

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

R. C. Weast, Ed., Handbook of Chemistry and Physics (CRC Press, Cleveland, 1977).

E. Aas, “The Refractive Index of Phytoplankton,” Inst. Rep. Ser., U. Oslo, 46 (1981), 61 pp.

E. Aas, “Influence of Shape and Structure on Light Scattering by Marine Particles,” Inst. Rep. Ser., U. Oslo, 53 (1984), 112 pp.

R. E. Norris, “Prasinophytes,” in Phytoflagellates, E. Cox, Ed. (Elsevier North-Holland, Amsterdam, 1980), p. 85.

H. H. Gran, “Pelagic Plant Life,” in Depths of the Ocean, X. Murray, X. Hjort, Eds. (Macmillan, London, 1912), p. 307.

J. L. Mueller, “The Influence of Phytoplankton on Ocean Color Spectra,” Ph.D. Thesis, Oregon State U. (1973).

A. Morel, A. Bricaud, “Theoretical Results Concerning the Optics of Phytoplankton, with Special Reference to Remote Sensing Applications,” in Oceanography from Space, J. F. R. Gower, Ed. (Plenum, New York, 1981), p. 313.
[CrossRef]

W. Wiscombe, A. Mugnai, “Exact Calculations of Scattering from Moderately Non-Spherical Tn-Particles: Comparison with Equivalent Spheres,” in Light Scattering by Irregularly Shaped Particles, D. W. Schuerman, Ed. (Plenum, New York, 1980), p. 141.
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Variations of the efficiency factor for attenuation Qc vs the parameter ρ = 2α(n − 1) for the same real part of the refractive index n (=1.05) and various imaginary parts n′.

Fig. 2
Fig. 2

Variations of the efficiency factor for backscattering Qbb as a function of the relative size α for two values of the real part of the refractive index (n = 1.020, n = 1.035) and increasing values of the imaginary part n′.

Fig. 3
Fig. 3

Variations of the mean efficiency factor for attenuation Q c ¯ of different polydispersed populations vs the parameter ρ ¯, the value of ρ corresponding to the maximum of the size distribution. Curves 1–4 correspond to distributions of increasing width. The dotted curve corresponds to a monodispersed population.

Fig. 4
Fig. 4

Variations of the mean efficiency factor for backscattering Q b b ¯ vs the modal relative size α ¯ (the value of α corresponding to the maximum of the size distribution) for a population obeying a lognormal law such as F ( α ¯ / 2 ) = F ( 2 α ¯ ) = 0.01 F ( α ¯ ). The different curves correspond to two values of the real part of the refractive index and, for each of these values, to five values of its imaginary part.

Fig. 5
Fig. 5

Theoretical spectral variations of the real part of the refractive index n (upper figure), estimated from those of the imaginary part n′ (lower figure), for the suspension of Platymonas suecica. The spectral values of n′, deduced from those of absorption [through Eqs. (9), (2′), and (10)], are plotted as a dashed line. This spectrum is decomposed into distinct oscillators, and the spectral values of n′ corresponding to each oscillator are plotted as dotted lines (lower figure), along with the associated variations of n (upper figure). The spectra recomposed from these oscillators for n and n′ are plotted as continuous lines.

Fig. 6
Fig. 6

Spectral variations of the efficiency factors for attenuation ( Q c ¯ ), scattering ( Q b ¯ ), and backscattering ( Q b b ¯ ) for a population with the same spectral values of the efficiency factor for absorption ( Q a ¯ ), and obeying the same size distribution law (identical to that used in Fig. 4) but with different values of central index, 1 + ɛ (indicated in the figure). The spectral variations of ( Q c ¯ ) for the nonabsorbing equivalent population ( Q c ¯ NAE ) are also given as dotted lines.

Fig. 7
Fig. 7

Size distribution laws F(d) of the three analyzed species as determined with a Coulter counter ZBI (P. suecica), TA2 (E. huxleyi), or with a video system (S. costatum) (left-hand side of figure). On the right are plotted the corresponding variations of the mean efficiency factor for attenuation for the nonabsorbing equivalent populations ( Q c ¯ ) NAE vs the parameter ρ ¯. As an example, for E. huxleyi, the portion of the Q c ¯ NAE curve which complies with the experimental Q c ¯ values (see text) corresponds to the range ρ ¯ = 1.765 3.530 for λ varying from 750 to 375 nm (hatched area). With d ¯ = 3.6 μ m, this provides the central value of the index n (1.044).

Fig. 8
Fig. 8

Modifications of the spectral values of Q c ¯, Q b ¯, and Q b b ¯ induced by a slight modification of the central value of the index 1 + ɛ around the previously estimated value (1.028) for the species S. costatum.

Fig. 9
Fig. 9

Theoretical spectral variations of Qa, Q c ¯, Q b ¯, Q b b ¯, and (dashed lines) and Q c ¯ NAE (dotted line) for each species. The experimental spectral values of Q a ¯, Q c ¯, Q b ¯ and are shown for comparison as continuous lines. (a) Platymonas suecica, (b) Skeletonema costatum, and (c) Emiliania huxleyi.

Fig. 10
Fig. 10

Volume scattering functions, normalized with respect to the total scattering coefficient, corresponding to the estimated values (at λ = 550 nm) of the refractive index and the size distribution for (a) the suspension of E. huxleyi and (b) the detached coccoliths. The values of the efficiency factor for scattering corresponding to these volume scattering functions are (a) 2.211 and (b) 1.618.

Tables (3)

Tables Icon

Table I Numerical Data Bank Generated Before Computing the Efficiency Factor for Backscattering Qbba

Tables Icon

Table III Information Concerning the Algal Cultures Studied.a

Equations (18)

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Q c ( ρ ) = 2 4 exp ( ρ tan ξ ) [ cos ξ ρ sin ( ρ ξ ) + ( cos ξ ρ ) 2 cos ( ρ 2 ξ ) ] + 4 ( cos ξ ρ ) 2 cos 2 ξ ,
Q a ( ρ ) = 1 + [ exp ( 2 ρ tan ξ ) ( 2 ρ tan ξ + 1 ) 1 ] / 2 ρ 2 tan 2 ξ ,
Q b ( ρ ) = Q c ( ρ ) Q a ( ρ ) ,
Q a ( ρ ) = 1 + 2 exp ( ρ ) ρ + 2 exp ( ρ ) 1 ρ 2 .
Q b b ( α , m ) = α 2 π 2 π i 1 ( θ , α , m ) + i 2 ( θ , α , m ) 2 sin θ d θ .
F ( d ) = 1 V d ( N ) d ( d ) ,
n = 1 + ɛ K ɛ υ 1 + υ 2 , .
n = K ɛ 1 1 + υ 2 ,
Q i ¯ = 0 Q i ( ρ ) F ( ρ ) ρ 2 d ρ 0 F ( ρ ) ρ 2 d ρ , with i = b , b b , c .
a = ( N V ) Q a s ,
a = π 4 0 Q a ( d ) d 2 F ( d ) d ( d )
a = π 4 Q a ¯ 0 F ( d ) d 2 d ( d ) .
a * = π 4 Q a ¯ C 0 F ( d ) d 2 d ( d ) = 3 2 Q a ¯ c i 0 F ( d ) d 2 d ( d ) 0 F ( d ) d 3 d ( d ) ,
c i = C [ π 6 0 F ( d ) d 3 d ( d ) ] 1 .
Q a ¯ ( ρ ¯ ) = 0 Q a ( ρ ) F ( ρ ) ρ 2 d ρ 0 F ( ρ ) ρ 2 d ρ .
n = 1 + ɛ ɛ j K j υ j 1 + υ j 2 ( with j = 1 9 ) .
a = 1 r ln ( 1 ϕ a ϕ ) , b = 1 r ln ( 1 ϕ b ϕ ) ,
d ¯ ( μ m )

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