Abstract

Based on anomalous diffraction approximation, analytical expressions for the scattering coefficient of marine particles with power-law size distribution in the infinite domain of sizes (0, ∞) were derived. Comparison with the exact Mie solution for the light scattering by spheres indicated that the obtained expressions can describe the relative spectral variability of the scattering coefficient well. This is demonstrated and discussed for the scattering spectra of main types of marine particulates characterized by different optical properties.

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References

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  1. J. T. O. Kirk, Light and Photosynthesis in Aquatic Ecosystems, (Cambridge University Press, 1994).
  2. M. Jonasz and G. R. Fournier, Light Scattering by Particles in Water, (Academic, Amsterdam, 2007).
  3. H. Bader, “The hyperbolic distribution of particles sizes,” J. Geophys. Res.75(15), 2822–2830 (1970).
    [CrossRef]
  4. C. J. Buonassissi and H. M. Dierssen, “A regional comparison of particle size distributions and the power law approximation in oceanic and estuarine surface waters,” J. Geophys. Res.115(C10), C10028 (2010).
    [CrossRef]
  5. R. A. Reynolds, D. Stramski, V. M. Wright, and S. B. Woźniak, “Measurements and characterization particle size distributions in coastal waters,” J. Geophys. Res.115(C8), C08024 (2010).
    [CrossRef]
  6. E. Boss, M. S. Twardowski, and S. Herring, “Shape of the particulate beam attenuation spectrum and its inversion to obtain the shape of the particulate size distribution,” Appl. Opt.40(27), 4885–4893 (2001).
    [CrossRef] [PubMed]
  7. G. Chang, A. Barnard, and J. R. V. Zaneveld, “Optical closure in a complex coastal environment: particle effects,” Appl. Opt.46(31), 7679–7692 (2007).
    [CrossRef] [PubMed]
  8. D. Doxaran, K. Ruddick, D. McKee, B. Gentili, D. Tailliez, M. Chami, and M. Babin, “Spectral variations of light scattering by marine particles in coastal waters from the visible to the near infrared,” Limnol. Oceanogr.54(4), 1257–1271 (2009).
    [CrossRef]
  9. G. Fournier and J. L. Forand, “Analytic phase function for ocean water,” Ocean Optics XII, J.S. Jaffe, ed., Proc. SPIE 2258, 194–201 (1994).
  10. C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters, (Academic, San Diego, Calif.,1994).
  11. A. Morel, “The scattering of light by sea water. Experimental results and theoretical approach,” AGARD lecture series 61 on Optics of the sea, Interface and in-water transmission and imaging, (Advisory Group for Aerospace Research and Development NATO, London, 1973).
  12. M. Babin, A. Morel, V. Fournier-Sicre, F. Fell, and D. Stramski, “Light scattering properties of marine particles in coastal and open ocean waters as related to the particle mass concentration,” Limnol. Oceanogr.48(2), 843–859 (2003).
    [CrossRef]
  13. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, (WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim, 2004).
  14. H. C. van de Hulst, 1981, Light Scattering by Small Particles, (Dover Publications, New York, 1981).
  15. D. Stramski and D. A. Kiefer, “Light scattering by microorganisms in the open ocean,” Prog. Oceanogr.28(4), 343–383 (1991).
    [CrossRef]
  16. F. D. Bryant and P. Latimer, “Optical efficiencies of large particles of arbitrary shape and orientation,” J. Colloid Interface Sci.30(3), 291–304 (1969).
    [CrossRef]
  17. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, (National Bureau of Standards, Applied Mathematics Series 55, 1964).
  18. D. Sarason, Complex Function Theory, (American Mathematical Society, 2007).
  19. Y. H. Ahn, A. Bricaud, and A. Morel, “Light backscattering efficiency and related properties of some phytoplankters,” Deep-Sea Res.39(11-12), 1835–1855 (1992).
    [CrossRef]
  20. D. Stramski, A. Bricaud, and A. Morel, “Modeling the inherent optical properties of the ocean based on the detailed composition of the planktonic community,” Appl. Opt.40(18), 2929–2945 (2001).
    [CrossRef] [PubMed]
  21. S. B. Woźniak and D. Stramski, “Modelling the optical properties of mineral particles suspended in seawater and their influence on ocean reflectance and chlorophyll estimation for remote sensing algorithm,” Appl. Opt.43(17), 3489–3503 (2004).
    [CrossRef] [PubMed]
  22. W. Slade and E. Boss, “Translation from Fortran to Matlab of the Bohren and Huffman Mie Code,” http://misclab.umeoce.maine.edu/software.php .

2010

C. J. Buonassissi and H. M. Dierssen, “A regional comparison of particle size distributions and the power law approximation in oceanic and estuarine surface waters,” J. Geophys. Res.115(C10), C10028 (2010).
[CrossRef]

R. A. Reynolds, D. Stramski, V. M. Wright, and S. B. Woźniak, “Measurements and characterization particle size distributions in coastal waters,” J. Geophys. Res.115(C8), C08024 (2010).
[CrossRef]

2009

D. Doxaran, K. Ruddick, D. McKee, B. Gentili, D. Tailliez, M. Chami, and M. Babin, “Spectral variations of light scattering by marine particles in coastal waters from the visible to the near infrared,” Limnol. Oceanogr.54(4), 1257–1271 (2009).
[CrossRef]

2007

2004

2003

M. Babin, A. Morel, V. Fournier-Sicre, F. Fell, and D. Stramski, “Light scattering properties of marine particles in coastal and open ocean waters as related to the particle mass concentration,” Limnol. Oceanogr.48(2), 843–859 (2003).
[CrossRef]

2001

1992

Y. H. Ahn, A. Bricaud, and A. Morel, “Light backscattering efficiency and related properties of some phytoplankters,” Deep-Sea Res.39(11-12), 1835–1855 (1992).
[CrossRef]

1991

D. Stramski and D. A. Kiefer, “Light scattering by microorganisms in the open ocean,” Prog. Oceanogr.28(4), 343–383 (1991).
[CrossRef]

1970

H. Bader, “The hyperbolic distribution of particles sizes,” J. Geophys. Res.75(15), 2822–2830 (1970).
[CrossRef]

1969

F. D. Bryant and P. Latimer, “Optical efficiencies of large particles of arbitrary shape and orientation,” J. Colloid Interface Sci.30(3), 291–304 (1969).
[CrossRef]

Ahn, Y. H.

Y. H. Ahn, A. Bricaud, and A. Morel, “Light backscattering efficiency and related properties of some phytoplankters,” Deep-Sea Res.39(11-12), 1835–1855 (1992).
[CrossRef]

Babin, M.

D. Doxaran, K. Ruddick, D. McKee, B. Gentili, D. Tailliez, M. Chami, and M. Babin, “Spectral variations of light scattering by marine particles in coastal waters from the visible to the near infrared,” Limnol. Oceanogr.54(4), 1257–1271 (2009).
[CrossRef]

M. Babin, A. Morel, V. Fournier-Sicre, F. Fell, and D. Stramski, “Light scattering properties of marine particles in coastal and open ocean waters as related to the particle mass concentration,” Limnol. Oceanogr.48(2), 843–859 (2003).
[CrossRef]

Bader, H.

H. Bader, “The hyperbolic distribution of particles sizes,” J. Geophys. Res.75(15), 2822–2830 (1970).
[CrossRef]

Barnard, A.

Boss, E.

Bricaud, A.

D. Stramski, A. Bricaud, and A. Morel, “Modeling the inherent optical properties of the ocean based on the detailed composition of the planktonic community,” Appl. Opt.40(18), 2929–2945 (2001).
[CrossRef] [PubMed]

Y. H. Ahn, A. Bricaud, and A. Morel, “Light backscattering efficiency and related properties of some phytoplankters,” Deep-Sea Res.39(11-12), 1835–1855 (1992).
[CrossRef]

Bryant, F. D.

F. D. Bryant and P. Latimer, “Optical efficiencies of large particles of arbitrary shape and orientation,” J. Colloid Interface Sci.30(3), 291–304 (1969).
[CrossRef]

Buonassissi, C. J.

C. J. Buonassissi and H. M. Dierssen, “A regional comparison of particle size distributions and the power law approximation in oceanic and estuarine surface waters,” J. Geophys. Res.115(C10), C10028 (2010).
[CrossRef]

Chami, M.

D. Doxaran, K. Ruddick, D. McKee, B. Gentili, D. Tailliez, M. Chami, and M. Babin, “Spectral variations of light scattering by marine particles in coastal waters from the visible to the near infrared,” Limnol. Oceanogr.54(4), 1257–1271 (2009).
[CrossRef]

Chang, G.

Dierssen, H. M.

C. J. Buonassissi and H. M. Dierssen, “A regional comparison of particle size distributions and the power law approximation in oceanic and estuarine surface waters,” J. Geophys. Res.115(C10), C10028 (2010).
[CrossRef]

Doxaran, D.

D. Doxaran, K. Ruddick, D. McKee, B. Gentili, D. Tailliez, M. Chami, and M. Babin, “Spectral variations of light scattering by marine particles in coastal waters from the visible to the near infrared,” Limnol. Oceanogr.54(4), 1257–1271 (2009).
[CrossRef]

Fell, F.

M. Babin, A. Morel, V. Fournier-Sicre, F. Fell, and D. Stramski, “Light scattering properties of marine particles in coastal and open ocean waters as related to the particle mass concentration,” Limnol. Oceanogr.48(2), 843–859 (2003).
[CrossRef]

Fournier-Sicre, V.

M. Babin, A. Morel, V. Fournier-Sicre, F. Fell, and D. Stramski, “Light scattering properties of marine particles in coastal and open ocean waters as related to the particle mass concentration,” Limnol. Oceanogr.48(2), 843–859 (2003).
[CrossRef]

Gentili, B.

D. Doxaran, K. Ruddick, D. McKee, B. Gentili, D. Tailliez, M. Chami, and M. Babin, “Spectral variations of light scattering by marine particles in coastal waters from the visible to the near infrared,” Limnol. Oceanogr.54(4), 1257–1271 (2009).
[CrossRef]

Herring, S.

Kiefer, D. A.

D. Stramski and D. A. Kiefer, “Light scattering by microorganisms in the open ocean,” Prog. Oceanogr.28(4), 343–383 (1991).
[CrossRef]

Latimer, P.

F. D. Bryant and P. Latimer, “Optical efficiencies of large particles of arbitrary shape and orientation,” J. Colloid Interface Sci.30(3), 291–304 (1969).
[CrossRef]

McKee, D.

D. Doxaran, K. Ruddick, D. McKee, B. Gentili, D. Tailliez, M. Chami, and M. Babin, “Spectral variations of light scattering by marine particles in coastal waters from the visible to the near infrared,” Limnol. Oceanogr.54(4), 1257–1271 (2009).
[CrossRef]

Morel, A.

M. Babin, A. Morel, V. Fournier-Sicre, F. Fell, and D. Stramski, “Light scattering properties of marine particles in coastal and open ocean waters as related to the particle mass concentration,” Limnol. Oceanogr.48(2), 843–859 (2003).
[CrossRef]

D. Stramski, A. Bricaud, and A. Morel, “Modeling the inherent optical properties of the ocean based on the detailed composition of the planktonic community,” Appl. Opt.40(18), 2929–2945 (2001).
[CrossRef] [PubMed]

Y. H. Ahn, A. Bricaud, and A. Morel, “Light backscattering efficiency and related properties of some phytoplankters,” Deep-Sea Res.39(11-12), 1835–1855 (1992).
[CrossRef]

Reynolds, R. A.

R. A. Reynolds, D. Stramski, V. M. Wright, and S. B. Woźniak, “Measurements and characterization particle size distributions in coastal waters,” J. Geophys. Res.115(C8), C08024 (2010).
[CrossRef]

Ruddick, K.

D. Doxaran, K. Ruddick, D. McKee, B. Gentili, D. Tailliez, M. Chami, and M. Babin, “Spectral variations of light scattering by marine particles in coastal waters from the visible to the near infrared,” Limnol. Oceanogr.54(4), 1257–1271 (2009).
[CrossRef]

Stramski, D.

R. A. Reynolds, D. Stramski, V. M. Wright, and S. B. Woźniak, “Measurements and characterization particle size distributions in coastal waters,” J. Geophys. Res.115(C8), C08024 (2010).
[CrossRef]

S. B. Woźniak and D. Stramski, “Modelling the optical properties of mineral particles suspended in seawater and their influence on ocean reflectance and chlorophyll estimation for remote sensing algorithm,” Appl. Opt.43(17), 3489–3503 (2004).
[CrossRef] [PubMed]

M. Babin, A. Morel, V. Fournier-Sicre, F. Fell, and D. Stramski, “Light scattering properties of marine particles in coastal and open ocean waters as related to the particle mass concentration,” Limnol. Oceanogr.48(2), 843–859 (2003).
[CrossRef]

D. Stramski, A. Bricaud, and A. Morel, “Modeling the inherent optical properties of the ocean based on the detailed composition of the planktonic community,” Appl. Opt.40(18), 2929–2945 (2001).
[CrossRef] [PubMed]

D. Stramski and D. A. Kiefer, “Light scattering by microorganisms in the open ocean,” Prog. Oceanogr.28(4), 343–383 (1991).
[CrossRef]

Tailliez, D.

D. Doxaran, K. Ruddick, D. McKee, B. Gentili, D. Tailliez, M. Chami, and M. Babin, “Spectral variations of light scattering by marine particles in coastal waters from the visible to the near infrared,” Limnol. Oceanogr.54(4), 1257–1271 (2009).
[CrossRef]

Twardowski, M. S.

Wozniak, S. B.

R. A. Reynolds, D. Stramski, V. M. Wright, and S. B. Woźniak, “Measurements and characterization particle size distributions in coastal waters,” J. Geophys. Res.115(C8), C08024 (2010).
[CrossRef]

S. B. Woźniak and D. Stramski, “Modelling the optical properties of mineral particles suspended in seawater and their influence on ocean reflectance and chlorophyll estimation for remote sensing algorithm,” Appl. Opt.43(17), 3489–3503 (2004).
[CrossRef] [PubMed]

Wright, V. M.

R. A. Reynolds, D. Stramski, V. M. Wright, and S. B. Woźniak, “Measurements and characterization particle size distributions in coastal waters,” J. Geophys. Res.115(C8), C08024 (2010).
[CrossRef]

Zaneveld, J. R. V.

Appl. Opt.

Deep-Sea Res.

Y. H. Ahn, A. Bricaud, and A. Morel, “Light backscattering efficiency and related properties of some phytoplankters,” Deep-Sea Res.39(11-12), 1835–1855 (1992).
[CrossRef]

J. Colloid Interface Sci.

F. D. Bryant and P. Latimer, “Optical efficiencies of large particles of arbitrary shape and orientation,” J. Colloid Interface Sci.30(3), 291–304 (1969).
[CrossRef]

J. Geophys. Res.

H. Bader, “The hyperbolic distribution of particles sizes,” J. Geophys. Res.75(15), 2822–2830 (1970).
[CrossRef]

C. J. Buonassissi and H. M. Dierssen, “A regional comparison of particle size distributions and the power law approximation in oceanic and estuarine surface waters,” J. Geophys. Res.115(C10), C10028 (2010).
[CrossRef]

R. A. Reynolds, D. Stramski, V. M. Wright, and S. B. Woźniak, “Measurements and characterization particle size distributions in coastal waters,” J. Geophys. Res.115(C8), C08024 (2010).
[CrossRef]

Limnol. Oceanogr.

M. Babin, A. Morel, V. Fournier-Sicre, F. Fell, and D. Stramski, “Light scattering properties of marine particles in coastal and open ocean waters as related to the particle mass concentration,” Limnol. Oceanogr.48(2), 843–859 (2003).
[CrossRef]

D. Doxaran, K. Ruddick, D. McKee, B. Gentili, D. Tailliez, M. Chami, and M. Babin, “Spectral variations of light scattering by marine particles in coastal waters from the visible to the near infrared,” Limnol. Oceanogr.54(4), 1257–1271 (2009).
[CrossRef]

Prog. Oceanogr.

D. Stramski and D. A. Kiefer, “Light scattering by microorganisms in the open ocean,” Prog. Oceanogr.28(4), 343–383 (1991).
[CrossRef]

Other

J. T. O. Kirk, Light and Photosynthesis in Aquatic Ecosystems, (Cambridge University Press, 1994).

M. Jonasz and G. R. Fournier, Light Scattering by Particles in Water, (Academic, Amsterdam, 2007).

G. Fournier and J. L. Forand, “Analytic phase function for ocean water,” Ocean Optics XII, J.S. Jaffe, ed., Proc. SPIE 2258, 194–201 (1994).

C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters, (Academic, San Diego, Calif.,1994).

A. Morel, “The scattering of light by sea water. Experimental results and theoretical approach,” AGARD lecture series 61 on Optics of the sea, Interface and in-water transmission and imaging, (Advisory Group for Aerospace Research and Development NATO, London, 1973).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, (WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim, 2004).

H. C. van de Hulst, 1981, Light Scattering by Small Particles, (Dover Publications, New York, 1981).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, (National Bureau of Standards, Applied Mathematics Series 55, 1964).

D. Sarason, Complex Function Theory, (American Mathematical Society, 2007).

W. Slade and E. Boss, “Translation from Fortran to Matlab of the Bohren and Huffman Mie Code,” http://misclab.umeoce.maine.edu/software.php .

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Figures (2)

Fig. 1
Fig. 1

Refractive index of marine particulates (relative to water).

Fig. 2
Fig. 2

Visible spectra of the scattering coefficient of spherical particles with power-law PSD determined by means of ADA in the infinite diameter interval (0, ∞ µm) and Mie solution in the (0.02, 500 µm) diameters interval.

Tables (2)

Tables Icon

Table 1 Results of the analysis of the relationship between ADA1 and Mie2 visible particles' scattering coefficients (Eq. (24).

Tables Icon

Table 2 Power approximation to the ADA particles’ visible scattering coefficients spectra (Eq. (25).

Equations (25)

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PSD=m ( D D o ) k ,
C e =Re P 2{ 1exp[ i 2π n w r λ ( nin'1 ) ] }dP ,
C a = P [ 1exp( 4π n w rn' λ ) ]dP ,
b= m D min D max ( C e C a ) D k dD ,
b=m P D 2 ( { Re D min D max 2[ 1exp( 0.5ADiBD ) ] D 2k dD } D min D max [ 1exp( AD ) ] D 2k dD )d( P D 2 ) ,
A=4π n w ( n' λ )( r D ),
B=2π n w ( n1 λ )( r D ),
I k (Z, D min )= D min [ 1exp( ZD ) ] D 2k dD ,
I (34) (Z, D min )= D min 3k k3 [ 1exp( ZD min ) ]+ Z k3 D min exp(ZD) D 3k dD .
Γ(x)= Z x 0 D x1 exp(ZD)dD
lim D min 0 I (34) (Z, D min )= Γ( 4k ) k3 Z k3 .
b=m Γ( 4k ) k3 P D 2 { Re[ 2 ( 0.5A+iB ) k3 ] A k3 }d( P D 2 ) .
( 0.5A+iB ) k3 = | 0.5A+iB | k3 exp[ ( k3 )φi ],
φ=arctan( B 0.5A ).
b=mS Γ( 4k ) k3 ( 2π n w n1 λ ) k3 ×{ 2 [ ( n' n1 ) 2 +1 ] ( k3 )/2 cos[ (k3)arctan( n1 n' ) ] ( 2 n' n1 ) k3 },
S= P D 2 ( r D ) k3 d( P D 2 ) .
lim k 4 Γ(4k) ){ 2 [ ( n' n1 ) 2 +1 ] ( k3 )/2 cos[ (k3)arctan( n1 n' ) ] ( 2 n' n1 ) k3 } =2arctan( n1 n' )+ n' n1 ln[ 4n ' 2 n ' 2 + (n1) 2 ].
b=mS( 2π n w n1 λ ){ 2arctan( n1 n' )+ n' n1 ln[ 4n ' 2 n ' 2 + (n1) 2 ] }.
I (45) (Z, D min )= D min 3k k3 [ 1exp( ZD min ) ]+ 1 k4 D min 4k k3 exp( ZD min ) 1 k4 Z k3 D min exp(ZD) D 4k dD .
lim D min 0 { Re[ 2 I (45) (0.5A+iB, D min ) ] I (45) (A, D min ) }= =Re[ 2 Γ( 5k ) k4 (0.5A+iB) k3 k3 ]+ Γ( 5k ) k4 A k3 k3 .
b=2mScos[ 0.5π(k3) ] Γ( 4k ) k3 ( 2π n w n1 λ ) k3
b=πmS( 2π n w n1 λ )
n'=0.007954exp(7.186λ)
b ADA (λ)=α b Mie (λ)
ln b ADA (λ)=εlnλ+const,

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