Abstract

Classical radiative transfer programs are based on the plane-parallel assumption. We show that the Gershun equation is valid if the irradiance is averaged over a sufficiently large area. We show that the equation is invalid for horizontal areas of the order of tens of meters in which horizontal gradients of irradiance in the presence of waves are much larger than vertical gradients. We calculate the distribution of irradiance beneath modeled two-dimensional surface waves. We show that many of the features typically observed in irradiance profiles can be explained by use of such models. We derive a method for determination of the diffuse attenuation coefficient that is based on the upward integration of the irradiance field beneath waves, starting at a depth at which the irradiance profile is affected only weakly by waves.

© 2001 Optical Society of America

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References

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  1. R. E. Walker, Marine Light Field Statistics (Wiley, New York, 1994).
  2. J. L. Mueller, R. W. Austin, “Ocean optics protocols for SeaWiFS validation, revision 1,” NASA Tech. Memo. 104566, Vol. 25S. B. Hooker, E. R. Firestone, J. G. Acker, eds. (NASA Goddard Space Flight Center, Greenbelt, Md., 1995).
  3. A. Gershun, “On the theory of the light field in a scattering medium,” C. R. (Dok.) Acad. Sci. URSS 49, 556–557 (1945).
  4. J. Lighthill, Waves in Fluid (Cambridge U. Press, Cambridge, UK, 1978).
  5. C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, San Diego, Calif., 1994).
  6. J. W. McLean, J. D. Freeman, “Effects of ocean waves on airborne lidar imaging,” Appl. Opt. 35, 3261–3269 (1996).
    [CrossRef] [PubMed]
  7. W. H. Wells, “Theory of small angle scattering,” in Optics of the Sea, AGARD Lecture Series No. 61 (Advisory Group for Aerospace Research and Development, Neuilly-Sur-Seine, France, 1973), Sect. 3-3.
  8. J. R. V. Zaneveld, “New developments of the theory of radiative transfer in the oceans,” in Optical Aspects of Oceanography, N. G. Jerlov, E. Nielsen, eds. (Academic, London, 1974), pp. 121–134.
  9. M. Stramska, T. D. Dickey, “Short-term variability of the underwater light field in the oligotrophic ocean in response to surface waves and clouds,” Deep-Sea Res. I 45, 1393–1410 (1998).
    [CrossRef]

1998 (1)

M. Stramska, T. D. Dickey, “Short-term variability of the underwater light field in the oligotrophic ocean in response to surface waves and clouds,” Deep-Sea Res. I 45, 1393–1410 (1998).
[CrossRef]

1996 (1)

1945 (1)

A. Gershun, “On the theory of the light field in a scattering medium,” C. R. (Dok.) Acad. Sci. URSS 49, 556–557 (1945).

Austin, R. W.

J. L. Mueller, R. W. Austin, “Ocean optics protocols for SeaWiFS validation, revision 1,” NASA Tech. Memo. 104566, Vol. 25S. B. Hooker, E. R. Firestone, J. G. Acker, eds. (NASA Goddard Space Flight Center, Greenbelt, Md., 1995).

Dickey, T. D.

M. Stramska, T. D. Dickey, “Short-term variability of the underwater light field in the oligotrophic ocean in response to surface waves and clouds,” Deep-Sea Res. I 45, 1393–1410 (1998).
[CrossRef]

Freeman, J. D.

Gershun, A.

A. Gershun, “On the theory of the light field in a scattering medium,” C. R. (Dok.) Acad. Sci. URSS 49, 556–557 (1945).

Lighthill, J.

J. Lighthill, Waves in Fluid (Cambridge U. Press, Cambridge, UK, 1978).

McLean, J. W.

Mobley, C. D.

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

Mueller, J. L.

J. L. Mueller, R. W. Austin, “Ocean optics protocols for SeaWiFS validation, revision 1,” NASA Tech. Memo. 104566, Vol. 25S. B. Hooker, E. R. Firestone, J. G. Acker, eds. (NASA Goddard Space Flight Center, Greenbelt, Md., 1995).

Stramska, M.

M. Stramska, T. D. Dickey, “Short-term variability of the underwater light field in the oligotrophic ocean in response to surface waves and clouds,” Deep-Sea Res. I 45, 1393–1410 (1998).
[CrossRef]

Walker, R. E.

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

Wells, W. H.

W. H. Wells, “Theory of small angle scattering,” in Optics of the Sea, AGARD Lecture Series No. 61 (Advisory Group for Aerospace Research and Development, Neuilly-Sur-Seine, France, 1973), Sect. 3-3.

Zaneveld, J. R. V.

J. R. V. Zaneveld, “New developments of the theory of radiative transfer in the oceans,” in Optical Aspects of Oceanography, N. G. Jerlov, E. Nielsen, eds. (Academic, London, 1974), pp. 121–134.

Appl. Opt. (1)

C. R. (Dok.) Acad. Sci. URSS (1)

A. Gershun, “On the theory of the light field in a scattering medium,” C. R. (Dok.) Acad. Sci. URSS 49, 556–557 (1945).

Deep-Sea Res. I (1)

M. Stramska, T. D. Dickey, “Short-term variability of the underwater light field in the oligotrophic ocean in response to surface waves and clouds,” Deep-Sea Res. I 45, 1393–1410 (1998).
[CrossRef]

Other (6)

W. H. Wells, “Theory of small angle scattering,” in Optics of the Sea, AGARD Lecture Series No. 61 (Advisory Group for Aerospace Research and Development, Neuilly-Sur-Seine, France, 1973), Sect. 3-3.

J. R. V. Zaneveld, “New developments of the theory of radiative transfer in the oceans,” in Optical Aspects of Oceanography, N. G. Jerlov, E. Nielsen, eds. (Academic, London, 1974), pp. 121–134.

J. Lighthill, Waves in Fluid (Cambridge U. Press, Cambridge, UK, 1978).

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

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

J. L. Mueller, R. W. Austin, “Ocean optics protocols for SeaWiFS validation, revision 1,” NASA Tech. Memo. 104566, Vol. 25S. B. Hooker, E. R. Firestone, J. G. Acker, eds. (NASA Goddard Space Flight Center, Greenbelt, Md., 1995).

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Figures (8)

Fig. 1
Fig. 1

Irradiance profile taken off the Oregon coast, September 1997, with a Satlantic irradiance profiler. The profiler drops at approximately 0.8 m/s and samples at 8 Hz. Conditions were calm.

Fig. 2
Fig. 2

Irradiance profile taken in the Gulf of California, October 1998, with the same profiler as used in Fig. 1. Conditions were calm.

Fig. 3
Fig. 3

Irradiance pattern beneath a random wave surface with a 1.1-m dominant wave: a = 0.06 m-1 and b = 0.15 m-1. A Petzold volume-scattering function was used (see text). Crosses are sampling points of an irradiance sensor dropping at 0.8 m/s with a sampling rate of 8 Hz.

Fig. 4
Fig. 4

Irradiance pattern beneath a generic sine wave. The vertical extent of the basic diamond pattern is a function of the wavelength and wave steepness only (see text).

Fig. 5
Fig. 5

Downwelling irradiance field beneath a sea surface composed of a gravity and a capillary wave. The gravity wave has a wavelength of 2.25 m and an amplitude of 0.1 m. The capillary wave has a wavelength of 0.05 m and an amplitude of 0.002 m.

Fig. 6
Fig. 6

Irradiance pattern beneath a superposition of sinusoidal waves with wavelengths of 2.25, 0.2, and 0.05 m and with amplitudes of 0.1, 0.01, and 0.002 m. Note that the addition of very small amplitude waves significantly alters the irradiance pattern.

Fig. 7
Fig. 7

Superposition of 20 modeled irradiance profiles with random offsets. a = 0.1 m-1 and b = 0 m-1. The surface consists of two waves with wavelengths of 2.25 and 0.2 m and amplitudes of 0.29 and 0.01 cm, respectively.

Fig. 8
Fig. 8

Modeled irradiance profile with waves of wavelengths 3, 1.2, 0.6, and 0.05 m and amplitudes of 0.1, 0.05, 0.01, and 0.0006 m, respectively. a = 0.07 m-1, b = 0.39 m-1, offset is 0.355 m, and drop speed is 0.6 m/s.

Equations (15)

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

·Ex, y, z=-ax, y, zE0x, y, z,
dEzdz=-azE0z,
KzEz=azE0z,
azKz=EzE0z=μ¯z,
x Ex, y, z  z Ex, y, z, yEx, y, z  z Ex, y, z.
x Ex, z+z Ex, z=-azE0x, z,
Ex1, z-Ex2, z+x1x2z Ex, zdx=-azx1x2 E0x, zdx.
Ex1, z-Ex2, zΔx+zEz¯=-azE0z¯.
zEz¯=-azE0z¯.
zc=L2tanπ2+a sin1nsina tan2πAL-a tan2πAL,
zfL=2 AL+18LA,
I=z00Edz¯dz=Ed0¯KdexpKdz0-1.
I=z00 EdzdzEd0¯KdexpKdz0-1.
Iz=z0z EdzdzIz=Ed0¯KdexpKdz0-expKdz.
Rrs=fQbba,

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