Abstract

Generation of random sea surfaces using wave variance spectra and Fourier transforms is formulated in a way that guarantees conservation of wave energy and fully resolves wave height and slope variances. Monte Carlo polarized ray tracing, which accounts for multiple scattering between light rays and wave facets, is used to compute effective Mueller matrices for reflection and transmission of air- or water-incident polarized radiance. Irradiance reflectances computed using a Rayleigh sky radiance distribution, sea surfaces generated with Cox–Munk statistics, and unpolarized ray tracing differ by 10%–18% compared with values computed using elevation- and slope-resolving surfaces and polarized ray tracing. Radiance reflectance factors, as used to estimate water-leaving radiance from measured upwelling and sky radiances, are shown to depend on sky polarization, and improved values are given.

© 2015 Optical Society of America

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References

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    [Crossref]
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2012 (3)

2011 (4)

2010 (3)

Z.-P. Lee, Y.-H. Ahn, C. D. Mobley, and R. Arnone, “Removal of surface-reflected light for the measurement of remote-sensing reflectance from an above-surface platform,” Opt. Express 18, 26313–26324 (2010).
[Crossref]

C. Emde, R. Buras, B. Mayer, and M. Blumthaler, “The impact of aerosols on polarized sky radiance: model development, validation, and applications,” Atmos. Chem. Phys. 10, 383–396 (2010).
[Crossref]

P.-W. Zhai, Y. Hu, J. Chowdhary, C. R. Trepte, P. L. Luker, and D. B. Josset, “A vector radiative transfer model for coupled atmosphere and ocean systems with a rough interface,” J. Quant. Spectrosc. Radiat. Transfer 111, 1025–1040 (2010).

2009 (2)

G. Zibordi, B. N. Holben, I. Slutsker, D. Giles, D. D’Alimonte, F. Mélin, J. F. Berthon, D. Vandemark, H. Feng, G. Schuster, B. Fabbri, S. Kaitala, and J. Seppale, “AERONET-OC: a network for the validation of ocean color primary radiometric products,” J. Atmos. Ocean. Technol. 26, 1634–1651 (2009).
[Crossref]

P.-W. Zhai, Y. Hu, C. R. Trepte, and P. L. Lucker, “A vector radiative transfer model for coupled atmosphere and ocean systems based on successive order of scattering method,” Opt. Express 17, 2057–2079 (2009).
[Crossref]

2008 (1)

2006 (1)

2004 (2)

2001 (2)

1999 (1)

1997 (1)

T. Elfouhaily, B. Chapron, K. Katsaros, and D. Vandemark, “A unified directional spectrum for long and short wind-driven waves,” J. Geophys. Res. 102, 15781–15796 (1997).
[Crossref]

1996 (1)

1991 (1)

M. P. Levesque, “Dynamic sea image generation,” Proc. SPIE 1486, 294–300 (1991).
[Crossref]

1990 (4)

J. W. McLean, “Modeling of ocean wave effects for LIDAR remote sensing,” Proc. SPIE 1302, 480–491 (1990).
[Crossref]

M. P. Levesque and D. St-Germain, “Generation of synthetic IR sea images,” Proc. SPIE 1331, 354–357 (1990).

M. L. Banner, “Equlibrium spectra of wind waves,” J. Phys. Oceanogr. 20, 966–984 (1990).
[Crossref]

W. W. Gregg and K. L. Carder, “A simple spectral solar irradiance model for cloudless maritime atmospheres,” Limnol. Oceanogr. 35, 1657–1675 (1990).
[Crossref]

1989 (1)

G. W. Kattawar and C. N. Adams, “Stokes vector calculations of the submarine light field in an atmosphere-ocean with scattering according to a Rayleigh phase matrix: effect of interface refractive index on radiance and polarization,” Limnol. Oceanogr. 34, 1453–1472 (1989).
[Crossref]

1988 (1)

A. W. Harrison and C. A. Coombes, “Angular distribution of clear sky short wavelength radiance,” Solar Energy 40, 57–63 (1988).
[Crossref]

1986 (1)

R. W. Preisendorfer and C. D. Mobley, “Albedos and glitter patterns of a wind-roughened sea surface,” J. Phys. Oceanogr. 16, 1293–1316 (1986).

1985 (1)

K. L. Carder and R. G. Steward, “A remote-sensing reflectance model of a red-tide dinoflagellate off West Florida,” Limnol. Oceanogr. 30, 286–298 (1985).
[Crossref]

1954 (1)

Adams, C. N.

G. W. Kattawar and C. N. Adams, “Stokes vector calculations of the submarine light field in an atmosphere-ocean with scattering according to a Rayleigh phase matrix: effect of interface refractive index on radiance and polarization,” Limnol. Oceanogr. 34, 1453–1472 (1989).
[Crossref]

Ahmed, S.

Ahn, Y.-H.

Arnone, R.

Banner, M. L.

M. L. Banner, “Equlibrium spectra of wind waves,” J. Phys. Oceanogr. 20, 966–984 (1990).
[Crossref]

Berthon, J. F.

G. Zibordi, B. N. Holben, I. Slutsker, D. Giles, D. D’Alimonte, F. Mélin, J. F. Berthon, D. Vandemark, H. Feng, G. Schuster, B. Fabbri, S. Kaitala, and J. Seppale, “AERONET-OC: a network for the validation of ocean color primary radiometric products,” J. Atmos. Ocean. Technol. 26, 1634–1651 (2009).
[Crossref]

S. B. Hooker, G. Zibordi, J. F. Berthon, and J. W. Brown, “Above-water radiometry in shallow coastal waters,” Appl. Opt. 43, 4254–4268 (2004).
[Crossref]

Blumthaler, M.

C. Emde, R. Buras, B. Mayer, and M. Blumthaler, “The impact of aerosols on polarized sky radiance: model development, validation, and applications,” Atmos. Chem. Phys. 10, 383–396 (2010).
[Crossref]

Boss, E. S.

Brown, J. W.

Buras, R.

C. Emde, R. Buras, B. Mayer, and M. Blumthaler, “The impact of aerosols on polarized sky radiance: model development, validation, and applications,” Atmos. Chem. Phys. 10, 383–396 (2010).
[Crossref]

Cairns, B.

Carder, K. L.

W. W. Gregg and K. L. Carder, “A simple spectral solar irradiance model for cloudless maritime atmospheres,” Limnol. Oceanogr. 35, 1657–1675 (1990).
[Crossref]

K. L. Carder and R. G. Steward, “A remote-sensing reflectance model of a red-tide dinoflagellate off West Florida,” Limnol. Oceanogr. 30, 286–298 (1985).
[Crossref]

Cartwright, D. E.

M. S. Longuet-Higgins, D. E. Cartwright, and N. D. Smith, “Observations of the directional spectrum of sea waves using the motions of a flotation buoy,” in Ocean Wave Spectra (Prentice-Hall, 1963), pp. 111–136.

Chaikovskaya, L. I.

Chami, M.

Chapron, B.

T. Elfouhaily, B. Chapron, K. Katsaros, and D. Vandemark, “A unified directional spectrum for long and short wind-driven waves,” J. Geophys. Res. 102, 15781–15796 (1997).
[Crossref]

Chowdhary, J.

Coombes, C. A.

A. W. Harrison and C. A. Coombes, “Angular distribution of clear sky short wavelength radiance,” Solar Energy 40, 57–63 (1988).
[Crossref]

Cox, C.

D’Alimonte, D.

G. Zibordi, B. N. Holben, I. Slutsker, D. Giles, D. D’Alimonte, F. Mélin, J. F. Berthon, D. Vandemark, H. Feng, G. Schuster, B. Fabbri, S. Kaitala, and J. Seppale, “AERONET-OC: a network for the validation of ocean color primary radiometric products,” J. Atmos. Ocean. Technol. 26, 1634–1651 (2009).
[Crossref]

Dilligeard, E.

Elfouhaily, T.

T. Elfouhaily, B. Chapron, K. Katsaros, and D. Vandemark, “A unified directional spectrum for long and short wind-driven waves,” J. Geophys. Res. 102, 15781–15796 (1997).
[Crossref]

Emde, C.

C. Emde, R. Buras, B. Mayer, and M. Blumthaler, “The impact of aerosols on polarized sky radiance: model development, validation, and applications,” Atmos. Chem. Phys. 10, 383–396 (2010).
[Crossref]

Fabbri, B.

G. Zibordi, B. N. Holben, I. Slutsker, D. Giles, D. D’Alimonte, F. Mélin, J. F. Berthon, D. Vandemark, H. Feng, G. Schuster, B. Fabbri, S. Kaitala, and J. Seppale, “AERONET-OC: a network for the validation of ocean color primary radiometric products,” J. Atmos. Ocean. Technol. 26, 1634–1651 (2009).
[Crossref]

Feng, H.

G. Zibordi, B. N. Holben, I. Slutsker, D. Giles, D. D’Alimonte, F. Mélin, J. F. Berthon, D. Vandemark, H. Feng, G. Schuster, B. Fabbri, S. Kaitala, and J. Seppale, “AERONET-OC: a network for the validation of ocean color primary radiometric products,” J. Atmos. Ocean. Technol. 26, 1634–1651 (2009).
[Crossref]

Freeman, J. D.

Gilerson, A.

Giles, D.

G. Zibordi, B. N. Holben, I. Slutsker, D. Giles, D. D’Alimonte, F. Mélin, J. F. Berthon, D. Vandemark, H. Feng, G. Schuster, B. Fabbri, S. Kaitala, and J. Seppale, “AERONET-OC: a network for the validation of ocean color primary radiometric products,” J. Atmos. Ocean. Technol. 26, 1634–1651 (2009).
[Crossref]

Gregg, W. W.

W. W. Gregg and K. L. Carder, “A simple spectral solar irradiance model for cloudless maritime atmospheres,” Limnol. Oceanogr. 35, 1657–1675 (1990).
[Crossref]

Gross, B.

Harmel, T.

Harrison, A. W.

A. W. Harrison and C. A. Coombes, “Angular distribution of clear sky short wavelength radiance,” Solar Energy 40, 57–63 (1988).
[Crossref]

Hedley, J.

Hlaing, S.

Holben, B. N.

G. Zibordi, B. N. Holben, I. Slutsker, D. Giles, D. D’Alimonte, F. Mélin, J. F. Berthon, D. Vandemark, H. Feng, G. Schuster, B. Fabbri, S. Kaitala, and J. Seppale, “AERONET-OC: a network for the validation of ocean color primary radiometric products,” J. Atmos. Ocean. Technol. 26, 1634–1651 (2009).
[Crossref]

Hooker, S. B.

Hu, Y.

P.-W. Zhai, G. W. Kattawar, and Y. Hu, “Comment on the transmission matrix for a dielectric interface,” J. Quant. Spectrosc. Radiat. Transfer 113, 1981–1984 (2012).
[Crossref]

P.-W. Zhai, Y. Hu, J. Chowdhary, C. R. Trepte, P. L. Luker, and D. B. Josset, “A vector radiative transfer model for coupled atmosphere and ocean systems with a rough interface,” J. Quant. Spectrosc. Radiat. Transfer 111, 1025–1040 (2010).

P.-W. Zhai, Y. Hu, C. R. Trepte, and P. L. Lucker, “A vector radiative transfer model for coupled atmosphere and ocean systems based on successive order of scattering method,” Opt. Express 17, 2057–2079 (2009).
[Crossref]

Ibrahim, A.

Josset, D. B.

P.-W. Zhai, Y. Hu, J. Chowdhary, C. R. Trepte, P. L. Luker, and D. B. Josset, “A vector radiative transfer model for coupled atmosphere and ocean systems with a rough interface,” J. Quant. Spectrosc. Radiat. Transfer 111, 1025–1040 (2010).

Kaitala, S.

G. Zibordi, B. N. Holben, I. Slutsker, D. Giles, D. D’Alimonte, F. Mélin, J. F. Berthon, D. Vandemark, H. Feng, G. Schuster, B. Fabbri, S. Kaitala, and J. Seppale, “AERONET-OC: a network for the validation of ocean color primary radiometric products,” J. Atmos. Ocean. Technol. 26, 1634–1651 (2009).
[Crossref]

Katsaros, K.

T. Elfouhaily, B. Chapron, K. Katsaros, and D. Vandemark, “A unified directional spectrum for long and short wind-driven waves,” J. Geophys. Res. 102, 15781–15796 (1997).
[Crossref]

Katsev, I. L.

Kattawar, G. W.

P.-W. Zhai, G. W. Kattawar, and Y. Hu, “Comment on the transmission matrix for a dielectric interface,” J. Quant. Spectrosc. Radiat. Transfer 113, 1981–1984 (2012).
[Crossref]

P.-W. Zhai, G. W. Kattawar, and P. Yang, “Impulse response solution of the three-dimensional vector radiative transfer equation in atmosphere-ocean systems. I. Monte Carlo method,” Appl. Opt. 47, 1037–1047 (2008).
[Crossref]

H. H. Tynes, G. W. Kattawar, E. P. Zege, I. L. Katsev, A. S. Prikhach, and L. I. Chaikovskaya, “Monte Carlo and multicomponent approximation methods for vector radiative transfer by use of effective Mueller matrix calculations,” Appl. Opt. 40, 400–412 (2001).
[Crossref]

G. W. Kattawar and C. N. Adams, “Stokes vector calculations of the submarine light field in an atmosphere-ocean with scattering according to a Rayleigh phase matrix: effect of interface refractive index on radiance and polarization,” Limnol. Oceanogr. 34, 1453–1472 (1989).
[Crossref]

Kay, S.

Lacis, A. A.

M. I. Mischenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002).

Lavender, S.

Lee, Z.-P.

Legbandt, T.

Levesque, M. P.

M. P. Levesque, “Dynamic sea image generation,” Proc. SPIE 1486, 294–300 (1991).
[Crossref]

M. P. Levesque and D. St-Germain, “Generation of synthetic IR sea images,” Proc. SPIE 1331, 354–357 (1990).

Longuet-Higgins, M. S.

M. S. Longuet-Higgins, D. E. Cartwright, and N. D. Smith, “Observations of the directional spectrum of sea waves using the motions of a flotation buoy,” in Ocean Wave Spectra (Prentice-Hall, 1963), pp. 111–136.

Lucker, P. L.

Luker, P. L.

P.-W. Zhai, Y. Hu, J. Chowdhary, C. R. Trepte, P. L. Luker, and D. B. Josset, “A vector radiative transfer model for coupled atmosphere and ocean systems with a rough interface,” J. Quant. Spectrosc. Radiat. Transfer 111, 1025–1040 (2010).

Massel, S. R.

S. R. Massel, “On the geometry of ocean surface waves,” Oceanologia 53, 521–548 (2011).

Mayer, B.

C. Emde, R. Buras, B. Mayer, and M. Blumthaler, “The impact of aerosols on polarized sky radiance: model development, validation, and applications,” Atmos. Chem. Phys. 10, 383–396 (2010).
[Crossref]

McLean, J. W.

J. W. McLean and J. D. Freeman, “Effects of ocean waves on airborne lidar imaging,” Appl. Opt. 35, 3261–3269 (1996).
[Crossref]

J. W. McLean, “Modeling of ocean wave effects for LIDAR remote sensing,” Proc. SPIE 1302, 480–491 (1990).
[Crossref]

Mélin, F.

G. Zibordi, B. N. Holben, I. Slutsker, D. Giles, D. D’Alimonte, F. Mélin, J. F. Berthon, D. Vandemark, H. Feng, G. Schuster, B. Fabbri, S. Kaitala, and J. Seppale, “AERONET-OC: a network for the validation of ocean color primary radiometric products,” J. Atmos. Ocean. Technol. 26, 1634–1651 (2009).
[Crossref]

Mischenko, M. I.

M. I. Mischenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002).

Mobley, C. D.

C. D. Mobley and E. S. Boss, “Improved irradiances for use in ocean heating, primary production, and photo-oxidation calculations,” Appl. Opt. 51, 6549–6560 (2012).
[Crossref]

Z.-P. Lee, Y.-H. Ahn, C. D. Mobley, and R. Arnone, “Removal of surface-reflected light for the measurement of remote-sensing reflectance from an above-surface platform,” Opt. Express 18, 26313–26324 (2010).
[Crossref]

C. D. Mobley, “Estimation of the remote-sensing reflectance from above-surface measurements,” Appl. Opt. 38, 7442–7455 (1999).
[Crossref]

R. W. Preisendorfer and C. D. Mobley, “Albedos and glitter patterns of a wind-roughened sea surface,” J. Phys. Oceanogr. 16, 1293–1316 (1986).

C. D. Mobley, “Modeling sea surfaces: a tutorial on Fourier transform techniques,” (Sequoia Scientific, Inc., 2014).

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

C. D. Mobley, “HydroPol mathematical documentation: invariant imbedding theory for the vector radiative transfer equation,” (Sequoia Scientific, 2014).

Moshary, F.

Munk, W.

Nimmo-Smith, A.

Preisendorfer, R. W.

R. W. Preisendorfer and C. D. Mobley, “Albedos and glitter patterns of a wind-roughened sea surface,” J. Phys. Oceanogr. 16, 1293–1316 (1986).

Prikhach, A. S.

Santer, R.

Schuster, G.

G. Zibordi, B. N. Holben, I. Slutsker, D. Giles, D. D’Alimonte, F. Mélin, J. F. Berthon, D. Vandemark, H. Feng, G. Schuster, B. Fabbri, S. Kaitala, and J. Seppale, “AERONET-OC: a network for the validation of ocean color primary radiometric products,” J. Atmos. Ocean. Technol. 26, 1634–1651 (2009).
[Crossref]

Schutgens, N. A. J.

L. G. Tilstra, N. A. J. Schutgens, and P. Stammes, “Analytical calculation of Stokes parameters Q and U of atmospheric radiation,” (Koninklijk Nederlands Meteorologisch Instituut, 2003).

Seppale, J.

G. Zibordi, B. N. Holben, I. Slutsker, D. Giles, D. D’Alimonte, F. Mélin, J. F. Berthon, D. Vandemark, H. Feng, G. Schuster, B. Fabbri, S. Kaitala, and J. Seppale, “AERONET-OC: a network for the validation of ocean color primary radiometric products,” J. Atmos. Ocean. Technol. 26, 1634–1651 (2009).
[Crossref]

Slutsker, I.

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[Crossref]

St-Germain, D.

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Tilstra, L. G.

L. G. Tilstra, N. A. J. Schutgens, and P. Stammes, “Analytical calculation of Stokes parameters Q and U of atmospheric radiation,” (Koninklijk Nederlands Meteorologisch Instituut, 2003).

Tonizzo, A.

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P.-W. Zhai, Y. Hu, J. Chowdhary, C. R. Trepte, P. L. Luker, and D. B. Josset, “A vector radiative transfer model for coupled atmosphere and ocean systems with a rough interface,” J. Quant. Spectrosc. Radiat. Transfer 111, 1025–1040 (2010).

P.-W. Zhai, Y. Hu, C. R. Trepte, and P. L. Lucker, “A vector radiative transfer model for coupled atmosphere and ocean systems based on successive order of scattering method,” Opt. Express 17, 2057–2079 (2009).
[Crossref]

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Tynes, H. H.

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G. Zibordi, B. N. Holben, I. Slutsker, D. Giles, D. D’Alimonte, F. Mélin, J. F. Berthon, D. Vandemark, H. Feng, G. Schuster, B. Fabbri, S. Kaitala, and J. Seppale, “AERONET-OC: a network for the validation of ocean color primary radiometric products,” J. Atmos. Ocean. Technol. 26, 1634–1651 (2009).
[Crossref]

T. Elfouhaily, B. Chapron, K. Katsaros, and D. Vandemark, “A unified directional spectrum for long and short wind-driven waves,” J. Geophys. Res. 102, 15781–15796 (1997).
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Yang, P.

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[Crossref]

P.-W. Zhai, Y. Hu, J. Chowdhary, C. R. Trepte, P. L. Luker, and D. B. Josset, “A vector radiative transfer model for coupled atmosphere and ocean systems with a rough interface,” J. Quant. Spectrosc. Radiat. Transfer 111, 1025–1040 (2010).

P.-W. Zhai, Y. Hu, C. R. Trepte, and P. L. Lucker, “A vector radiative transfer model for coupled atmosphere and ocean systems based on successive order of scattering method,” Opt. Express 17, 2057–2079 (2009).
[Crossref]

P.-W. Zhai, G. W. Kattawar, and P. Yang, “Impulse response solution of the three-dimensional vector radiative transfer equation in atmosphere-ocean systems. I. Monte Carlo method,” Appl. Opt. 47, 1037–1047 (2008).
[Crossref]

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S. B. Hooker, G. Zibordi, J. F. Berthon, and J. W. Brown, “Above-water radiometry in shallow coastal waters,” Appl. Opt. 43, 4254–4268 (2004).
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H. M. Tulldahl and K. O. Steinvall, “Simulation of sea surface wave influence on small target detection with airborne laser depth sounding,” Appl. Opt. 43, 2462–2483 (2004).
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[Crossref]

J. Chowdhary, B. Cairns, and L. D. Travis, “Contribution of water-leaving radiances to multiangle, multispectral polarimetric observations over the open ocean: bio-optical model results for case 1 waters,” Appl. Opt. 45, 5542–5567 (2006).
[Crossref]

P.-W. Zhai, G. W. Kattawar, and P. Yang, “Impulse response solution of the three-dimensional vector radiative transfer equation in atmosphere-ocean systems. I. Monte Carlo method,” Appl. Opt. 47, 1037–1047 (2008).
[Crossref]

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S. B. Hooker, G. Zibordi, J. F. Berthon, and J. W. Brown, “Above-water radiometry in shallow coastal waters,” Appl. Opt. 43, 4254–4268 (2004).
[Crossref]

T. Harmel, A. Gilerson, S. Hlaing, A. Tonizzo, T. Legbandt, A. Weidemann, R. Arnone, and S. Ahmed, “Long Island Sound Coastal Observatory: assessment of above-water radiometric measurement uncertainties using collocated multi and hyperspectral systems,” Appl. Opt. 50, 5842–5860 (2011).
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[Crossref]

J. Atmos. Ocean. Technol. (1)

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[Crossref]

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T. Elfouhaily, B. Chapron, K. Katsaros, and D. Vandemark, “A unified directional spectrum for long and short wind-driven waves,” J. Geophys. Res. 102, 15781–15796 (1997).
[Crossref]

J. Opt. Soc. Am. (1)

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[Crossref]

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[Crossref]

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[Crossref]

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Oceanologia (1)

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Opt. Express (3)

Proc. SPIE (3)

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[Crossref]

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C. D. Mobley, “Surface reflectance factors,” http://www.oceanopticsbook.info/view/remote_sensing/level_3/surface_reflectance_factors (2013).

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J. Tessendorf, “Simulating ocean water,” (SIGGRAPH Course Notes, 2004).

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

C. D. Mobley, “HydroPol mathematical documentation: invariant imbedding theory for the vector radiative transfer equation,” (Sequoia Scientific, 2014).

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Figures (19)

Fig. 1.
Fig. 1. Example sampling of elevation and slope spectra. The red dots on the lines in the upper panels show the bounds of the elevation and slope spectra sampled using 1024 points with L = 200 m . The light red lines are the true spectra S , and the truncated heavy blue lines are the adjusted spectra S ˜ obtained from Eqs. (9 )–(11). The dashed lines at 370 rad / m indicate the conventional boundary between gravity and capillary waves. The lower panels show random realizations of short segments along the x direction of the sea surface elevation and slope for the true (light red line) and adjusted (heavy blue line) spectra.
Fig. 2.
Fig. 2. Approach to fully resolved mean square slopes (diamonds, left ordinate) and elevations (circles, right ordinate) as a function of the number of grid points N , for winds speeds of W = 5 (red) and 10 m s 1 (blue).
Fig. 3.
Fig. 3. 2D example sea surface random realization for L x = L y = 100 m and N x = N y = 512 and a wind speed of 10 m s 1 in the + x direction. Light color is high surface elevation; dark color is low elevation. Inset numbers give the significant wave height H 1 / 3 , mean square slopes (total, mss; along wind, mss x ; and cross wind, mss y ), and mean surface tilt angles from the vertical in the along-wind ( θ x ) and cross-wind ( θ y ) directions.
Fig. 4.
Fig. 4. Example mapping of a rectangular FFT grid (light blue lines) to a hexagonal grid of triangles as used in ray tracing. Red dots show the triangle vertices. FFT grid points at the right and top, shown by the dotted lines, are obtained by the inherent spatial periodicity of the surface as determined by FFT techniques. Thus, the surface elevation of point 35 is the same as that of point 27, point 57 is the same as point 1, etc. One of the triangles generated from the FFT grid is shaded in red.
Fig. 5.
Fig. 5. Four cases for determining Stokes vector rotation angles. v 1 is the initial direction, which is rotated into the final direction v 2 by a counterclockwise rotation about direction ξ .
Fig. 6.
Fig. 6. Reflected energy pattern (glitter pattern) for unpolarized air-incident light at a 50 deg incident angle from the zenith and a 10 m s 1 wind speed. The light source is in an otherwise black sky. The viewing direction is looking downward at the surface, facing the glitter pattern. The quad outlined in gray indicates the specular reflection quad that would receive all reflected light for a level surface. The tan color in the lower-right panel indicates values that are identically zero.
Fig. 7.
Fig. 7. Transmitted energy pattern for air-incident light corresponding to Fig. 6. The viewing direction is looking upward at the surface from within the water.
Fig. 8.
Fig. 8. Reflected energy pattern (underwater glitter pattern) for water-incident light at a 50 deg incident angle from the nadir and a 10 m s 1 wind speed. The light source is unpolarized in an otherwise black ocean.
Fig. 9.
Fig. 9. Transmitted energy pattern for water-incident light corresponding to Fig. 8. The viewing direction is looking downward at the surface from the air side.
Fig. 10.
Fig. 10. Dependence of energy reflectance along the midline of the glitter pattern ( φ = 0 ) in Fig. 6, as a function of the number of grid points N x used for surface generation.
Fig. 11.
Fig. 11. Percent of incident rays that undergo multiple interactions with the surface, as a function of wind speed and incident angle from the zenith, for FFT and Cox–Munk surfaces.
Fig. 12.
Fig. 12. Sky radiance distribution for single scattering by atmospheric molecules according to the Rayleigh scattering equations. The upper-left panel shows the total radiance magnitude I relative to 1 in the Sun’s direction. The Sun’s location is at the left side of the plotted hemisphere of sky directions. The other panels show the horizontal versus vertical, ± 45 deg , and degree of total polarization in percent.
Fig. 13.
Fig. 13. Surface-reflected radiance for a single-scattering Rayleigh sky and a wind speed of 10 m s 1 . The Sun zenith angle is 50 deg and the viewing geometry corresponds to Fig. 6.
Fig. 14.
Fig. 14. Surface-transmitted radiance for a single-scattering Rayleigh sky and a wind speed of 10 m s 1 . The Sun zenith angle is 50 deg, and the viewing geometry corresponds to Fig. 7.
Fig. 15.
Fig. 15. Comparison of surface irradiance reflectances R surf as functions of solar zenith angle and wind speed for a polarized sky and various combinations of sea surface model (FFT versus Cox–Munk), ray tracing (polarized versus unpolarized), single versus multiple scattering, Sun azimuthal angle, and wave age.
Fig. 16.
Fig. 16. Radiance reflectance factors ρ ( θ v , ϕ v ) for a level sea surface and a Sun zenith angle of 50 deg. The top panel is for unpolarized ray tracing; the bottom panel is for polarized ray tracing. In both cases, the incident radiance is that of the single-scattering Rayleigh sky shown in Fig. 12. The dashed and dotted lines show the Fresnel reflectance for incident radiance, which is linearly polarized perpendicular ( R ) or parallel ( R ) to the incident meridian plane. The solid line without symbols is the Fresnel reflectance for unpolarized incident radiance.
Fig. 17.
Fig. 17. Radiance reflectance factors ρ computed using Cox–Munk surfaces and unpolarized ray tracing. The curves are for Sun zenith angles θ sun = 30 and 60 deg, azimuthal viewing directions of ϕ v = 90 and 135 deg, and wind speeds of 2, 5, 10, and 15 m s 1 ; θ v is the off-nadir viewing direction.
Fig. 18.
Fig. 18. Radiance reflectance factors ρ computed using FFT surfaces and polarized ray tracing. The solid curves correspond to the viewing and wave conditions of Fig. 17. These curves are for a fully developed sea and the Sun’s incident rays parallel to the wind speed ( ϕ sun = 0 ). The dotted curves and diamond symbols show the curves for θ sun = 30 and 60 deg, azimuthal viewing direction of ϕ v = 135 , but with the Sun’s azimuthal angle at ϕ sun = 90 , so that the incoming rays are perpendicular to the wind direction. The dashed curves and box symbols show the cases of θ sun = 30 and 60 deg azimuthal viewing direction of ϕ v = 135 , ϕ sun = 0 , for a very young sea with Ω c = 5 .
Fig. 19.
Fig. 19. ρ for a Rayleigh sky and a wind speed of 10 m s 1 . The upper-left panel shows ρ as a function of off-nadir and azimuthal viewing directions (relative to the Sun’s azimuth at ϕ v = 0 ) for a single Sun zenith angle of θ sun = 40 deg , computed using FFT surfaces and polarized ray tracing. The upper-right panel shows ρ as a function of the Sun’s zenith angle (radial distance in the plot) and viewing azimuth for a fixed off-nadir viewing direction of θ v = 40 deg . The black dots indicate the ρ ( θ v , ϕ v ) = ( 40 , 135 ) viewing direction. The white regions indicate the specular viewing direction for θ sun = 40 deg . The lower two panels show the ratios of ρ computed for FFT surfaces and polarized ray tracing to the values computed for Cox–Munk surfaces and unpolarized ray tracing, corresponding to the respective left and right upper panels.

Tables (2)

Tables Icon

Table 1. Mean Square Slopes as Given by Cox–Munk Formulas in Eq. (1) and by Random Realizations of FFT Surfaces a

Tables Icon

Table 2. Differences in Surface Irradiance Reflectance R surf as Computed in Various Ways a

Equations (41)

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σ a 2 = 0.003 + 0.00316 W 12.5 ± 0.002 ,
σ c 2 = 0.00192 W 12.5 ± 0.004 ,
σ 2 = 0.003 + 0.00512 W 12.5 ± 0.004 ,
var { z } = z 2 = Ψ ( k x , k y ) d k x d k y ,
Ψ 1 S ( k , φ ) = 1 k S ( k ) Φ ( k , φ ) .
Φ ( k , φ ) = 1 2 π [ 1 + Δ ( k ) cos ( 2 φ ) ] .
σ a 2 = k x 2 Ψ 2 S ( k x , k y ) d k x d k y = 0 π π k 2 cos 2 φ Ψ 2 S ( k , φ ) k d k d φ .
σ 2 = σ a 2 + σ c 2 = 0 k 2 S ( k ) d k .
z ^ o ( k uv ) 1 2 [ ρ ( k uv ) + i σ ( k uv ) ] × [ Ψ 1 S ( k = k uv ) 2 Δ k x Δ k y ] 1 / 2 ,
= 1 2 [ ρ ( k uv ) + i σ ( k uv ) ] Ψ 2 S ( k uv ) .
z ^ o ( u , v ) z ^ o * ( u , v ) = { 1 2 [ ρ ( u , v ) + i σ ( u , v ) ] Ψ 2 S ( u , v ) } × { 1 2 [ ρ ( u , v ) i σ ( u , v ) ] Ψ 2 S ( u , v ) } = Ψ 2 S ( u , v ) 2 [ ρ 2 ( u , v ) + σ 2 ( u , v ) ] = Ψ 2 S ( u , v ) ,
z ^ ( k uv , t ) 1 2 [ z ^ o ( + k uv ) exp ( i ω uv t ) + z ^ o * ( k uv ) exp ( + i ω uv t ) ]
z 2 = k 0 k h S ( k ) d k ,
σ 2 = k 0 k h k 2 S ( k ) d k .
f E z 2 ( N ) z 2 = 0.4219 0.4296 = 0.982 ,
S ˜ ( k ) [ 1 + δ ( k ) ] S ( k ) ,
δ ( k ) { 0 if k k p δ Ny ( k k p k Ny k p ) if k > k p ,
σ 2 k 0 k h k 2 S ( k ) d k = k 0 k Ny k 2 S ( k ) d k + k Ny k h k 2 S ( k ) d k k 0 k Ny k 2 S ˜ ( k ) d k = k 0 k Ny k 2 S ( k ) d k + δ Ny k p k Ny k 2 ( k k p k Ny k p ) S ( k ) d k .
δ Ny = k Ny k h k 2 S ( k ) d k k p k Ny k 2 ( k k p k Ny k p ) S ( k ) d k .
h i = z × ξ i | z × ξ i | , v i = ξ i × h i , ξ i = h i × v i .
s = ξ i × n | ξ i × n | , p = ξ i × s , ξ i = s × p .
α i = | cos 1 ( v 1 · v 2 ) | = | cos 1 ( h i · s ) | .
α i = 2 π | cos 1 ( v 1 · v 2 ) | = 2 π | cos 1 ( h i · s ) | .
R ̲ ( α ) = [ 1 0 0 0 0 cos 2 α sin 2 α 0 0 sin 2 α cos 2 α 0 0 0 0 1 ] .
S ̲ f = R ̲ ( α m , f ) M ̲ ( ψ m ) R ̲ ( α m 1 , m ) R ̲ ( α 1 , 2 ) M ̲ ( ψ 1 ) R ̲ ( α i , 1 ) S ̲ i ,
S ̲ f E ̲ ( ξ i ξ f ) S ̲ i ,
R ̲ ( Q i j Q k l ) = E ̲ ( Q i j Q k l ) | μ i | Ω i j | μ k | Ω k l .
E ̲ = [ X X X 0 X X X 0 X X X 0 0 0 0 X ] ,
E ̲ = [ X X X 0 X X X X X X X X 0 X X X ] ,
S ̲ refl ( Q k l ) = i j R ̲ aw ( Q i j Q k l ) S ̲ sky ( Q i j ) ,
E d ( sky ) = i j I sky ( Q i j ) μ i Ω i j .
R ̲ ( 40 , ϕ , 40 , ϕ ) = [ 2.566 × 10 2 1.963 × 10 2 3.521 × 10 7 0.0 1.963 × 10 2 2.566 × 10 2 7.489 × 10 7 0.0 3.521 × 10 7 7.489 × 10 7 1.616 × 10 2 0.0 0.0 0.0 0.0 1.616 × 10 2 ] .
S ̲ sky ( 40 , 90 ) = [ 4.931 × 10 2 2.403 × 10 2 1.583 × 10 2 0.0 ] ,
S ̲ sky ( 40 , 135 ) = [ 3.932 × 10 2 1.418 × 10 2 3.262 × 10 2 0.0 ] .
S ̲ sr ( 40 , 90 ) = [ 9.547 × 10 4 5.617 × 10 4 3.883 × 10 4 0.0 ]
S ̲ sr ( 40 , 135 ) = [ 1.287 × 10 3 1.136 × 10 3 5.269 × 10 4 0.0 ] .
ρ ( 40 , 90 ) = I sr I sky = 9.547 × 10 4 4.931 × 10 2 = 0.0194 ρ ( 40 , 135 ) = I sr I sky = 1.287 × 10 3 3.932 × 10 2 = 0.0327 ,
S ̲ sr ( 40 , 90 ) = [ 1.265 × 10 3 , 0 , 0 , 0 ] T
S ̲ sr ( 40 , 135 ) = [ 1.009 × 10 3 , 0 , 0 , 0 ] T .
ρ ( 40 , 90 ) = 1.265 × 10 3 4.931 × 10 2 = 0.0257
ρ ( 40 , 135 ) = 1.009 × 10 3 3.932 × 10 2 = 0.0257 ,

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