Abstract

A model based on Mie theory is described for predicting scattering phase functions at forward angles (0.1°-90°) with particle size distribution (PSD) slope and bulk refractive index as input parameters. The PSD slope 'ξ ' is calculated from the hyperbolic slope of the particle attenuation spectrum measured in different waters. The bulk refractive index 'n' is evaluated by an inversion model, using measured backscattering ratio (Bp) and PSD slope values. For predicting the desired phase function in a certain water type, in situ measurements of the coefficients of particulate backscattering, scattering and beam attenuation are needed. These parameters are easily measurable using commercially available instruments which provide data with high sampling rates. Thus numerical calculation of the volume scattering function is carried out extensively by varying the optical characteristics of particulates in water. The entire range of forward scattering angles (0.1°-90°) is divided into two subsets, i.e., 0.1° to 5° and 5° to 90°. The particulates-in-water phase function is then modeled for both the ranges. Results of the present model are evaluated based on the well-established Petzold average particle phase function and by comparison with those predicted by other phase function models. For validation, the backscattering ratio is modeled as a function of the bulk refractive index and PSD slope, which is subsequently inverted to give a methodology to estimate the bulk refractive index from easily measurable optical parameters. The new phase function model which is based on the exact numerical solution obtained through Mie theory is mathematically less complex and predicts forward scattering phase functions within the desired accuracy.

© 2015 Optical Society of America

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References

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  2. C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic Press, 1994).
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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  26. M. S. Twardowski, E. Boss, J. B. Macdonald, W. S. Pegau, A. H. Barnard, and J. R. V. Zaneveld, “A model for estimating bulk refractive index from the optical backscattering ratio and the implications for understanding particle composition in case I and case II waters,” J. Geophys. Res. 106(C7), 14,129–14,142 (2001).
    [Crossref]
  27. Z. P. Lee, “Remote Sensing of Inherent Optical Properties: Fundamentals, Tests of Algorithms, and Applications,” 5 (IOCCG, Dartmouth, Canada, 2005), pp. 126.
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    [Crossref] [PubMed]
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    [Crossref]

2014 (1)

H. J. Nasiha, P. Shanmugam, and V. G. Hariharasudhan, “A new inversion model to estimate bulk refractive index of particles in costal oceanic waters: Implications for remote sensing,” IEEE J. Sel. Topics in App, Earth. Obs. Remote Sensing 7, 3069–3083 (2014).

2010 (1)

2009 (1)

2007 (1)

2005 (1)

M. Chami, E. B. Shybanov, T. Y. Churilova, G. A. Khomenko, M. E. G. Lee, O. V. Martynov, G. A. Berseneva, and G. K. Korotaev, “Optical properties of the particles in the Crimea coastal waters (Black Sea),” J. Geophys. Res. 110(C11), C11020 (2005).
[Crossref]

2002 (2)

2001 (2)

M. S. Twardowski, E. Boss, J. B. Macdonald, W. S. Pegau, A. H. Barnard, and J. R. V. Zaneveld, “A model for estimating bulk refractive index from the optical backscattering ratio and the implications for understanding particle composition in case I and case II waters,” J. Geophys. Res. 106(C7), 14,129–14,142 (2001).
[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]

1994 (1)

1993 (1)

1986 (1)

M. Jonasz and H. Prandke, “Comparison of measured and computed light scattering in the Baltic,” Tellus 38B(2), 144–157 (1986).
[Crossref]

1980 (2)

L. O. Reynolds and N. J. McCormik, “Approximate two-parameter phase function for light scattering,” J.O.S.A. 70(10), 1206–1212 (1980).
[Crossref]

P. Diehl and H. Haardt, “Measurement of the spectral attenuation to support biological research in a ‘plankton tube’ experiment,” Oceanol. Acta 3, 89–96 (1980).

1973 (1)

1970 (2)

1941 (1)

L. C. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[Crossref]

Agrawal, Y. C.

Bader, H.

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

Barnard, A. H.

M. S. Twardowski, E. Boss, J. B. Macdonald, W. S. Pegau, A. H. Barnard, and J. R. V. Zaneveld, “A model for estimating bulk refractive index from the optical backscattering ratio and the implications for understanding particle composition in case I and case II waters,” J. Geophys. Res. 106(C7), 14,129–14,142 (2001).
[Crossref]

Berseneva, G. A.

M. Chami, E. B. Shybanov, T. Y. Churilova, G. A. Khomenko, M. E. G. Lee, O. V. Martynov, G. A. Berseneva, and G. K. Korotaev, “Optical properties of the particles in the Crimea coastal waters (Black Sea),” J. Geophys. Res. 110(C11), C11020 (2005).
[Crossref]

Boss, E.

A. L. Whitmire, E. Boss, T. J. Cowles, and W. S. Pegau, “Spectral variability of the particulate backscattering ratio,” Opt. Express 15(11), 7019–7031 (2007).
[Crossref] [PubMed]

C. D. Mobley, L. K. Sundman, and E. Boss, “Phase function effects on oceanic light fields,” Appl. Opt. 41(6), 1035–1050 (2002).
[Crossref] [PubMed]

M. S. Twardowski, E. Boss, J. B. Macdonald, W. S. Pegau, A. H. Barnard, and J. R. V. Zaneveld, “A model for estimating bulk refractive index from the optical backscattering ratio and the implications for understanding particle composition in case I and case II waters,” J. Geophys. Res. 106(C7), 14,129–14,142 (2001).
[Crossref]

Bricaud, A.

Brown, O. B.

Chami, M.

A. Sokolov, M. Chami, E. Dmitriev, and G. Khomenko, “Parameterization of volume scattering function of coastal waters based on the statistical approach,” Opt. Express 18(5), 4615–4636 (2010).
[Crossref] [PubMed]

M. Chami, E. B. Shybanov, T. Y. Churilova, G. A. Khomenko, M. E. G. Lee, O. V. Martynov, G. A. Berseneva, and G. K. Korotaev, “Optical properties of the particles in the Crimea coastal waters (Black Sea),” J. Geophys. Res. 110(C11), C11020 (2005).
[Crossref]

Churilova, T. Y.

M. Chami, E. B. Shybanov, T. Y. Churilova, G. A. Khomenko, M. E. G. Lee, O. V. Martynov, G. A. Berseneva, and G. K. Korotaev, “Optical properties of the particles in the Crimea coastal waters (Black Sea),” J. Geophys. Res. 110(C11), C11020 (2005).
[Crossref]

Cowles, T. J.

Diehl, P.

P. Diehl and H. Haardt, “Measurement of the spectral attenuation to support biological research in a ‘plankton tube’ experiment,” Oceanol. Acta 3, 89–96 (1980).

Dmitriev, E.

Gagne, G.

Gentili, B.

Gordon, H. R.

Greenstein, J. L.

L. C. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[Crossref]

Haardt, H.

P. Diehl and H. Haardt, “Measurement of the spectral attenuation to support biological research in a ‘plankton tube’ experiment,” Oceanol. Acta 3, 89–96 (1980).

Haltrin, V. I.

Hariharasudhan, V. G.

H. J. Nasiha, P. Shanmugam, and V. G. Hariharasudhan, “A new inversion model to estimate bulk refractive index of particles in costal oceanic waters: Implications for remote sensing,” IEEE J. Sel. Topics in App, Earth. Obs. Remote Sensing 7, 3069–3083 (2014).

Henyey, L. C.

L. C. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[Crossref]

Holland, A. C.

Jin, Z.

Jonasz, M.

M. Jonasz and H. Prandke, “Comparison of measured and computed light scattering in the Baltic,” Tellus 38B(2), 144–157 (1986).
[Crossref]

Kattawar, G. W.

Khomenko, G.

Khomenko, G. A.

M. Chami, E. B. Shybanov, T. Y. Churilova, G. A. Khomenko, M. E. G. Lee, O. V. Martynov, G. A. Berseneva, and G. K. Korotaev, “Optical properties of the particles in the Crimea coastal waters (Black Sea),” J. Geophys. Res. 110(C11), C11020 (2005).
[Crossref]

Korotaev, G. K.

M. Chami, E. B. Shybanov, T. Y. Churilova, G. A. Khomenko, M. E. G. Lee, O. V. Martynov, G. A. Berseneva, and G. K. Korotaev, “Optical properties of the particles in the Crimea coastal waters (Black Sea),” J. Geophys. Res. 110(C11), C11020 (2005).
[Crossref]

Lee, M. E. G.

M. Chami, E. B. Shybanov, T. Y. Churilova, G. A. Khomenko, M. E. G. Lee, O. V. Martynov, G. A. Berseneva, and G. K. Korotaev, “Optical properties of the particles in the Crimea coastal waters (Black Sea),” J. Geophys. Res. 110(C11), C11020 (2005).
[Crossref]

Macdonald, J. B.

M. S. Twardowski, E. Boss, J. B. Macdonald, W. S. Pegau, A. H. Barnard, and J. R. V. Zaneveld, “A model for estimating bulk refractive index from the optical backscattering ratio and the implications for understanding particle composition in case I and case II waters,” J. Geophys. Res. 106(C7), 14,129–14,142 (2001).
[Crossref]

Martynov, O. V.

M. Chami, E. B. Shybanov, T. Y. Churilova, G. A. Khomenko, M. E. G. Lee, O. V. Martynov, G. A. Berseneva, and G. K. Korotaev, “Optical properties of the particles in the Crimea coastal waters (Black Sea),” J. Geophys. Res. 110(C11), C11020 (2005).
[Crossref]

McCormik, N. J.

L. O. Reynolds and N. J. McCormik, “Approximate two-parameter phase function for light scattering,” J.O.S.A. 70(10), 1206–1212 (1980).
[Crossref]

Mikkelsen, O. A.

Mobley, C. D.

Morel, A.

Nasiha, H. J.

H. J. Nasiha, P. Shanmugam, and V. G. Hariharasudhan, “A new inversion model to estimate bulk refractive index of particles in costal oceanic waters: Implications for remote sensing,” IEEE J. Sel. Topics in App, Earth. Obs. Remote Sensing 7, 3069–3083 (2014).

Pegau, W. S.

A. L. Whitmire, E. Boss, T. J. Cowles, and W. S. Pegau, “Spectral variability of the particulate backscattering ratio,” Opt. Express 15(11), 7019–7031 (2007).
[Crossref] [PubMed]

M. S. Twardowski, E. Boss, J. B. Macdonald, W. S. Pegau, A. H. Barnard, and J. R. V. Zaneveld, “A model for estimating bulk refractive index from the optical backscattering ratio and the implications for understanding particle composition in case I and case II waters,” J. Geophys. Res. 106(C7), 14,129–14,142 (2001).
[Crossref]

Platt, T.

Prandke, H.

M. Jonasz and H. Prandke, “Comparison of measured and computed light scattering in the Baltic,” Tellus 38B(2), 144–157 (1986).
[Crossref]

Reinersman, P.

Reynolds, L. O.

L. O. Reynolds and N. J. McCormik, “Approximate two-parameter phase function for light scattering,” J.O.S.A. 70(10), 1206–1212 (1980).
[Crossref]

Sathyendranath, S.

Shanmugam, P.

H. J. Nasiha, P. Shanmugam, and V. G. Hariharasudhan, “A new inversion model to estimate bulk refractive index of particles in costal oceanic waters: Implications for remote sensing,” IEEE J. Sel. Topics in App, Earth. Obs. Remote Sensing 7, 3069–3083 (2014).

Shybanov, E. B.

M. Chami, E. B. Shybanov, T. Y. Churilova, G. A. Khomenko, M. E. G. Lee, O. V. Martynov, G. A. Berseneva, and G. K. Korotaev, “Optical properties of the particles in the Crimea coastal waters (Black Sea),” J. Geophys. Res. 110(C11), C11020 (2005).
[Crossref]

Sokolov, A.

Stamnes, K.

Stavn, R. H.

Stramski, D.

Sundman, L. K.

Twardowski, M. S.

M. S. Twardowski, E. Boss, J. B. Macdonald, W. S. Pegau, A. H. Barnard, and J. R. V. Zaneveld, “A model for estimating bulk refractive index from the optical backscattering ratio and the implications for understanding particle composition in case I and case II waters,” J. Geophys. Res. 106(C7), 14,129–14,142 (2001).
[Crossref]

Ulloa, O.

Whitmire, A. L.

Zaneveld, J. R. V.

M. S. Twardowski, E. Boss, J. B. Macdonald, W. S. Pegau, A. H. Barnard, and J. R. V. Zaneveld, “A model for estimating bulk refractive index from the optical backscattering ratio and the implications for understanding particle composition in case I and case II waters,” J. Geophys. Res. 106(C7), 14,129–14,142 (2001).
[Crossref]

Appl. Opt. (7)

Astrophys. J. (1)

L. C. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[Crossref]

IEEE J. Sel. Topics in App, Earth. Obs. Remote Sensing (1)

H. J. Nasiha, P. Shanmugam, and V. G. Hariharasudhan, “A new inversion model to estimate bulk refractive index of particles in costal oceanic waters: Implications for remote sensing,” IEEE J. Sel. Topics in App, Earth. Obs. Remote Sensing 7, 3069–3083 (2014).

J. Geophys. Res. (3)

M. Chami, E. B. Shybanov, T. Y. Churilova, G. A. Khomenko, M. E. G. Lee, O. V. Martynov, G. A. Berseneva, and G. K. Korotaev, “Optical properties of the particles in the Crimea coastal waters (Black Sea),” J. Geophys. Res. 110(C11), C11020 (2005).
[Crossref]

M. S. Twardowski, E. Boss, J. B. Macdonald, W. S. Pegau, A. H. Barnard, and J. R. V. Zaneveld, “A model for estimating bulk refractive index from the optical backscattering ratio and the implications for understanding particle composition in case I and case II waters,” J. Geophys. Res. 106(C7), 14,129–14,142 (2001).
[Crossref]

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

J.O.S.A. (1)

L. O. Reynolds and N. J. McCormik, “Approximate two-parameter phase function for light scattering,” J.O.S.A. 70(10), 1206–1212 (1980).
[Crossref]

Oceanol. Acta (1)

P. Diehl and H. Haardt, “Measurement of the spectral attenuation to support biological research in a ‘plankton tube’ experiment,” Oceanol. Acta 3, 89–96 (1980).

Opt. Express (3)

Tellus (1)

M. Jonasz and H. Prandke, “Comparison of measured and computed light scattering in the Baltic,” Tellus 38B(2), 144–157 (1986).
[Crossref]

Other (11)

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

H. C. Van De Hulst, Light Scattering by Small Particles (Dover Publications Inc., 1981).

N. G. Jerlov and E. S. Nielsen, Optical Aspects of Oceanography (Academic Press, 1974), Chap. 1.

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

T. J. Petzold, “Volume Scattering Functions for selected ocean waters,” SIO Ref. 72–78, (Scripps Institute of Oceanography, (1972).

J. C. Kitchen, “Particle size distributions and the vertical distribution of suspended matter in the upwelling region off Oregon,” 77–10 (Oregon State Univ., Corvallis, 1977), pp. 118.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic Press, 1969).

C. Mätzler, “MATLAB functions for Mie scattering and absorption,” 2002–08 (University of Bern, 2002), pp. 18.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley, 1983).

B. Aernouts, R. Watté, J. Lammertyn, and W. Saeys, “A flexible tool for simulating the bulk optical properties of polydisperse suspensions of spherical particles in an absorbing host medium,” in Optical Modeling and Design II, Frank Wyrowski, John T. Sheridan, Jani Tervo, Youri Meuret, eds., Proc. of SPIE 8429,1–11 (2012).

Z. P. Lee, “Remote Sensing of Inherent Optical Properties: Fundamentals, Tests of Algorithms, and Applications,” 5 (IOCCG, Dartmouth, Canada, 2005), pp. 126.

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Figures (6)

Fig. 1
Fig. 1 (a) Variation of the Backscattering Ratio (BP) with 1/(n-1)2. (b) Variation of the intermediate parameters P1 and P2 with PSD slope (ξ).
Fig. 2
Fig. 2 Variation of the intermediate parameters Pm (P1- P4) with PSD slope values.
Fig. 3
Fig. 3 Variation of the particle backscattering coefficient with (a) maximum particle diameter 'Dmax' with fixed Dmin = 0.01 μm, λ = 530 nm, m = 1.05-0.0001i and PSD slope (ξ) = 3, 3.5, 4, 5, (b) minimum particle diameter ' Dmin' with fixed Dmax = 200 μm, λ = 530 nm, m = 1.05-0.0001i and PSD slope (ξ) = 4, (c) wavelength ' λ ' with fixed Dmin = 0.01 μm, Dmax = 200 μm, and PSD slope (ξ) = 4, and (d) PSD slope with fixed Dmin = 0.01 μm, Dmax = 200 μm, λ = 530 nm and real refractive indices values n = 1.04- 0.02- 1.2.
Fig. 4
Fig. 4 Comparison of the phase functions obtained through different models for the scattering angles (a) from 0.1° to 90° and (b) 0.1° to 10° (extended 10° instead of 5° for better clarity) to emphasize the large variation and deviation of different phase functions at small angles.
Fig. 5
Fig. 5 Contours of the backscattering ratio Bp from Eq. (21).
Fig. 6
Fig. 6 Comparison of the present study with the data presented by Sokolov et al. [21]

Tables (4)

Tables Icon

Table 1 Model parameters for calculating the backscattering ratio.

Tables Icon

Table 2 Model parameters for computing the phase function in the range 0.1ᵒ-5ᵒ.

Tables Icon

Table 3 Model parameters for computing the phase function in the range 5°- 90°.

Tables Icon

Table 4 Particulate optical properties obtained for different types of waters. Results also show the refractive index values derived for these waters.

Equations (36)

Equations on this page are rendered with MathJax. Learn more.

β ˜ ( θ )= β( θ ) b .
b=2π 0 π β(θ)sinθdθ .
Q = bp 2 α 2 n=1 ( 2n+1 )( | a n | 2 + | b n | 2 ) .
Q = ext 2 α 2 n=1 ( 2n+1 )Re( a n + b n ) .
Q = bbp 1 α 2 | n=1 ( 2n+1 ) (1) n ( a n b n ) | 2 .
α= πD λ m .
Q ext = Q bp + Q abs .
n max =α+4 α 1 3 +1.
S 1 ( cosθ )= n=1 2n+1 n(n+1) ( a n π n + b n τ n ) .
S 2 ( cosθ )= n=1 2n+1 n(n+1) ( a n τ n + b n π n ) .
S 11 ( cosθ )= 1 2 ( | S 1 ( cosθ ) | 2 + | S 2 ( cosθ ) | 2 ).
S 11 ( cosθ )= ( | S 1 ( cosθ ) | 2 + | S 2 ( cosθ ) | 2 ) n=1 ( 2n+1 )( | a n | 2 + | b n | 2 ) .
b p = λ 3 8 π 2 α min α max Q bp (α)f(α) α 2 dα .
b bp = λ 3 8 π 2 α min α max Q bbp (α)f(α) α 2 dα .
B p = α min α max Q bbp (α)f(α) α 2 dα α min α max Q bp (α)f(α) α 2 dα .
β ˜ ( θ )= α min α max S 11 ( cosθ )f(α) α 2 dα .
f(D)=K D ξ .
K= ( ξ1 ) ( D min 1ξ D max 1ξ ) .
B p = P 1 ( x ) P 2 .
x= 1 ( n1 ) 2 .
P m = a m ( ξ3 ) 2 + b m ( ξ3 )+ c m .
n=1+ ( P 1 B P ) 1 2 P 2 .
log( β ˜ ( θ ) )= P 1 ( ln(θ) ) 2 + P 2 ( ln(θ) )+ P 3 .
P m = a m exp( x )+ b m ( x )+ c m
a m = d m y 2 + e m sin( 5y )+ f m b m = h m y 2 + i m sin( 5y )+ j m c m = k m y 2 + l m sin( 5y )+ o m ]. x=ξ3. y=n1andm=1,2,3
log( β ˜ ( θ ) )= P 1 ( ln(θ) ) 3 + P 2 ( ln(θ) ) 2 + P 3 ( ln(θ) )+ P 4 .
P m = a m exp( x )+ b m ( x )+ c m .
a m = d m y 2 + e m sin( 5y )+ f m b m = h m y 2 + i m sin( 5y )+ j m c m = k m y 2 + l m sin( 5y )+ o m ]. x=ξ3. y=n1andm=1,2,3,4
β ˜ FF ( θ )= 1 4π (1δ) 2 δ v { [ ν( 1δ )( 1 δ v ) ]+[ δ( 1 δ v )ν( 1δ ) ] sin 2 ( θ 2 ) } + 1 δ v 180 ( 16π( δ 180 1 ) δ v 180 ) ( 3 cos 2 θ1 ).
ν= 3ξ 2 .  and  δ= 4 3 (n1) 2 sin 2 ( θ 2 ).
B P =1 1 δ 90 ν+1 0.5( 1 δ 90 ν ) (1 δ 90 ) δ 90 ν .
β ˜ OTHG ( θ )= 1 4π 1 g 2 ( 1+ g 2 2gcosθ ) 3/2 .
B P = 1g 2g ( 1g 1+ g 2 1 ).
β ˜ TTHG ( θ )=α β ˜ OTHG ( θ, g 1 )+( 1α ) β ˜ OTHG ( θ, g 2 ).
g 2 =0.30614+1.0006 g 1 0.01826 g 1 2 +0.03644 g 1 3 .
α= g 2 ( 1+ g 2 ) ( g 1 + g 2 )( 1+ g 2 g 1 ) .

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