Abstract

We develop a computationally fast radiative transfer model for simulating the fluctuations of the underwater downwelling irradiance Ed at near-surface depths, which occur due to focusing of sunlight by wind-driven surface waves. The model is based on the hybrid matrix operator–Monte Carlo method, which was specifically designed for simulating radiative transfer in a coupled atmosphere–surface–ocean system involving a dynamic ocean surface. In the current version of the model, we use a simplified description of surface waves, which accounts for surface slope statistics, but not surface wave elevation, as a direct source of underwater light fluctuations. We compare the model results with measurements made in the Santa Barbara Channel. The model-simulated and measured time series of Ed(t) show remarkable similarity. Major features of the probability distribution of instantaneous irradiance, the frequency content of irradiance fluctuations, and the statistical properties of light flashes produced by wave focusing are also generally consistent between the model simulations and measurements for a few near-surface depths and light wavelengths examined. Despite the simplification in the representation of surface waves, this model provides a reasonable first-order approximation to modeling the wave focusing effects at near-surface depths, which require high temporal and spatial resolution (of the order of 1ms and 1mm, respectively) to be adequately resolved.

© 2010 Optical Society of America

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References

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  1. J. Dera and J. Olszewski, “Experimental study of short-period irradiance fluctuations under an undulated sea surface,” Oceanologia 10, 27-49 (1978).
  2. J. Dera and D. Stramski, “Maximum effects of sunlight focusing under a wind-disturbed sea surface,” Oceanologia 23, 15-42 (1986).
  3. J. Dera, R. Hapter, and B. Malewicz, “Fluctuation of light in the euphotic zone and its influence on primary production,” Merentutkimuslait. Julk./Havsforskningsinst. 239, 58-66(1975).
  4. B. Quéguiner and L. Legendre, “Phytoplankton photosynthetic adaptation to high-frequency light fluctuations simulating those induced by sea-surface waves,” Mar. Biol. 90, 483-491 (1986).
    [CrossRef]
  5. D. Stramski, G. Rosenberg, and L. Legendre, “Photosynthetic and optical-properties of the marine chlorophyte Dunaliella tertiolecta grown under fluctuating light caused by surface-wave focusing,” Mar. Biol. 115, 363-372 (1993).
    [CrossRef]
  6. D. Stramski, “The effect of daylight diffuseness on the focusing of sunlight by sea surface waves,” Oceanologia 24, 11-27(1986).
  7. D. Stramski, “Fluctuations of solar irradiance induced by surface waves in the Baltic,” Bull. Pol. Acad. Sci. Earth Sci. 34, 333-344 (1986).
  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. Schenck, “On the focusing of sunlight by ocean waves,” J. Opt. Soc. Am. 47, 653-657 (1957).
    [CrossRef]
  10. J. R. V. Zaneveld, E. Boss, and A. Barnard, “Influence of surface waves on measured and modeled irradiance profiles,” Appl. Opt. 40, 1442-1449 (2001).
    [CrossRef]
  11. R. E. Walker, Marine Light Field Statistics (Wiley, 1994).
  12. R. L. Snyder and J. Dera, “Wave-induced light-field fluctuations in the sea,” J. Opt. Soc. Am. 60, 1072-1079 (1970).
    [CrossRef]
  13. P.-W. Zhai, G. W. Kattawar, and P. Yang, “Impulse response solution to the three-dimensional vector radiative transfer equation in atmosphere-ocean systems. I. Monte Carlo method,” Appl. Opt. 47, 1037-1047 (2008).
    [CrossRef] [PubMed]
  14. P.-W. Zhai, G. W. Kattawar, and P. Yang, “Impulse response solution to the three-dimensional vector radiative transfer equation in atmosphere-ocean systems. II. The hybrid matrix operator-Monte Carlo method,” Appl. Opt. 47, 1063-1071(2008).
    [CrossRef] [PubMed]
  15. R. Deckert and K. J. Michael, “Lensing effect on underwater levels of UV radiation,” J. Geophys. Res. 111, C05014 (2006).
    [CrossRef]
  16. G. N. Plass, G. W. Kattawar, and F. E. Catchings, “Matrix operator theory of radiative transfer. 1: Rayleigh scattering,” Appl. Opt. 12, 314-329 (1973).
    [CrossRef] [PubMed]
  17. G. W. Kattawar, G. N. Plass, and F. E. Catchings, “Matrix operator theory of radiative transfer. 2: Scattering from maritime haze,” Appl. Opt. 12, 1071-1084 (1973).
    [CrossRef] [PubMed]
  18. M. Darecki, D. Stramski, and M. Sokólski, “An Underwater Porcupine Radiometer System for measuring high-frequency fluctuations in light field induced by sea surface waves,” poster paper presented at the Ocean Optics XIX Conference (2008).
  19. C. Cox and W. Munk, “Statistics of the sea surface derived from sun glitter,” J. Mar. Res. 13, 198-227 (1954).
  20. G. R. Fournier and J. L. Forand, “Analytic phase function for ocean water,” Proc. SPIE 194-201 (1994).
    [CrossRef]
  21. J. E. Hansen and L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527-610 (1974).
    [CrossRef]
  22. M. St. Denis and W. J. Pierson, “On the motions of ships in confused seas,” Trans. Soc. Nav. Archit. Mar. Eng. 61, 280-357 (1953).
  23. W. J. Pierson and L. Moskowitz, “Proposed spectral form for fully developed wind seas based on similarity theory of S. A. Kitaigorodskii,” J. Geophys. Res. 69, 5181-5190 (1964).
    [CrossRef]
  24. P. A. Hwang and O. H. Shendin, 'The dependence of sea surface slope on atmospheric stability and swell conditions,” J. Geophys. Res. 93, 13903-13912 (1988).
    [CrossRef]
  25. D. Stramski and L. Legendre, “Laboratory simulation of light focusing by water surface waves,” Mar. Biol. 114, 341-348(1992).
    [CrossRef]
  26. 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).
  27. P. Gernez and D. Antoine, “Field characterization of wave-induced underwater light field fluctuations,” J. Geophys. Res. 114, C06025 (2009).
    [CrossRef]

2009 (1)

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

2008 (2)

2006 (1)

R. Deckert and K. J. Michael, “Lensing effect on underwater levels of UV radiation,” J. Geophys. Res. 111, C05014 (2006).
[CrossRef]

2001 (1)

1994 (1)

G. R. Fournier and J. L. Forand, “Analytic phase function for ocean water,” Proc. SPIE 194-201 (1994).
[CrossRef]

1993 (2)

D. Stramski, G. Rosenberg, and L. Legendre, “Photosynthetic and optical-properties of the marine chlorophyte Dunaliella tertiolecta grown under fluctuating light caused by surface-wave focusing,” Mar. Biol. 115, 363-372 (1993).
[CrossRef]

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 (1)

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

1988 (1)

P. A. Hwang and O. H. Shendin, 'The dependence of sea surface slope on atmospheric stability and swell conditions,” J. Geophys. Res. 93, 13903-13912 (1988).
[CrossRef]

1986 (4)

D. Stramski, “The effect of daylight diffuseness on the focusing of sunlight by sea surface waves,” Oceanologia 24, 11-27(1986).

D. Stramski, “Fluctuations of solar irradiance induced by surface waves in the Baltic,” Bull. Pol. Acad. Sci. Earth Sci. 34, 333-344 (1986).

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

B. Quéguiner and L. Legendre, “Phytoplankton photosynthetic adaptation to high-frequency light fluctuations simulating those induced by sea-surface waves,” Mar. Biol. 90, 483-491 (1986).
[CrossRef]

1978 (1)

J. Dera and J. Olszewski, “Experimental study of short-period irradiance fluctuations under an undulated sea surface,” Oceanologia 10, 27-49 (1978).

1975 (1)

J. Dera, R. Hapter, and B. Malewicz, “Fluctuation of light in the euphotic zone and its influence on primary production,” Merentutkimuslait. Julk./Havsforskningsinst. 239, 58-66(1975).

1974 (1)

J. E. Hansen and L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527-610 (1974).
[CrossRef]

1973 (2)

1971 (1)

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 (1)

1964 (1)

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

1957 (1)

1954 (1)

C. Cox and W. Munk, “Statistics of the sea surface derived from sun glitter,” J. Mar. Res. 13, 198-227 (1954).

1953 (1)

M. St. Denis and W. J. Pierson, “On the motions of ships in confused seas,” Trans. Soc. Nav. Archit. Mar. Eng. 61, 280-357 (1953).

Antoine, D.

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

Barnard, A.

Boss, E.

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).

Catchings, F. E.

Cox, C.

C. Cox and W. Munk, “Statistics of the sea surface derived from sun glitter,” J. Mar. Res. 13, 198-227 (1954).

Darecki, M.

M. Darecki, D. Stramski, and M. Sokólski, “An Underwater Porcupine Radiometer System for measuring high-frequency fluctuations in light field induced by sea surface waves,” poster paper presented at the Ocean Optics XIX Conference (2008).

Deckert, R.

R. Deckert and K. J. Michael, “Lensing effect on underwater levels of UV radiation,” J. Geophys. Res. 111, C05014 (2006).
[CrossRef]

Denis, M. St.

M. St. Denis and W. J. Pierson, “On the motions of ships in confused seas,” Trans. Soc. Nav. Archit. Mar. Eng. 61, 280-357 (1953).

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).

J. Dera and J. Olszewski, “Experimental study of short-period irradiance fluctuations under an undulated sea surface,” Oceanologia 10, 27-49 (1978).

J. Dera, R. Hapter, and B. Malewicz, “Fluctuation of light in the euphotic zone and its influence on primary production,” Merentutkimuslait. Julk./Havsforskningsinst. 239, 58-66(1975).

R. L. Snyder and J. Dera, “Wave-induced light-field fluctuations in the sea,” J. Opt. Soc. Am. 60, 1072-1079 (1970).
[CrossRef]

Forand, J. L.

G. R. Fournier and J. L. Forand, “Analytic phase function for ocean water,” Proc. SPIE 194-201 (1994).
[CrossRef]

Fournier, G. R.

G. R. Fournier and J. L. Forand, “Analytic phase function for ocean water,” Proc. SPIE 194-201 (1994).
[CrossRef]

Gernez, P.

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

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).

Hansen, J. E.

J. E. Hansen and L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527-610 (1974).
[CrossRef]

Hapter, R.

J. Dera, R. Hapter, and B. Malewicz, “Fluctuation of light in the euphotic zone and its influence on primary production,” Merentutkimuslait. Julk./Havsforskningsinst. 239, 58-66(1975).

Hwang, P. A.

P. A. Hwang and O. H. Shendin, 'The dependence of sea surface slope on atmospheric stability and swell conditions,” J. Geophys. Res. 93, 13903-13912 (1988).
[CrossRef]

Kattawar, G. W.

Legendre, L.

D. Stramski, G. Rosenberg, and L. Legendre, “Photosynthetic and optical-properties of the marine chlorophyte Dunaliella tertiolecta grown under fluctuating light caused by surface-wave focusing,” Mar. Biol. 115, 363-372 (1993).
[CrossRef]

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

B. Quéguiner and L. Legendre, “Phytoplankton photosynthetic adaptation to high-frequency light fluctuations simulating those induced by sea-surface waves,” Mar. Biol. 90, 483-491 (1986).
[CrossRef]

Malewicz, B.

J. Dera, R. Hapter, and B. Malewicz, “Fluctuation of light in the euphotic zone and its influence on primary production,” Merentutkimuslait. Julk./Havsforskningsinst. 239, 58-66(1975).

Michael, K. J.

R. Deckert and K. J. Michael, “Lensing effect on underwater levels of UV radiation,” J. Geophys. Res. 111, C05014 (2006).
[CrossRef]

Moskowitz, L.

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

Munk, W.

C. Cox and W. Munk, “Statistics of the sea surface derived from sun glitter,” J. Mar. Res. 13, 198-227 (1954).

Olszewski, J.

J. Dera and J. Olszewski, “Experimental study of short-period irradiance fluctuations under an undulated sea surface,” Oceanologia 10, 27-49 (1978).

Pierson, W. J.

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

M. St. Denis and W. J. Pierson, “On the motions of ships in confused seas,” Trans. Soc. Nav. Archit. Mar. Eng. 61, 280-357 (1953).

Plass, G. N.

Quéguiner, B.

B. Quéguiner and L. Legendre, “Phytoplankton photosynthetic adaptation to high-frequency light fluctuations simulating those induced by sea-surface waves,” Mar. Biol. 90, 483-491 (1986).
[CrossRef]

Rosenberg, G.

D. Stramski, G. Rosenberg, and L. Legendre, “Photosynthetic and optical-properties of the marine chlorophyte Dunaliella tertiolecta grown under fluctuating light caused by surface-wave focusing,” Mar. Biol. 115, 363-372 (1993).
[CrossRef]

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).

Schenck, H.

Shendin, O. H.

P. A. Hwang and O. H. Shendin, 'The dependence of sea surface slope on atmospheric stability and swell conditions,” J. Geophys. Res. 93, 13903-13912 (1988).
[CrossRef]

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.

Sokólski, M.

M. Darecki, D. Stramski, and M. Sokólski, “An Underwater Porcupine Radiometer System for measuring high-frequency fluctuations in light field induced by sea surface waves,” poster paper presented at the Ocean Optics XIX Conference (2008).

Stramski, D.

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, G. Rosenberg, and L. Legendre, “Photosynthetic and optical-properties of the marine chlorophyte Dunaliella tertiolecta grown under fluctuating light caused by surface-wave focusing,” Mar. Biol. 115, 363-372 (1993).
[CrossRef]

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

D. Stramski, “The effect of daylight diffuseness on the focusing of sunlight by sea surface waves,” Oceanologia 24, 11-27(1986).

D. Stramski, “Fluctuations of solar irradiance induced by surface waves in the Baltic,” Bull. Pol. Acad. Sci. Earth Sci. 34, 333-344 (1986).

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

M. Darecki, D. Stramski, and M. Sokólski, “An Underwater Porcupine Radiometer System for measuring high-frequency fluctuations in light field induced by sea surface waves,” poster paper presented at the Ocean Optics XIX Conference (2008).

Travis, L. D.

J. E. Hansen and L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527-610 (1974).
[CrossRef]

Walker, R. E.

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

Yang, P.

Zaneveld, J. R. V.

Zhai, P.-W.

Appl. Opt. (5)

Bull. Mar. Sci. (1)

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).

Bull. Pol. Acad. Sci. Earth Sci. (1)

D. Stramski, “Fluctuations of solar irradiance induced by surface waves in the Baltic,” Bull. Pol. Acad. Sci. Earth Sci. 34, 333-344 (1986).

J. Geophys. Res. (4)

R. Deckert and K. J. Michael, “Lensing effect on underwater levels of UV radiation,” J. Geophys. Res. 111, C05014 (2006).
[CrossRef]

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

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

P. A. Hwang and O. H. Shendin, 'The dependence of sea surface slope on atmospheric stability and swell conditions,” J. Geophys. Res. 93, 13903-13912 (1988).
[CrossRef]

J. Mar. Res. (1)

C. Cox and W. Munk, “Statistics of the sea surface derived from sun glitter,” J. Mar. Res. 13, 198-227 (1954).

J. Opt. Soc. Am. (2)

Mar. Biol. (3)

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

B. Quéguiner and L. Legendre, “Phytoplankton photosynthetic adaptation to high-frequency light fluctuations simulating those induced by sea-surface waves,” Mar. Biol. 90, 483-491 (1986).
[CrossRef]

D. Stramski, G. Rosenberg, and L. Legendre, “Photosynthetic and optical-properties of the marine chlorophyte Dunaliella tertiolecta grown under fluctuating light caused by surface-wave focusing,” Mar. Biol. 115, 363-372 (1993).
[CrossRef]

Merentutkimuslait. Julk./Havsforskningsinst. (1)

J. Dera, R. Hapter, and B. Malewicz, “Fluctuation of light in the euphotic zone and its influence on primary production,” Merentutkimuslait. Julk./Havsforskningsinst. 239, 58-66(1975).

Oceanologia (4)

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, “The effect of daylight diffuseness on the focusing of sunlight by sea surface waves,” Oceanologia 24, 11-27(1986).

J. Dera and J. Olszewski, “Experimental study of short-period irradiance fluctuations under an undulated sea surface,” Oceanologia 10, 27-49 (1978).

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

Proc. SPIE (1)

G. R. Fournier and J. L. Forand, “Analytic phase function for ocean water,” Proc. SPIE 194-201 (1994).
[CrossRef]

Space Sci. Rev. (1)

J. E. Hansen and L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527-610 (1974).
[CrossRef]

Trans. Soc. Nav. Archit. Mar. Eng. (1)

M. St. Denis and W. J. Pierson, “On the motions of ships in confused seas,” Trans. Soc. Nav. Archit. Mar. Eng. 61, 280-357 (1953).

Other (2)

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

M. Darecki, D. Stramski, and M. Sokólski, “An Underwater Porcupine Radiometer System for measuring high-frequency fluctuations in light field induced by sea surface waves,” poster paper presented at the Ocean Optics XIX Conference (2008).

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Figures (10)

Fig. 1
Fig. 1

Schematic diagrams of (a) the coupled atmosphere–surface–ocean system, (b) the atmospheric downwelling transmission response function T 01 ( ρ r , ρ i ) , (c) the upwelling reflection response function R 10 ( ρ r , ρ i ) , (d) the detector response function D 2 z ( ρ r , ρ i ) , (e) the coupled terms D 2 z · T 12 ( t ) · T 01 , and (f) the coupled terms D 2 z · T 12 ( t ) · R 10 · R 12 ( t ) · T 01 . The arguments ρ r and ρ i represent the position and direction associated with the response and impulse radiances, respectively.

Fig. 2
Fig. 2

Schematic diagram of the three parts of the fast HMOMC model. Thick solid arrows represent the refracted direct sunlight, thin solid arrows represent the oceanic-only diffuse radiance, and dashed arrows represent all other diffuse radiance contributions.

Fig. 3
Fig. 3

Time series of normalized downwelling irradiance E d ( t ) / E d (a) from the model simulations and (b) from field measurements in the Santa Barbara Channel. Shown are the results for the green spectral band, λ = 532 nm , and the three near-surface depths as indicated.

Fig. 4
Fig. 4

Modeled and measured probability density functions of the normalized downwelling irradiance E d ( t ) / E d at λ = 532 nm for the three near-surface depths as indicated.

Fig. 5
Fig. 5

(a) Comparison of the flash frequency N as a function of the threshold irradiance E d ( thr ) as computed from model simulations (circles) and from measurements (squares) at λ = 532 nm for the three near-surface depths as indicated. The solid curves show the corresponding exponential fits to the simulated data obtained from the least squares fitting and the dashed curves are the fits for the experimental data. (b) Model-simulated frequency of flashes as a function of the threshold irradiance for the three depths (circles) along with the error bars indicating the flash frequency for the additional depths that differ by 0.14 or 0.13 m from the three basic depths (see text for detailed explanation). The corresponding exponential fits for the flash frequency at the three basic depths (solid curves) and the additional depths (dashed curves) are also shown.

Fig. 6
Fig. 6

Comparison of the frequency of flashes at two threshold irradiances, (a)  N ( 1.5 E d ) and (b)  N ( 2 E d ) , calculated from the model simulations and the measurements. The results are shown for the three wavelengths and three near-surface depths as indicated.

Fig. 7
Fig. 7

Comparisons of the slope parameter A calculated from the model simulations and the measurements. The results are shown for the three wavelengths and three near-surface depths as indicated.

Fig. 8
Fig. 8

Comparison of the mean flash duration for the threshold irradiance 1.5 E d calculated from the model simulations and the measurements. The results are shown for the three wavelengths and three near-surface depths as indicated.

Fig. 9
Fig. 9

Comparison of the probability density functions of the flash durations for the threshold irradiance 1.5 E d as calculated from the model simulations and from the measurement at 532 nm . The results are shown for the three near-surface depths as indicated.

Fig. 10
Fig. 10

Power spectral densities of the irradiance time series at λ = 532 nm , computed from the measured irradiance time-series (gray curves) and the model-simulated irradiance time-series (black curves). The results are shown for the three near-surface depths as indicated.

Tables (1)

Tables Icon

Table 1 Ocean Inherent Optical Properties Used in the Simulations a

Equations (9)

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L r ( ρ r ) = ρ i F ( ρ r , ρ i ) · L i ( ρ i ) ,
L z ( ρ r ; t ) = ρ i D 0 z ( ρ r , ρ i ; t ) · L 0 ( ρ i ; t ) .
D 0 z ( t ) = D 2 z · T 12 ( t ) · T 01 + D 2 z · T 12 ( t ) · R 10 · R 12 ( t ) · T 01 + D 2 z · R 21 ( t ) · R 23 · T 12 ( t ) · T 01 + higher order terms ,
E d ( r r ; t ) = Ω d L z ( ρ r ; t ) | cos θ | d Ω ( n r ) ,
T 01 = T 01 , ( dir ) + T 01 , ( dif ) , D 2 z = D 2 z , ( dir ) + D 2 z , ( dif ) ,
D 0 z ( t ) = D 2 z , ( dir ) · T 12 ( t ) · T 01 , ( dir ) + D 2 z , ( dif ) · T 12 ( t ) · T 01 , ( dir ) + D 0 z , ( dif ) ,
h ( r , t ) = k S ( k ) 2 π cos 2 Θ e i ( k · r ω ( k ) t + Φ k ) ,
S ( k ) = a k 3 e b ( g / k W 2 ) 2 ,
N = N 0 e A E d ( thr ) ,

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