Abstract

We present a statistical model that analytically quantifies the probability density function (PDF) of the downwelling light irradiance under random ocean waves modeling the surface as independent and identically distributed flat facets. The model can incorporate the separate effects of surface short waves and volume light scattering. The theoretical model captures the characteristics of the PDF, from skewed to near-Gaussian shape as the depth increases from shallow to deep water. The model obtains a closed-form asymptotic for the probability that diminishes at a rate between exponential and Gaussian with increasing extreme values. The model is validated by comparisons with existing field measurements and Monte Carlo simulation.

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  1. W. McFarland and E. Loew, “Wave produced changes in underwater light and their relations to vision,” Environ. Biol. Fish 8, 173–184 (1983).
    [CrossRef]
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    [CrossRef]
  5. H. R. Gordon, J. S. Smith, and O. B. Brown, “Spectra of underwater light-field fluctuations in the photic zone,” Bull. Mar. Sci 21, 466–470 (1971).
  6. J. Dera and D. Stramski, “Maximum effects of sunlight focusing under a wind-disturbed sea surface,” Oceanologia 23, 15–42 (1986).
  7. D. Stramski and L. Legendre, “Laboratory simulation of light focusing by water surface waves,” Mar. Biol 114, 341–348 (1992).
    [CrossRef]
  8. J. Dera, S. Sagan, and D. Stramski, “Focusing of sunlight by sea surface waves: new results from the Black Sea,” Oceanologia 34, 13–25 (1993).
  9. H. W. Wijesekera, W. S. Pegau, and T. J. Boyd, “Effect of surface waves on the irradiance distribution in the upper ocean,” Opt. Express 13(23), 9257–9264 (2005).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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  13. C. D. Mobley, Light and Water Radiative Transfer in Natural Waters (Academic Press, 1994).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  19. V. L. Veber, “On the spatial fluctuations of underwater illumination,” Izv. Atmos. Oceanic Phys. 18, 735–741 (1982).
  20. V. I. Shevernev, “Statistical structure of the illumination field under a wavy surface,” Izv. Atmos. Oceanic Phys. 18, 735–741 (1973).
  21. B. R. Frieden, Probability, Statistical Optics, and Data Testing: A Problem Solving Approach (Springer, 1983).
    [CrossRef]
  22. C. Cox and W. H. Munk, “Measurement of the roughness of the sea surface from photographs of the sun’s glitter,” J. Opt. Soc. Am. B 44(11), 838–850 (1954).
    [CrossRef]
  23. R. G. Gallager, Discrete Stochastic Processes (Springer, 1996).
  24. R. E. Walker, Marine Light Field Statistics (Wiley-Interscience, 1994).
  25. Z. Xu, “A DNS capability for obtaining underwater light field and retrieving upper ocean conditions via in-water light measurements,” thesis (MIT, 2011).

2010

2009

P. Gernez and D. Antoine, “Field characterization of wave-induced underwater light field fluctuations,” J. Geophys. Res. 114(15), C06025 (2009).
[CrossRef]

2008

2005

2001

1993

J. Dera, S. Sagan, and D. Stramski, “Focusing of sunlight by sea surface waves: new results from the Black Sea,” Oceanologia 34, 13–25 (1993).

1992

1986

J. Dera and D. Stramski, “Maximum effects of sunlight focusing under a wind-disturbed sea surface,” Oceanologia 23, 15–42 (1986).

1983

W. McFarland and E. Loew, “Wave produced changes in underwater light and their relations to vision,” Environ. Biol. Fish 8, 173–184 (1983).
[CrossRef]

1982

V. L. Veber, “On the spatial fluctuations of underwater illumination,” Izv. Atmos. Oceanic Phys. 18, 735–741 (1982).

1973

V. I. Shevernev, “Statistical structure of the illumination field under a wavy surface,” Izv. Atmos. Oceanic Phys. 18, 735–741 (1973).

1971

H. R. Gordon, J. S. Smith, and O. B. Brown, “Spectra of underwater light-field fluctuations in the photic zone,” Bull. Mar. Sci 21, 466–470 (1971).

1970

R. L. Snyder and J. Dera, “Wave-induced light-field fluctuations in sea,” J. Opt. Soc. Am. B 60(8), 1072–1083 (1970).
[CrossRef]

1954

C. Cox and W. H. Munk, “Measurement of the roughness of the sea surface from photographs of the sun’s glitter,” J. Opt. Soc. Am. B 44(11), 838–850 (1954).
[CrossRef]

Antoine, D.

P. Gernez and D. Antoine, “Field characterization of wave-induced underwater light field fluctuations,” J. Geophys. Res. 114(15), C06025 (2009).
[CrossRef]

Boss, E.

Boyd, T. J.

Brown, O. B.

H. R. Gordon, J. S. Smith, and O. B. Brown, “Spectra of underwater light-field fluctuations in the photic zone,” Bull. Mar. Sci 21, 466–470 (1971).

Brown, W. C.

Cox, C.

C. Cox and W. H. Munk, “Measurement of the roughness of the sea surface from photographs of the sun’s glitter,” J. Opt. Soc. Am. B 44(11), 838–850 (1954).
[CrossRef]

Darecki, M.

Y. You, D. Stramski, M. Darecki, and G. W. Kattawar, “Modeling of wave-induced irradiance fluctuations at near-surface depths in the ocean: a comparison with measurements,” Appl. Opt. 49(6), 1041–1053 (2010).
[CrossRef] [PubMed]

P. Gernez, D. Stramski, and M. Darecki, “Vertical changes in the probability distribution of downwelling irradiance within the near-surface ocean under clear sky conditions,” presented at Ocean Optics XX, Anchorage, Alaska, 27 September 2010.

Dera, J.

J. Dera, S. Sagan, and D. Stramski, “Focusing of sunlight by sea surface waves: new results from the Black Sea,” Oceanologia 34, 13–25 (1993).

J. Dera and D. Stramski, “Maximum effects of sunlight focusing under a wind-disturbed sea surface,” Oceanologia 23, 15–42 (1986).

R. L. Snyder and J. Dera, “Wave-induced light-field fluctuations in sea,” J. Opt. Soc. Am. B 60(8), 1072–1083 (1970).
[CrossRef]

Freeman, J. D.

J. W. McLean and J. D. Freeman, “Effects of ocean waves on airborne lidar imaging,” Appl. Opt.35(18), 3261–3269 (1996).
[CrossRef] [PubMed]

Frieden, B. R.

B. R. Frieden, Probability, Statistical Optics, and Data Testing: A Problem Solving Approach (Springer, 1983).
[CrossRef]

Gallager, R. G.

R. G. Gallager, Discrete Stochastic Processes (Springer, 1996).

Gernez, P.

P. Gernez and D. Antoine, “Field characterization of wave-induced underwater light field fluctuations,” J. Geophys. Res. 114(15), C06025 (2009).
[CrossRef]

P. Gernez, D. Stramski, and M. Darecki, “Vertical changes in the probability distribution of downwelling irradiance within the near-surface ocean under clear sky conditions,” presented at Ocean Optics XX, Anchorage, Alaska, 27 September 2010.

Gordon, H. R.

H. R. Gordon, J. S. Smith, and O. B. Brown, “Spectra of underwater light-field fluctuations in the photic zone,” Bull. Mar. Sci 21, 466–470 (1971).

Hedley, J.

Hwang, P. A.

Kattawar, G. W.

Legendre, L.

D. Stramski and L. Legendre, “Laboratory simulation of light focusing by water surface waves,” Mar. Biol 114, 341–348 (1992).
[CrossRef]

Loew, E.

W. McFarland and E. Loew, “Wave produced changes in underwater light and their relations to vision,” Environ. Biol. Fish 8, 173–184 (1983).
[CrossRef]

Majumdar, A. K.

McFarland, W.

W. McFarland and E. Loew, “Wave produced changes in underwater light and their relations to vision,” Environ. Biol. Fish 8, 173–184 (1983).
[CrossRef]

McLean, J. W.

J. W. McLean and J. D. Freeman, “Effects of ocean waves on airborne lidar imaging,” Appl. Opt.35(18), 3261–3269 (1996).
[CrossRef] [PubMed]

Mobley, C. D.

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

Munk, W. H.

C. Cox and W. H. Munk, “Measurement of the roughness of the sea surface from photographs of the sun’s glitter,” J. Opt. Soc. Am. B 44(11), 838–850 (1954).
[CrossRef]

Pegau, W. S.

Sagan, S.

J. Dera, S. Sagan, and D. Stramski, “Focusing of sunlight by sea surface waves: new results from the Black Sea,” Oceanologia 34, 13–25 (1993).

Shevernev, V. I.

V. I. Shevernev, “Statistical structure of the illumination field under a wavy surface,” Izv. Atmos. Oceanic Phys. 18, 735–741 (1973).

Smith, J. S.

H. R. Gordon, J. S. Smith, and O. B. Brown, “Spectra of underwater light-field fluctuations in the photic zone,” Bull. Mar. Sci 21, 466–470 (1971).

Snyder, R. L.

R. L. Snyder and J. Dera, “Wave-induced light-field fluctuations in sea,” J. Opt. Soc. Am. B 60(8), 1072–1083 (1970).
[CrossRef]

Stramski, D.

Y. You, D. Stramski, M. Darecki, and G. W. Kattawar, “Modeling of wave-induced irradiance fluctuations at near-surface depths in the ocean: a comparison with measurements,” Appl. Opt. 49(6), 1041–1053 (2010).
[CrossRef] [PubMed]

J. Dera, S. Sagan, and D. Stramski, “Focusing of sunlight by sea surface waves: new results from the Black Sea,” Oceanologia 34, 13–25 (1993).

D. Stramski and L. Legendre, “Laboratory simulation of light focusing by water surface waves,” Mar. Biol 114, 341–348 (1992).
[CrossRef]

J. Dera and D. Stramski, “Maximum effects of sunlight focusing under a wind-disturbed sea surface,” Oceanologia 23, 15–42 (1986).

P. Gernez, D. Stramski, and M. Darecki, “Vertical changes in the probability distribution of downwelling irradiance within the near-surface ocean under clear sky conditions,” presented at Ocean Optics XX, Anchorage, Alaska, 27 September 2010.

Veber, V. L.

V. L. Veber, “On the spatial fluctuations of underwater illumination,” Izv. Atmos. Oceanic Phys. 18, 735–741 (1982).

Walker, R. E.

R. E. Walker, Marine Light Field Statistics (Wiley-Interscience, 1994).

Wijesekera, H. W.

Xu, Z.

Z. Xu, “A DNS capability for obtaining underwater light field and retrieving upper ocean conditions via in-water light measurements,” thesis (MIT, 2011).

Yang, P.

You, Y.

Zaneveld, J. R. V.

Zhai, P.

Appl. Opt.

Bull. Mar. Sci

H. R. Gordon, J. S. Smith, and O. B. Brown, “Spectra of underwater light-field fluctuations in the photic zone,” Bull. Mar. Sci 21, 466–470 (1971).

Environ. Biol. Fish

W. McFarland and E. Loew, “Wave produced changes in underwater light and their relations to vision,” Environ. Biol. Fish 8, 173–184 (1983).
[CrossRef]

Izv. Atmos. Oceanic Phys.

V. L. Veber, “On the spatial fluctuations of underwater illumination,” Izv. Atmos. Oceanic Phys. 18, 735–741 (1982).

V. I. Shevernev, “Statistical structure of the illumination field under a wavy surface,” Izv. Atmos. Oceanic Phys. 18, 735–741 (1973).

J. Geophys. Res.

P. Gernez and D. Antoine, “Field characterization of wave-induced underwater light field fluctuations,” J. Geophys. Res. 114(15), C06025 (2009).
[CrossRef]

J. Opt. Soc. Am. B

C. Cox and W. H. Munk, “Measurement of the roughness of the sea surface from photographs of the sun’s glitter,” J. Opt. Soc. Am. B 44(11), 838–850 (1954).
[CrossRef]

R. L. Snyder and J. Dera, “Wave-induced light-field fluctuations in sea,” J. Opt. Soc. Am. B 60(8), 1072–1083 (1970).
[CrossRef]

Mar. Biol

D. Stramski and L. Legendre, “Laboratory simulation of light focusing by water surface waves,” Mar. Biol 114, 341–348 (1992).
[CrossRef]

Oceanologia

J. Dera, S. Sagan, and D. Stramski, “Focusing of sunlight by sea surface waves: new results from the Black Sea,” Oceanologia 34, 13–25 (1993).

J. Dera and D. Stramski, “Maximum effects of sunlight focusing under a wind-disturbed sea surface,” Oceanologia 23, 15–42 (1986).

Opt. Express

Other

B. R. Frieden, Probability, Statistical Optics, and Data Testing: A Problem Solving Approach (Springer, 1983).
[CrossRef]

R. G. Gallager, Discrete Stochastic Processes (Springer, 1996).

R. E. Walker, Marine Light Field Statistics (Wiley-Interscience, 1994).

Z. Xu, “A DNS capability for obtaining underwater light field and retrieving upper ocean conditions via in-water light measurements,” thesis (MIT, 2011).

P. Gernez, D. Stramski, and M. Darecki, “Vertical changes in the probability distribution of downwelling irradiance within the near-surface ocean under clear sky conditions,” presented at Ocean Optics XX, Anchorage, Alaska, 27 September 2010.

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

J. W. McLean and J. D. Freeman, “Effects of ocean waves on airborne lidar imaging,” Appl. Opt.35(18), 3261–3269 (1996).
[CrossRef] [PubMed]

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Figures (4)

Fig. 1
Fig. 1

Geometry of the method: faceted ocean surface.

Fig. 2
Fig. 2

Comparison of the GP theoretical model (—) for the probability density function (PDF) of the normalized downwelling irradiance χ = E/〈E〉 with Monte Carlo (MC) simulation (···) and experimental data (- - -) at different depths below the ocean surface: D= (a) 0.86m; (b) 1.7m; (c) 2.85m; (d) 4.72m. The conditions used correspond to those in the experiment case [11] with inherent optical properties (IOPs) (attenuation coefficient) c=0.6982 m−1, (absorption coefficient) a=0.0886 m−1, and (scattering coefficient) b=0.6096 m−1; wind speed at 10 meters above the ocean surface U 10 ≈5.5 m/s; solar zenith angle θs = 30°; and light wavelength 532 nm.

Fig. 3
Fig. 3

Standard deviation of the downwelling normalized irradiance σχ as a function of the scattering depth Dcω 0. Results of GP model (—) is compared with Monte Carlo (MC) simulations (···) and experimental data (⋄). The physical conditions are the same as in Fig. 2.

Fig. 4
Fig. 4

Wind speed effect on σχ . (a) Depth dependence of σχ for U 10=2m/s. (b) Dependence on wind speed for different depths: Dcω 0= 0.524 (—); 1.04 (- - -); 1.74 (- · - ·); and Dcω 0 → ∞ (···).

Equations (33)

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ξ i x i D = m 1 m s i [ 1 + O ( ɛ 2 ) ] ,
λ i = λ ¯ = L [ 1 + O ( ɛ 2 ) ] ,
α i = α ¯ = 1 + O ( ɛ 2 ) .
E = i = 1 N r E 0 α ¯ = N r E 0 [ 1 + O ( ɛ 2 ) ] .
χ = E E = N r N r ,
P ξ ¯ i ( λ ¯ 2 ξ i λ ¯ 2 ) = λ ¯ 2 λ ¯ 2 p ξ ¯ i ( ξ i ) d ξ i .
N r P ξ ¯ i ( λ ¯ 2 ξ i λ ¯ 2 ) ,
p N ¯ r ( N r ) = exp ( N r ) N r N r Γ ( N r + 1 ) ,
N r = i = P ξ ¯ i ( λ ¯ 2 ξ i λ ¯ 2 ) = 1 + O ( L σ ξ ) 2 ,
σ ξ = ( m 1 ) D m σ s ,
D * m 1 m D L σ s 1 .
p χ ¯ ( χ ) = exp ( 1 ) Γ ( χ + 1 ) .
R ( ζ ) = exp [ ζ 2 2 ( σ 2 / σ 4 ) 2 ] ,
l s = 2 σ 2 σ 4 .
D * = m 1 m D l s σ s 1 .
σ s S 2 = k * S η ( k ) k 2 d k .
˜ = D ( m 1 ) m σ s S ,
λ i = λ ˜ = 2 ˜ + λ ¯ = 2 ˜ + l s ,
α i = α ˜ = 2 ˜ / l s + α ¯ = 2 ˜ / l s + 1 .
N r = i = P ξ ¯ i ( λ ˜ / 2 ξ i λ ˜ / 2 ) = α ˜ [ 1 + O ( l s σ ξ ) 2 ] ,
p χ ¯ ( χ ) = exp ( α ˜ ) α ˜ α ˜ χ + 1 Γ ( α ˜ χ + 1 ) .
λ i = λ ^ = 2 ^ + l s , α i = α ^ = 2 ^ / l s + 1 ,
H ^ ( k , D ) = exp { D c ω 0 g 0 [ 1 1 exp ( D k α 0 D k α 0 ] } ,
^ = c g 0 ω 0 α 0 D 2 .
h ( ξ ) = h ˜ ( ξ ) * h ^ ( ξ ) ,
^ ˜ = ( ˜ 2 + ^ 2 ) 1 2 .
α = 2 ^ ˜ / l s + 1 .
p χ ¯ ( χ ) = exp ( 1 σ χ 2 ) ( 1 σ χ 2 ) χ / σ χ 2 + 1 Γ ( χ / σ χ 2 + 1 ) ,
σ χ 2 = 1 α .
Γ ( χ / σ χ 2 + 1 ) = 1 2 π ( χ / σ χ 2 = 1 ) χ / σ χ 2 + 1 2 e ( χ / σ χ 2 + 1 ) [ 1 + O ( 1 1 + χ / σ χ 2 ) ] .
p χ ¯ ( χ ) exp [ χ σ χ 2 ln ( χ σ χ 2 ) ] for χ / σ χ 2 1 .
σ χ 2 l s [ ( D 2 + ( c D 2 ) 2 ] 1 / 2 .
c ω 0 D c r = m 1 m σ s S g 0 α 0 .

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