Abstract

The behavior of light at the air-sea interface has been investigated using ray tracing methods with numerically realized surfaces that incorporate features on scales from 3 millimeters to 200 meters. The directional reflection of light at the surface realizations was tested using Monte Carlo code. Estimated directionally reflected radiances were generally in good agreement with those from existing methods that model the slope statistics but not the shape of the sea surface. However, significant differences were found for some incident and exitant directions. The model was used to quantitatively estimate the pixel-to-pixel variation in ocean color images caused by spatial variation in the sea surface shape.

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2010

P. Zhai, Y. Hu, J. Chowdhary, C. R. Trepte, P. L. Lucker, and D. B. Josset, “A vector radiative transfer model for coupled atmosphere and ocean systems with a rough interface,” J. Quant. Spectrosc. Radiat. Transf. 111(7-8), 1025–1040 (2010).
[CrossRef]

W. J. Plant, W. C. Keller, K. Hayes, G. Chatham, and N. Lederer, “Normalized radar cross section of the sea for backscatter: 2. Modulation by internal waves,” J. Geophys. Res. 115(C9), C09033 (2010).
[CrossRef]

2009

F. Nouguier, C. Guérin, and B. Chapron, “““Choppy wave” model for nonlinear gravity waves,” J. Geophys. Res. 114(C9), C09012 (2009).
[CrossRef]

S. Kay, J. D. Hedley, and S. Lavender, “Sun glint correction of high and low spatial resolution images of aquatic scenes: a review of methods for visible and near-infrared wavelengths,” Remote Sens. 1(4), 697–730 (2009).
[CrossRef]

Y. You, P. W. Zhai, G. W. Kattawar, and P. Yang, “Polarized radiance fields under a dynamic ocean surface: a three-dimensional radiative transfer solution,” Appl. Opt. 48(16), 3019–3029 (2009).
[CrossRef] [PubMed]

S. Fauqueux, K. Caillault, P. Simoneau, and L. Labarre, “Multiresolution infrared optical properties for Gaussian sea surfaces: theoretical validation in the one-dimensional case,” Appl. Opt. 48(28), 5337–5347 (2009).
[CrossRef] [PubMed]

2008

S. Y. Kotchenova, E. F. Vermote, R. Levy, and A. Lyapustin, “Radiative transfer codes for atmospheric correction and aerosol retrieval: intercomparison study,” Appl. Opt. 47(13), 2215–2226 (2008).
[CrossRef] [PubMed]

M. Ottaviani, R. Spurr, K. Stamnes, W. Li, W. Su, and W. Wiscombe, “Improving the description of sunglint for accurate prediction of remotely sensed radiances,” J. Quant. Spectrosc. Radiat. Transf. 109(14), 2364–2375 (2008).
[CrossRef]

D. Hauser, G. Caudal, S. Guimbard, and A. A. Mouche, “A study of the slope probability density function of the ocean waves from radar observations,” J. Geophys. Res. 113(C2), C02006 (2008).
[CrossRef]

2007

V. Ross and D. Dion, “Sea surface slope statistics derived from Sun glint radiance measurements and their apparent dependence on sensor elevation,” J. Geophys. Res. 112(C9), C09015 (2007).
[CrossRef]

C. E. Long and D. T. Resio, “Wind wave spectral observations in Currituck Sound, North Carolina,” J. Geophys. Res. 112(C5), C05001 (2007).
[CrossRef]

2006

D. Lyzenga, N. Malinas, and F. Tanis, “Multispectral bathymetry using a simple physically based algorithm,” IEEE Trans. Geosci. Rem. Sens. 44(8), 2251–2259 (2006).
[CrossRef]

Z. Jin, T. P. Charlock, K. Rutledge, K. Stamnes, and Y. Wang, “Analytical solution of radiative transfer in the coupled atmosphere-ocean system with a rough surface,” Appl. Opt. 45(28), 7443–7455 (2006).
[CrossRef] [PubMed]

F. Bréon and N. Henriot, “Spaceborne observations of ocean glint reflectance and modeling of wave slope distributions,” J. Geophys. Res. 111(C6), C06005 (2006).
[CrossRef]

M. Heron, W. Skirving, and K. Michael, “Short-wave ocean wave slope models for use in remote sensing data analysis,” IEEE Trans. Geosci. Rem. Sens. 44(7), 1962–1973 (2006).
[CrossRef]

2005

M. Frigo and S. G. Johnson, “The Design and Implementation of FFTW3,” Proc. IEEE 93(2), 216–231 (2005).
[CrossRef]

2003

E. Hochberg, S. Andrefouet, and M. Tyler, “Sea surface correction of high spatial resolution Ikonos images to improve bottom mapping in near-shore environments,” IEEE Trans. Geosci. Rem. Sens. 41(7), 1724–1729 (2003).
[CrossRef]

F. Schwenger and E. Repasi, “Sea surface simulation for testing of multiband imaging sensors,” Proc. SPIE 5075, 72–84 (2003).
[CrossRef]

2001

J. R. Zaneveld, E. Boss, and P. Hwang, “The influence of coherent waves on the remotely sensed reflectance,” Opt. Express 9(6), 260–266 (2001).
[CrossRef] [PubMed]

D. Wang and P. Hwang, “Evolution of the bimodal directional distribution of ocean waves,” J. Phys. Oceanogr. 31(5), 1200–1221 (2001).
[CrossRef]

1998

K. Ewans, “Observations of the directional spectrum of fetch-limited waves,” J. Phys. Oceanogr. 28(3), 495–512 (1998).
[CrossRef]

1997

T. Elfouhaily, B. Chapron, K. Katsaros, and D. Vandemark, “A unified directional spectrum for long and short wind-driven waves,” J. Geophys. Res. 102(C7), 15781–15796 (1997).
[CrossRef]

1996

R. H. Grant, G. M. Heisler, and W. Gao, “Photosynthetically-active radiation: sky radiance distributions under clear and overcast conditions,” Agric. For. Meteorol. 82(1-4), 267–292 (1996).
[CrossRef]

1994

J. Apel, “An Improved Model of the Ocean Surface-Wave Vector Spectrum and Its Effects on Radar Backscatter,” J. Geophys. Res. 99(C8), 16269–16291 (1994).
[CrossRef]

1992

1988

E. Thorsos, “The validity of the Kirchhoff approximation for rough-surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 83(1), 78–92 (1988).
[CrossRef]

1986

R. Preisendorfer and C. Mobley, “Albedos and glitter patterns of a wind-roughened sea-surface,” J. Phys. Oceanogr. 16(7), 1293–1316 (1986).
[CrossRef]

1983

T. Nakajima and M. Tanaka, “Effect of wind-generated waves on the transfer of solar-radiation in the atmosphere ocean system,” J. Quant. Spectrosc. Radiat. Transf. 29(6), 521–537 (1983).
[CrossRef]

1975

1973

K. Hasselmann, T. Barnett, E. Bouws, D. E. Carlson, D. E. Cartwright, K. Enke, J. Ewing, and ., “Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP),” Ergnzungsheft zur Deutschen Hydrographischen Zeitschrift Reihe 8, 95 (1973).

1972

E. Raschke, “Multiple-scattering calculations of transfer of solar-radiation in an atmosphere-ocean system,” Bull. Am. Meteorol. Soc. 53, 501 (1972).

1969

M. Sancer, “Shadow-corrected electromagnetic scattering from a randomly rough surface,” IEEE Trans. Antenn. Propag. 17(5), 577–585 (1969).
[CrossRef]

1967

B. Smith, ““Geometrical shadowing of a random rough surface,” IEEE Trans. Antennas Propag, A 668–671, 15 (1967).

1964

W. Pierson and L. Moskowitz, “Proposed spectral form for fully developed wind seas based on similarity theory of S. A. Kitaigorodskii,” J. Geophys. Res. 69(24), 5181–5190 (1964).
[CrossRef]

1954

Andrefouet, S.

E. Hochberg, S. Andrefouet, and M. Tyler, “Sea surface correction of high spatial resolution Ikonos images to improve bottom mapping in near-shore environments,” IEEE Trans. Geosci. Rem. Sens. 41(7), 1724–1729 (2003).
[CrossRef]

Apel, J.

J. Apel, “An Improved Model of the Ocean Surface-Wave Vector Spectrum and Its Effects on Radar Backscatter,” J. Geophys. Res. 99(C8), 16269–16291 (1994).
[CrossRef]

Barnett, T.

K. Hasselmann, T. Barnett, E. Bouws, D. E. Carlson, D. E. Cartwright, K. Enke, J. Ewing, and ., “Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP),” Ergnzungsheft zur Deutschen Hydrographischen Zeitschrift Reihe 8, 95 (1973).

Boss, E.

Bouws, E.

K. Hasselmann, T. Barnett, E. Bouws, D. E. Carlson, D. E. Cartwright, K. Enke, J. Ewing, and ., “Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP),” Ergnzungsheft zur Deutschen Hydrographischen Zeitschrift Reihe 8, 95 (1973).

Bréon, F.

F. Bréon and N. Henriot, “Spaceborne observations of ocean glint reflectance and modeling of wave slope distributions,” J. Geophys. Res. 111(C6), C06005 (2006).
[CrossRef]

Caillault, K.

Carlson, D. E.

K. Hasselmann, T. Barnett, E. Bouws, D. E. Carlson, D. E. Cartwright, K. Enke, J. Ewing, and ., “Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP),” Ergnzungsheft zur Deutschen Hydrographischen Zeitschrift Reihe 8, 95 (1973).

Cartwright, D. E.

K. Hasselmann, T. Barnett, E. Bouws, D. E. Carlson, D. E. Cartwright, K. Enke, J. Ewing, and ., “Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP),” Ergnzungsheft zur Deutschen Hydrographischen Zeitschrift Reihe 8, 95 (1973).

Caudal, G.

D. Hauser, G. Caudal, S. Guimbard, and A. A. Mouche, “A study of the slope probability density function of the ocean waves from radar observations,” J. Geophys. Res. 113(C2), C02006 (2008).
[CrossRef]

Chapron, B.

F. Nouguier, C. Guérin, and B. Chapron, “““Choppy wave” model for nonlinear gravity waves,” J. Geophys. Res. 114(C9), C09012 (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(C7), 15781–15796 (1997).
[CrossRef]

Charlock, T. P.

Chatham, G.

W. J. Plant, W. C. Keller, K. Hayes, G. Chatham, and N. Lederer, “Normalized radar cross section of the sea for backscatter: 2. Modulation by internal waves,” J. Geophys. Res. 115(C9), C09033 (2010).
[CrossRef]

Chowdhary, J.

P. Zhai, Y. Hu, J. Chowdhary, C. R. Trepte, P. L. Lucker, and D. B. Josset, “A vector radiative transfer model for coupled atmosphere and ocean systems with a rough interface,” J. Quant. Spectrosc. Radiat. Transf. 111(7-8), 1025–1040 (2010).
[CrossRef]

Cox, C.

Dion, D.

V. Ross and D. Dion, “Sea surface slope statistics derived from Sun glint radiance measurements and their apparent dependence on sensor elevation,” J. Geophys. Res. 112(C9), C09015 (2007).
[CrossRef]

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(C7), 15781–15796 (1997).
[CrossRef]

Enke, K.

K. Hasselmann, T. Barnett, E. Bouws, D. E. Carlson, D. E. Cartwright, K. Enke, J. Ewing, and ., “Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP),” Ergnzungsheft zur Deutschen Hydrographischen Zeitschrift Reihe 8, 95 (1973).

Ewans, K.

K. Ewans, “Observations of the directional spectrum of fetch-limited waves,” J. Phys. Oceanogr. 28(3), 495–512 (1998).
[CrossRef]

Ewing, J.

K. Hasselmann, T. Barnett, E. Bouws, D. E. Carlson, D. E. Cartwright, K. Enke, J. Ewing, and ., “Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP),” Ergnzungsheft zur Deutschen Hydrographischen Zeitschrift Reihe 8, 95 (1973).

Fauqueux, S.

Frigo, M.

M. Frigo and S. G. Johnson, “The Design and Implementation of FFTW3,” Proc. IEEE 93(2), 216–231 (2005).
[CrossRef]

Gao, W.

R. H. Grant, G. M. Heisler, and W. Gao, “Photosynthetically-active radiation: sky radiance distributions under clear and overcast conditions,” Agric. For. Meteorol. 82(1-4), 267–292 (1996).
[CrossRef]

Gordon, H. R.

Grant, R. H.

R. H. Grant, G. M. Heisler, and W. Gao, “Photosynthetically-active radiation: sky radiance distributions under clear and overcast conditions,” Agric. For. Meteorol. 82(1-4), 267–292 (1996).
[CrossRef]

Guérin, C.

F. Nouguier, C. Guérin, and B. Chapron, “““Choppy wave” model for nonlinear gravity waves,” J. Geophys. Res. 114(C9), C09012 (2009).
[CrossRef]

Guimbard, S.

D. Hauser, G. Caudal, S. Guimbard, and A. A. Mouche, “A study of the slope probability density function of the ocean waves from radar observations,” J. Geophys. Res. 113(C2), C02006 (2008).
[CrossRef]

Guinn, J. A.

Hasselmann, K.

K. Hasselmann, T. Barnett, E. Bouws, D. E. Carlson, D. E. Cartwright, K. Enke, J. Ewing, and ., “Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP),” Ergnzungsheft zur Deutschen Hydrographischen Zeitschrift Reihe 8, 95 (1973).

Hauser, D.

D. Hauser, G. Caudal, S. Guimbard, and A. A. Mouche, “A study of the slope probability density function of the ocean waves from radar observations,” J. Geophys. Res. 113(C2), C02006 (2008).
[CrossRef]

Hayes, K.

W. J. Plant, W. C. Keller, K. Hayes, G. Chatham, and N. Lederer, “Normalized radar cross section of the sea for backscatter: 2. Modulation by internal waves,” J. Geophys. Res. 115(C9), C09033 (2010).
[CrossRef]

Hedley, J. D.

S. Kay, J. D. Hedley, and S. Lavender, “Sun glint correction of high and low spatial resolution images of aquatic scenes: a review of methods for visible and near-infrared wavelengths,” Remote Sens. 1(4), 697–730 (2009).
[CrossRef]

Heisler, G. M.

R. H. Grant, G. M. Heisler, and W. Gao, “Photosynthetically-active radiation: sky radiance distributions under clear and overcast conditions,” Agric. For. Meteorol. 82(1-4), 267–292 (1996).
[CrossRef]

Henriot, N.

F. Bréon and N. Henriot, “Spaceborne observations of ocean glint reflectance and modeling of wave slope distributions,” J. Geophys. Res. 111(C6), C06005 (2006).
[CrossRef]

Heron, M.

M. Heron, W. Skirving, and K. Michael, “Short-wave ocean wave slope models for use in remote sensing data analysis,” IEEE Trans. Geosci. Rem. Sens. 44(7), 1962–1973 (2006).
[CrossRef]

Hochberg, E.

E. Hochberg, S. Andrefouet, and M. Tyler, “Sea surface correction of high spatial resolution Ikonos images to improve bottom mapping in near-shore environments,” IEEE Trans. Geosci. Rem. Sens. 41(7), 1724–1729 (2003).
[CrossRef]

Hu, Y.

P. Zhai, Y. Hu, J. Chowdhary, C. R. Trepte, P. L. Lucker, and D. B. Josset, “A vector radiative transfer model for coupled atmosphere and ocean systems with a rough interface,” J. Quant. Spectrosc. Radiat. Transf. 111(7-8), 1025–1040 (2010).
[CrossRef]

Hwang, P.

D. Wang and P. Hwang, “Evolution of the bimodal directional distribution of ocean waves,” J. Phys. Oceanogr. 31(5), 1200–1221 (2001).
[CrossRef]

J. R. Zaneveld, E. Boss, and P. Hwang, “The influence of coherent waves on the remotely sensed reflectance,” Opt. Express 9(6), 260–266 (2001).
[CrossRef] [PubMed]

Jin, Z.

Johnson, S. G.

M. Frigo and S. G. Johnson, “The Design and Implementation of FFTW3,” Proc. IEEE 93(2), 216–231 (2005).
[CrossRef]

Josset, D. B.

P. Zhai, Y. Hu, J. Chowdhary, C. R. Trepte, P. L. Lucker, and D. B. Josset, “A vector radiative transfer model for coupled atmosphere and ocean systems with a rough interface,” J. Quant. Spectrosc. Radiat. Transf. 111(7-8), 1025–1040 (2010).
[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(C7), 15781–15796 (1997).
[CrossRef]

Kattawar, G. W.

Kay, S.

S. Kay, J. D. Hedley, and S. Lavender, “Sun glint correction of high and low spatial resolution images of aquatic scenes: a review of methods for visible and near-infrared wavelengths,” Remote Sens. 1(4), 697–730 (2009).
[CrossRef]

Keller, W. C.

W. J. Plant, W. C. Keller, K. Hayes, G. Chatham, and N. Lederer, “Normalized radar cross section of the sea for backscatter: 2. Modulation by internal waves,” J. Geophys. Res. 115(C9), C09033 (2010).
[CrossRef]

Kotchenova, S. Y.

Labarre, L.

Lavender, S.

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B. Smith, ““Geometrical shadowing of a random rough surface,” IEEE Trans. Antennas Propag, A 668–671, 15 (1967).

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E. Hochberg, S. Andrefouet, and M. Tyler, “Sea surface correction of high spatial resolution Ikonos images to improve bottom mapping in near-shore environments,” IEEE Trans. Geosci. Rem. Sens. 41(7), 1724–1729 (2003).
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D. Lyzenga, N. Malinas, and F. Tanis, “Multispectral bathymetry using a simple physically based algorithm,” IEEE Trans. Geosci. Rem. Sens. 44(8), 2251–2259 (2006).
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V. Ross and D. Dion, “Sea surface slope statistics derived from Sun glint radiance measurements and their apparent dependence on sensor elevation,” J. Geophys. Res. 112(C9), C09015 (2007).
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F. Bréon and N. Henriot, “Spaceborne observations of ocean glint reflectance and modeling of wave slope distributions,” J. Geophys. Res. 111(C6), C06005 (2006).
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Figures (10)

Fig. 1
Fig. 1

(a) Effective shape of the surface when using a slope-statistics method; the facets are orientated according to a given probability distribution but there is no fixed spatial scale. (b) Visualization of a surface created using the current method.

Fig. 2
Fig. 2

Ratio of mean square slopes in the upwind and crosswind directions calculated using data from sun glint observations by Cox and Munk [17] (circles), Ross and Dion [31] (crosses), and Bréon and Henriot [32] (dotted line); slope ratio calculated from the spectrum used in the current work with the spreading function of Heron et al. [33] (solid line) and Elfouhaily et al. [27] (dash-dot line).

Fig. 3
Fig. 3

Omnidirectional (a) slope and (b) elevation variance spectrum of Elfouhaily et al. [27], labeled to show the wavenumber range included in the current model.

Fig. 4
Fig. 4

(a) Visualization of a sea surface created using the method described above, with a wind of 3 m s−1 from the left. Image width is 50 m; grey levels show elevation. (b) Elevation against position for a 5 m line on the surface. (c) A 20 cm section of the line enlarged.

Fig. 5
Fig. 5

Wind speed dependence of (a) mean square slope and (b) elevation standard deviation of the surface realizations (blue crosses) compared to empirical models (red lines), the value obtained by integrating the spectrum across all wavenumbers (green) and by integration across the wavenumber range used in surface creation (purple). 7 surfaces were created for wind speeds 3 and 7 m s−1, 11 for 9, 11 and 13 m s−1, 19 for 10 m s−1, 20 for 5 m s−1 and 22 for 15 m s−1.

Fig. 6
Fig. 6

Diagram to illustrate the division of the sphere used in ray tracing. The lower half of the sphere is divided in the same way.

Fig. 7
Fig. 7

Reflected radiance predicted by the new model (blue) and the Cox-Munk slope-statistics model (CM, red) for wind speeds 5 and 10 m s−1 and solar zenith angle (a-d) −10°, (e-h) −30° and (i-l) −50°. The wind and the sun are aligned and illumination is for clear sky conditions. Neither model was azimuthally averaged. In each case “fwd” shows the reflected radiance in the plane of the sun and “cross” shows reflected radiance in the plane at 90° to the solar plane. Error bars show ± 1 standard error (n = 10 for both data sets).

Fig. 8
Fig. 8

Reflected radiance distribution estimated by (a, b) the new model and (c,d) the slope-statistics model. All results are for a clear sky scenario with wind speed 10 m s−1 and the wind aligned with the sun. In (a, c) the solar zenith angle is −10°, in (b, d) it is −30°.

Fig. 9
Fig. 9

Polar plots of reflected radiance as a function of viewing azimuth for viewing zenith 20°, solar zenith 10°, solar azimuth 180° and wind speed 5 m s−1. Wind direction is (a, d) 180° (b, e) −135° (c, f) −90°. (a-c) show the non-azimuthally averaged models, (d-f) show the azimuthally averaged versions. In both cases the blue line shows the new model and the red lines shows the slope statistics model (CM). Error bars show ± 1 standard error (n = 10). Note that the process of azimuthal averaging removes some of the variation between surfaces, so the standard errors in (d-f) are very small.

Fig. 10
Fig. 10

Upwelling radiance just above the surface as output by the new model, plotted as a function of viewing zenith angle for various wind speeds and directions. Incoming radiance is distributed as for a clear sky [40]; in (a-c) the sun is at zenith 30°, azimuth 135° to the viewing angle, and in (d-f) it is at zenith 70°, azimuth 150°. Wind speeds are (a,d) 5 m s−1, (b,e) 10 m s−1and (c,f) 15 m s−1. Wind-sun angles are 0° (red), 45° (green) and 90° (blue). Points ▼▲ show maximum and minimum radiance for 10 surfaces.

Equations (6)

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

η ( r , t ) = k X ( k , t ) e i k . r
X ( k ) = X o ( k ) + X o ( k )
X o ( k ) = 1 2 ( ξ r + i ξ i ) Ψ ( k ) Δ k x Δ k y
Ψ ( k ) = 1 k S E l f ( k ) D H e r o n ( k , θ )
mean square slope = 0.003 + 0.00512 U 10 ± 0.004
elevation standard deviation (m)  = 0.005 U 10 2

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