Abstract

The presence of bubbles can significantly change the radiative properties of seawater and these changes will affect remote sensing and underwater target detection. In this work, the spectral reflectance and bidirectional reflectance characteristics of the bubble layer in the upper ocean are investigated using the Monte Carlo method. The Hall–Novarini (HN) bubble population model, which considers the effect of wind speed and depth on the bubble size distribution, is used. The scattering coefficients and the scattering phase functions of bubbles in seawater are calculated using Mie theory, and the inherent optical properties of seawater for wavelengths between 300 nm and 800 nm are related to chlorophyll concentration (Chl). The effects of bubble coating, Chl, and bubble number density on the spectral reflectance of the bubble layer are studied. The bidirectional reflectance distribution function (BRDF) of the bubble layer for both normal and oblique incidence is also investigated. The results show that bubble populations in clear waters under high wind speed conditions significantly influence the reflection characteristics of the bubble layer. Furthermore, the contribution of bubble populations to the reflection characteristics is mainly due to the strong backscattering of bubbles that are coated with an organic film.

© 2015 Optical Society of America

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References

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2015 (1)

2014 (1)

M. Christensen, J. Zhang, J. S. Reid, X. Zhang, E. J. Hyer, and A. Smirnov, “A theoretical study of the effect of subsurface oceanic bubbles on the enhanced aerosol optical depth band over the southern oceans as detected from MODIS,” Atmos. Meas. Tech. Discuss 7(12), 12795–12825 (2014).
[Crossref]

2013 (1)

F. Q. Wang, Y. Shuai, H. P. Tan, X. F. Zhang, and Q. J. Mao, “Heat transfer analyses of porous media receiver with multi-dish collector by coupling MCRT and FVM method,” Sol. Energy 93, 158–168 (2013).
[Crossref]

2012 (1)

W. Freda and J. Piskozub, “Revisiting the role of oceanic phase function in remote sensing reflectance,” Oceanologia 54(1), 29–38 (2012).
[Crossref]

2011 (3)

G. de Leeuw, E. L. Andreas, M. D. Anguelova, C. W. Fairall, E. R. Lewis, C. O’Dowd, M. Schulz, and S. E. Schwartz, “Production flux of sea spray aerosol,” Rev. Geophys. 49(2), RG2001 (2011).
[Crossref]

C. E. Blenkinsopp and J. R. Chaplin, “Void fraction measurements and scale effects in breaking waves in freshwater and seawater,” Coast. Eng. 58(5), 417–428 (2011).
[Crossref]

H. Czerski, M. Twardowski, X. Zhang, and S. Vagle, “Resolving size distributions of bubbles with radii less than 30 μm with optical and acoustical methods,” J. Geophys. Res. 116, C00H11 (2011).
[Crossref]

2010 (2)

M. A. Sayer, G. E. Thomas, and R. G. Grainger, “A sea surface reflectance model for (A) ATSR, and application to aerosol retrievals,” Atmos. Meas. Tech. 3(4), 813–838 (2010).
[Crossref]

D. D’Alimonte, G. Zibordi, T. Kajiyama, and J. C. Cunha, “Monte Carlo code for high spatial resolution ocean color simulations,” Appl. Opt. 49(26), 4936–4950 (2010).
[Crossref] [PubMed]

2009 (3)

2007 (1)

C. D. O’Dowd and G. de Leeuw, “Marine aerosol production: a review of the current knowledge,” Philos Trans A Math Phys Eng Sci 365(1856), 1753–1774 (2007).
[Crossref] [PubMed]

2006 (1)

I. Leifer and G. de Leeuw, “Bubbles generated from wind-steepened breaking waves: 1. Bubble plume bubbles,” J. Geophys. Res. 111, C06020 (2006).

2005 (3)

M. A. Ainslie, “Effect of wind-generated bubbles on fixed range acoustic attenuation in shallow water at 1–4 kHz,” J. Acoust. Soc. Am. 118(6), 3513–3523 (2005).
[Crossref]

V. Ross, D. Dion, and G. Potvin, “Detailed analytical approach to the Gaussian surface bidirectional reflectance distribution function specular component applied to the sea surface,” J. Opt. Soc. Am. A 22(11), 2442–2453 (2005).
[Crossref] [PubMed]

X. L. Xia, D. P. Ren, and H. P. Tan, “Effects of medium absorption and scattering on bi-directional reflection of semitransparent coatings,” J. Infrared. Millim. W. 24, 361–365 (2005).

2004 (2)

X. Zhang, M. Lewis, W. P. Bissett, B. Johnson, and D. Kohler, “Optical influence of ship wakes,” Appl. Opt. 43(15), 3122–3132 (2004).
[Crossref] [PubMed]

L. H. Liu, H. C. Zhang, and H. P. Tan, “Monte Carlo discrete curved ray-tracing method for radiative transfer in an absorbing-emitting semitransparent slab with variable spatial refractive index,” J. Quant. Spectrosc. Radiat. Transf. 84(3), 357–362 (2004).
[Crossref]

2003 (1)

C. D. Mobley, H. Zhang, and K. J. Voss, “Effects of optically shallow bottoms on upwelling radiances: Bidirectional reflectance distribution function effects,” Limnol. Oceanogr. 48(1part2), 337–345 (2003).
[Crossref]

2002 (3)

2001 (1)

D. Stramski and J. Tegowski, “Effects of intermittent entrainment of air bubbles by breaking wind waves on ocean reflectance and underwater light field,” J. Geophys. Res. 106(C12), 31345–31360 (2001).
[Crossref]

1999 (1)

1998 (2)

X. Zhang, M. Lewis, and B. Johnson, “Influence of bubbles on scattering of light in the ocean,” Appl. Opt. 37(27), 6525–6536 (1998).
[Crossref] [PubMed]

D. Shooter, R. J. Davies-Colley, and J. T. O. Kirk, “Light absorption and scattering by ocean waters in the vicinity of the Chatham Rise, South Pacific Ocean,” Mar. Freshw. Res. 49(6), 455–461 (1998).
[Crossref]

1997 (1)

G. B. Deane, “Sound generation and air entrainment by breaking waves in the surf zone,” J. Acoust. Soc. Am. 102(5), 2671–2689 (1997).
[Crossref]

1996 (1)

1995 (2)

X. Quan and E. S. Fry, “Empirical equation for the index of refraction of seawater,” Appl. Opt. 34(18), 3477–3480 (1995).
[Crossref] [PubMed]

R. S. Keiffer, J. C. Novarini, and G. V. Norton, “The impact of the background bubble layer on reverberation-derived scattering strengths in the low to moderate frequency range,” J. Acoust. Soc. Am. 97(1), 227–234 (1995).
[Crossref]

1991 (1)

1983 (1)

R. E. Glazman, “Effects of adsorbed films on gas bubble radial oscillations,” J. Acoust. Soc. Am. 74(3), 980–986 (1983).
[Crossref]

Ainslie, M. A.

M. A. Ainslie, “Effect of wind-generated bubbles on fixed range acoustic attenuation in shallow water at 1–4 kHz,” J. Acoust. Soc. Am. 118(6), 3513–3523 (2005).
[Crossref]

Andreas, E. L.

G. de Leeuw, E. L. Andreas, M. D. Anguelova, C. W. Fairall, E. R. Lewis, C. O’Dowd, M. Schulz, and S. E. Schwartz, “Production flux of sea spray aerosol,” Rev. Geophys. 49(2), RG2001 (2011).
[Crossref]

Anguelova, M. D.

G. de Leeuw, E. L. Andreas, M. D. Anguelova, C. W. Fairall, E. R. Lewis, C. O’Dowd, M. Schulz, and S. E. Schwartz, “Production flux of sea spray aerosol,” Rev. Geophys. 49(2), RG2001 (2011).
[Crossref]

Asher, W. E.

R. Wanninkhof, W. E. Asher, D. T. Ho, C. Sweeney, and W. R. McGillis, “Advances in quantifying air-sea gas exchange and environmental forcing,” Annu. Rev. Mar. Sci. 1(1), 213–244 (2009).
[Crossref] [PubMed]

Bissett, W. P.

Blenkinsopp, C. E.

C. E. Blenkinsopp and J. R. Chaplin, “Void fraction measurements and scale effects in breaking waves in freshwater and seawater,” Coast. Eng. 58(5), 417–428 (2011).
[Crossref]

Boss, E.

Chaplin, J. R.

C. E. Blenkinsopp and J. R. Chaplin, “Void fraction measurements and scale effects in breaking waves in freshwater and seawater,” Coast. Eng. 58(5), 417–428 (2011).
[Crossref]

Chen, B.

Christensen, M.

M. Christensen, J. Zhang, J. S. Reid, X. Zhang, E. J. Hyer, and A. Smirnov, “A theoretical study of the effect of subsurface oceanic bubbles on the enhanced aerosol optical depth band over the southern oceans as detected from MODIS,” Atmos. Meas. Tech. Discuss 7(12), 12795–12825 (2014).
[Crossref]

Cunha, J. C.

Czerski, H.

H. Czerski, M. Twardowski, X. Zhang, and S. Vagle, “Resolving size distributions of bubbles with radii less than 30 μm with optical and acoustical methods,” J. Geophys. Res. 116, C00H11 (2011).
[Crossref]

D’Alimonte, D.

Davies-Colley, R. J.

D. Shooter, R. J. Davies-Colley, and J. T. O. Kirk, “Light absorption and scattering by ocean waters in the vicinity of the Chatham Rise, South Pacific Ocean,” Mar. Freshw. Res. 49(6), 455–461 (1998).
[Crossref]

de Leeuw, G.

G. de Leeuw, E. L. Andreas, M. D. Anguelova, C. W. Fairall, E. R. Lewis, C. O’Dowd, M. Schulz, and S. E. Schwartz, “Production flux of sea spray aerosol,” Rev. Geophys. 49(2), RG2001 (2011).
[Crossref]

C. D. O’Dowd and G. de Leeuw, “Marine aerosol production: a review of the current knowledge,” Philos Trans A Math Phys Eng Sci 365(1856), 1753–1774 (2007).
[Crossref] [PubMed]

I. Leifer and G. de Leeuw, “Bubbles generated from wind-steepened breaking waves: 1. Bubble plume bubbles,” J. Geophys. Res. 111, C06020 (2006).

Deane, G. B.

G. B. Deane, “Sound generation and air entrainment by breaking waves in the surf zone,” J. Acoust. Soc. Am. 102(5), 2671–2689 (1997).
[Crossref]

Dion, D.

Fairall, C. W.

G. de Leeuw, E. L. Andreas, M. D. Anguelova, C. W. Fairall, E. R. Lewis, C. O’Dowd, M. Schulz, and S. E. Schwartz, “Production flux of sea spray aerosol,” Rev. Geophys. 49(2), RG2001 (2011).
[Crossref]

Flatau, P.

Freda, W.

W. Freda and J. Piskozub, “Revisiting the role of oceanic phase function in remote sensing reflectance,” Oceanologia 54(1), 29–38 (2012).
[Crossref]

Fry, E. S.

Gentili, B.

Glazman, R. E.

R. E. Glazman, “Effects of adsorbed films on gas bubble radial oscillations,” J. Acoust. Soc. Am. 74(3), 980–986 (1983).
[Crossref]

Grainger, R. G.

M. A. Sayer, G. E. Thomas, and R. G. Grainger, “A sea surface reflectance model for (A) ATSR, and application to aerosol retrievals,” Atmos. Meas. Tech. 3(4), 813–838 (2010).
[Crossref]

Ho, D. T.

R. Wanninkhof, W. E. Asher, D. T. Ho, C. Sweeney, and W. R. McGillis, “Advances in quantifying air-sea gas exchange and environmental forcing,” Annu. Rev. Mar. Sci. 1(1), 213–244 (2009).
[Crossref] [PubMed]

Hu, L.

Hu, Y.

Hyer, E. J.

M. Christensen, J. Zhang, J. S. Reid, X. Zhang, E. J. Hyer, and A. Smirnov, “A theoretical study of the effect of subsurface oceanic bubbles on the enhanced aerosol optical depth band over the southern oceans as detected from MODIS,” Atmos. Meas. Tech. Discuss 7(12), 12795–12825 (2014).
[Crossref]

Johnson, B.

Josset, D. B.

Kajiyama, T.

Keiffer, R. S.

R. S. Keiffer, J. C. Novarini, and G. V. Norton, “The impact of the background bubble layer on reverberation-derived scattering strengths in the low to moderate frequency range,” J. Acoust. Soc. Am. 97(1), 227–234 (1995).
[Crossref]

Kirk, J. T. O.

D. Shooter, R. J. Davies-Colley, and J. T. O. Kirk, “Light absorption and scattering by ocean waters in the vicinity of the Chatham Rise, South Pacific Ocean,” Mar. Freshw. Res. 49(6), 455–461 (1998).
[Crossref]

Kohler, D.

Korotaev, G.

X. Zhang, M. Lewis, M. Lee, B. Johnson, and G. Korotaev, “The volume scattering function of natural bubble populations,” Limnol. Oceanogr. 47(5), 1273–1282 (2002).
[Crossref]

Lee, M.

X. Zhang, M. Lewis, M. Lee, B. Johnson, and G. Korotaev, “The volume scattering function of natural bubble populations,” Limnol. Oceanogr. 47(5), 1273–1282 (2002).
[Crossref]

Lee, Z.

Leifer, I.

I. Leifer and G. de Leeuw, “Bubbles generated from wind-steepened breaking waves: 1. Bubble plume bubbles,” J. Geophys. Res. 111, C06020 (2006).

Lewis, E. R.

G. de Leeuw, E. L. Andreas, M. D. Anguelova, C. W. Fairall, E. R. Lewis, C. O’Dowd, M. Schulz, and S. E. Schwartz, “Production flux of sea spray aerosol,” Rev. Geophys. 49(2), RG2001 (2011).
[Crossref]

Lewis, M.

Liu, L. H.

L. H. Liu, H. C. Zhang, and H. P. Tan, “Monte Carlo discrete curved ray-tracing method for radiative transfer in an absorbing-emitting semitransparent slab with variable spatial refractive index,” J. Quant. Spectrosc. Radiat. Transf. 84(3), 357–362 (2004).
[Crossref]

Lucker, P. L.

Mao, Q. J.

F. Q. Wang, Y. Shuai, H. P. Tan, X. F. Zhang, and Q. J. Mao, “Heat transfer analyses of porous media receiver with multi-dish collector by coupling MCRT and FVM method,” Sol. Energy 93, 158–168 (2013).
[Crossref]

McGillis, W. R.

R. Wanninkhof, W. E. Asher, D. T. Ho, C. Sweeney, and W. R. McGillis, “Advances in quantifying air-sea gas exchange and environmental forcing,” Annu. Rev. Mar. Sci. 1(1), 213–244 (2009).
[Crossref] [PubMed]

Melville, W. K.

Mobley, C. D.

C. D. Mobley, H. Zhang, and K. J. Voss, “Effects of optically shallow bottoms on upwelling radiances: Bidirectional reflectance distribution function effects,” Limnol. Oceanogr. 48(1part2), 337–345 (2003).
[Crossref]

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]

Morel, A.

Norton, G. V.

R. S. Keiffer, J. C. Novarini, and G. V. Norton, “The impact of the background bubble layer on reverberation-derived scattering strengths in the low to moderate frequency range,” J. Acoust. Soc. Am. 97(1), 227–234 (1995).
[Crossref]

Novarini, J. C.

R. S. Keiffer, J. C. Novarini, and G. V. Norton, “The impact of the background bubble layer on reverberation-derived scattering strengths in the low to moderate frequency range,” J. Acoust. Soc. Am. 97(1), 227–234 (1995).
[Crossref]

O’Dowd, C.

G. de Leeuw, E. L. Andreas, M. D. Anguelova, C. W. Fairall, E. R. Lewis, C. O’Dowd, M. Schulz, and S. E. Schwartz, “Production flux of sea spray aerosol,” Rev. Geophys. 49(2), RG2001 (2011).
[Crossref]

O’Dowd, C. D.

C. D. O’Dowd and G. de Leeuw, “Marine aerosol production: a review of the current knowledge,” Philos Trans A Math Phys Eng Sci 365(1856), 1753–1774 (2007).
[Crossref] [PubMed]

Piskozub, J.

Potvin, G.

Quan, X.

Reid, J. S.

M. Christensen, J. Zhang, J. S. Reid, X. Zhang, E. J. Hyer, and A. Smirnov, “A theoretical study of the effect of subsurface oceanic bubbles on the enhanced aerosol optical depth band over the southern oceans as detected from MODIS,” Atmos. Meas. Tech. Discuss 7(12), 12795–12825 (2014).
[Crossref]

Ren, D. P.

X. L. Xia, D. P. Ren, and H. P. Tan, “Effects of medium absorption and scattering on bi-directional reflection of semitransparent coatings,” J. Infrared. Millim. W. 24, 361–365 (2005).

Ross, V.

Sayer, M. A.

M. A. Sayer, G. E. Thomas, and R. G. Grainger, “A sea surface reflectance model for (A) ATSR, and application to aerosol retrievals,” Atmos. Meas. Tech. 3(4), 813–838 (2010).
[Crossref]

Schulz, M.

G. de Leeuw, E. L. Andreas, M. D. Anguelova, C. W. Fairall, E. R. Lewis, C. O’Dowd, M. Schulz, and S. E. Schwartz, “Production flux of sea spray aerosol,” Rev. Geophys. 49(2), RG2001 (2011).
[Crossref]

Schwartz, S. E.

G. de Leeuw, E. L. Andreas, M. D. Anguelova, C. W. Fairall, E. R. Lewis, C. O’Dowd, M. Schulz, and S. E. Schwartz, “Production flux of sea spray aerosol,” Rev. Geophys. 49(2), RG2001 (2011).
[Crossref]

Shooter, D.

D. Shooter, R. J. Davies-Colley, and J. T. O. Kirk, “Light absorption and scattering by ocean waters in the vicinity of the Chatham Rise, South Pacific Ocean,” Mar. Freshw. Res. 49(6), 455–461 (1998).
[Crossref]

Shuai, Y.

F. Q. Wang, Y. Shuai, H. P. Tan, X. F. Zhang, and Q. J. Mao, “Heat transfer analyses of porous media receiver with multi-dish collector by coupling MCRT and FVM method,” Sol. Energy 93, 158–168 (2013).
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F. Q. Wang, Y. Shuai, H. P. Tan, X. F. Zhang, and Q. J. Mao, “Heat transfer analyses of porous media receiver with multi-dish collector by coupling MCRT and FVM method,” Sol. Energy 93, 158–168 (2013).
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[Crossref]

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D. Stramski and J. Tegowski, “Effects of intermittent entrainment of air bubbles by breaking wind waves on ocean reflectance and underwater light field,” J. Geophys. Res. 106(C12), 31345–31360 (2001).
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F. Q. Wang, Y. Shuai, H. P. Tan, X. F. Zhang, and Q. J. Mao, “Heat transfer analyses of porous media receiver with multi-dish collector by coupling MCRT and FVM method,” Sol. Energy 93, 158–168 (2013).
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C. D. Mobley, H. Zhang, and K. J. Voss, “Effects of optically shallow bottoms on upwelling radiances: Bidirectional reflectance distribution function effects,” Limnol. Oceanogr. 48(1part2), 337–345 (2003).
[Crossref]

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L. H. Liu, H. C. Zhang, and H. P. Tan, “Monte Carlo discrete curved ray-tracing method for radiative transfer in an absorbing-emitting semitransparent slab with variable spatial refractive index,” J. Quant. Spectrosc. Radiat. Transf. 84(3), 357–362 (2004).
[Crossref]

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M. Christensen, J. Zhang, J. S. Reid, X. Zhang, E. J. Hyer, and A. Smirnov, “A theoretical study of the effect of subsurface oceanic bubbles on the enhanced aerosol optical depth band over the southern oceans as detected from MODIS,” Atmos. Meas. Tech. Discuss 7(12), 12795–12825 (2014).
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M. Christensen, J. Zhang, J. S. Reid, X. Zhang, E. J. Hyer, and A. Smirnov, “A theoretical study of the effect of subsurface oceanic bubbles on the enhanced aerosol optical depth band over the southern oceans as detected from MODIS,” Atmos. Meas. Tech. Discuss 7(12), 12795–12825 (2014).
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M. A. Sayer, G. E. Thomas, and R. G. Grainger, “A sea surface reflectance model for (A) ATSR, and application to aerosol retrievals,” Atmos. Meas. Tech. 3(4), 813–838 (2010).
[Crossref]

Atmos. Meas. Tech. Discuss (1)

M. Christensen, J. Zhang, J. S. Reid, X. Zhang, E. J. Hyer, and A. Smirnov, “A theoretical study of the effect of subsurface oceanic bubbles on the enhanced aerosol optical depth band over the southern oceans as detected from MODIS,” Atmos. Meas. Tech. Discuss 7(12), 12795–12825 (2014).
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D. Stramski and J. Tegowski, “Effects of intermittent entrainment of air bubbles by breaking wind waves on ocean reflectance and underwater light field,” J. Geophys. Res. 106(C12), 31345–31360 (2001).
[Crossref]

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X. L. Xia, D. P. Ren, and H. P. Tan, “Effects of medium absorption and scattering on bi-directional reflection of semitransparent coatings,” J. Infrared. Millim. W. 24, 361–365 (2005).

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J. Quant. Spectrosc. Radiat. Transf. (1)

L. H. Liu, H. C. Zhang, and H. P. Tan, “Monte Carlo discrete curved ray-tracing method for radiative transfer in an absorbing-emitting semitransparent slab with variable spatial refractive index,” J. Quant. Spectrosc. Radiat. Transf. 84(3), 357–362 (2004).
[Crossref]

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X. Zhang, M. Lewis, M. Lee, B. Johnson, and G. Korotaev, “The volume scattering function of natural bubble populations,” Limnol. Oceanogr. 47(5), 1273–1282 (2002).
[Crossref]

C. D. Mobley, H. Zhang, and K. J. Voss, “Effects of optically shallow bottoms on upwelling radiances: Bidirectional reflectance distribution function effects,” Limnol. Oceanogr. 48(1part2), 337–345 (2003).
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F. Q. Wang, Y. Shuai, H. P. Tan, X. F. Zhang, and Q. J. Mao, “Heat transfer analyses of porous media receiver with multi-dish collector by coupling MCRT and FVM method,” Sol. Energy 93, 158–168 (2013).
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Figures (14)

Fig. 1
Fig. 1 Variations in the bubble number density with depth at different wind speeds.
Fig. 2
Fig. 2 Scattering coefficients of bubble populations in each layer at different wind speeds.
Fig. 3
Fig. 3 Scattering phase functions of coated bubbles and clean bubbles at λ = 550 nm.
Fig. 4
Fig. 4 (a) Total spectral absorption coefficients and (b) total spectral scattering coefficients of the water body as a function of wavelength.
Fig. 5
Fig. 5 Schematic diagram of (a) light transfer in the bubble layer and (b) coordinates system.
Fig. 6
Fig. 6 Determination of the scattering angle from random number for clean bubbles, coated bubbles, phytoplankton particles, and pure seawater.
Fig. 7
Fig. 7 BRDF·cosθr distribution as a function of view zenith angle.
Fig. 8
Fig. 8 BRDF distribution at different incident angles.
Fig. 9
Fig. 9 Spectral reflectance of the bubble layer for solar zenith angle θi = 0° with Chls = 0.01, 0.1, 1.0, 10.0 mg·m−3. N1~4 corresponds to the bubble number density at wind speed v10 = 6, 10, 14 and 16.5 m/s, respectively.
Fig. 10
Fig. 10 Ratio of the spectral reflectance under four wind speed conditions (v10 = 6, 10, 14, 16.5 m/s) to that for the case with no bubbles in seawater. N1~4 corresponds to the bubble number density at wind speed v10 = 6, 10, 14 and 16.5 m/s, respectively.
Fig. 11
Fig. 11 BRDF·cosθr of the bubble layer at λ = 550 nm for solar zenith angle θi = 0° and Chls = 0.01, 0.1, 1.0, and 10.0 mg·m−3. N1~4 corresponds to the bubble number density at wind speed v10 = 6, 10, 14 and 16.5 m/s, respectively.
Fig. 12
Fig. 12 Ratio of BRDF·cosθr under four wind speed conditions (v10 = 6, 10, 14, 16.5 m/s) to that for the case with no bubbles in seawater. N1~4 corresponds to the bubble number density at wind speed v10 = 6, 10, 14 and 16.5 m/s, respectively.
Fig. 13
Fig. 13 BRDF·cosθr distribution in the incident plane in clear waters with Chl = 0.1 mg·m−3 at λ = 550 nm for solar zenith angle of θi = 0°, 30°, 45° and 60°. N1~4 corresponds to the bubble number density at wind speed v10 = 6, 10, 14 and 16.5 m/s, respectively.
Fig. 14
Fig. 14 BRDF·cosθr distribution in clear waters with Chl = 0.1 mg·m−3 at λ = 550 nm for solar zenith angle of θi = 0°, 30°, 45° and 60°. The solar azimuth angle is ϕi = 180° and the view azimuth angle ranges from ϕr = 0° to 180°. N1~4 corresponds to the bubble number density at wind speed v10 = 6, 10, 14 and 16.5 m/s, respectively.

Equations (21)

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n ( r , z ) = ( 1.6 × 10 4 ) G ( r , z ) ( v 10 13 ) 3 exp [ z L ( v 10 ) ] ,
L ( v ) = { 0.4 v 10 7.5 0.4 + 0 .115 ( v 10 7.5 ) v 10 > 7.5 ,
G ( r , z ) = { [ r ref ( z ) / r ] 4 r min r r ref ( z ) [ r ref ( z ) / r ] 4.37 + (z/2 .55) 2 r ref ( z ) < r r max ,
N ( z ) = r min r max n ( r , z ) d r ( 1.6 × 10 4 ) r ref 4 3 r min 3 ( v 10 13 ) 3 exp [ z L ( v 10 ) ] .
N ¯ i = z 1 i z 2 i N ( z ) d z / ( z 2 i z 1 i ) , i = 1 8 ,
b b u b ( z ) = 10 - 12 N ( z ) Q ¯ sca S ¯ , Q ¯ sca = 2.0 ,
S ¯ = r min r max n ( r , z ) N ( z ) π r 2 d r = r min r max 3 r min 3 r ref 4 G ( r , z ) π r 2 d r 3 π r min 2 ,
β ˜ b u b ( z , θ ) = 1 b b u b r min r max Q β ( r , θ ) π r 2 n ( r , z ) d r ,
a t ( λ ) = a w ( λ ) + a p h ( λ ) + a CDOM ( λ ) .
a CDOM ( λ ) = a CDOM ( 440 ) e 0.014 ( λ 440 ) ,
b t ( λ ) = b w ( λ ) + b p h ( λ ) = 0.00193 ( 550 λ ) 4.32 + ( 550 λ ) 0.30 Chl 0.62 .
β w ( θ ) = β w ( 90 ° ) ( 1 + 0.925 cos 2 θ ) ,
R = E u E d = 2 π M r ( Ω r ) M 0 ,
BRDF( θ i , ϕ i , θ r , ϕ r ) = d L r ( θ r , ϕ r ) L i ( θ i , ϕ i ) cos θ i d Ω i L r ( θ r , ϕ r ) E d = M r ( Ω r ) M 0 Ω r cos θ r ,
η 1 = b b u b b b u b + b p h + b w ,
η 2 = b b u b + b p h b b u b + b p h + b w .
P w ( θ ) = 0.118 cos 3 θ 0.382 cos θ + 0.5 , 0 P w ( θ ) 1.
2 π 0 π β ˜ ( θ ) sin ( θ ) d θ = 1 ,
p ( θ ) = 2 π β ˜ ( θ ) sin ( θ ) , 0 θ π .
P ( θ ) = 0 θ p ( θ ) d θ = 2 π 0 θ β ˜ ( θ ) sin ( θ ) d θ = ξ , 0 P ( θ ) 1 ,
ρ ( θ i , θ t ) = { 1 2 { [ sin ( θ i θ t ) sin ( θ i + θ t ) ] 2 + [ tan ( θ i θ t ) tan ( θ i + θ t ) ] 2 } , θ i 0 ( n i n t n i + n t ) 2 , θ i = 0

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