Abstract

It is shown that where μ̅s is the average cosine of scattering, then for any set of photons that undergoes exactly n scatterings per photon, the average cosine after scattering is μ̅0μ̅s n, where μ̅0 is the average cosine of the photon flux before scattering. For a set of photons that has traversed distance d through a medium with scattering coefficient b, the average cosine is μ̅0 exp[-bd(1 - μ̅s)]. For water bodies in which loss of upward-scattered photons through the surface is small enough to be disregarded, the value of μ̅c (the average cosine of all the photons instantaneously present in the water column) for any given incoming flux of photons with average cosine μ̅0 is determined entirely by the inherent optical properties of the water in accordance with μ̅c= μ̅0/[1 + (b/a)(1 - μ̅s)], where a and b are the absorption and scattering coefficients.

© 1999 Optical Society of America

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References

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  1. J. T. O. Kirk, “Volume scattering function, average cosines, and the underwater light field,” Limnol. Oceanogr. 36, 455–467 (1991).
    [CrossRef]
  2. R. A. Maffione, K. J. Voss, R. C. Honey, “Measurement of the spectral absorption coefficient in the ocean with an isotropic source,” Appl. Opt. 32, 3273–3279 (1993).
    [CrossRef] [PubMed]
  3. R. A. Maffione, J. S. Jaffe, “The average cosine due to an isotropic light source in the ocean,” J. Geophys. Res. 100, 13,179–13,192 (1995).
    [CrossRef]
  4. R. W. Preisendorfer, “Application of radiative transfer theory to light measurements in the sea,” Int. Union Geodesy Geophys. Monogr. 10, 11–30 (1961).
  5. N. G. Jerlov, Marine Optics (Elsevier, Amsterdam, 1976).
  6. J. T. O. Kirk, “Dependence of relationship between inherent and apparent optical properties of water on solar altitude,” Limnol. Oceanogr. 29, 350–356 (1984).
    [CrossRef]
  7. J. Berwald, D. Stramski, C. D. Mobley, D. A. Kiefer, “Influences of absorption and scattering on vertical changes in the average cosine of the underwater light field,” Limnol. Oceanogr. 40, 1347–1357 (1995).
    [CrossRef]
  8. T. J. Petzold, “Volume scattering functions for selected ocean waters,” (Scripps Institution of Oceanography, University of California, San Diego, La Jolla, Calif., 1972).
  9. J. T. O. Kirk, “Characteristics of the light field in highly turbid waters: a Monte Carlo study,” Limnol. Oceanogr. 39, 702–706 (1994).
    [CrossRef]

1995 (2)

R. A. Maffione, J. S. Jaffe, “The average cosine due to an isotropic light source in the ocean,” J. Geophys. Res. 100, 13,179–13,192 (1995).
[CrossRef]

J. Berwald, D. Stramski, C. D. Mobley, D. A. Kiefer, “Influences of absorption and scattering on vertical changes in the average cosine of the underwater light field,” Limnol. Oceanogr. 40, 1347–1357 (1995).
[CrossRef]

1994 (1)

J. T. O. Kirk, “Characteristics of the light field in highly turbid waters: a Monte Carlo study,” Limnol. Oceanogr. 39, 702–706 (1994).
[CrossRef]

1993 (1)

1991 (1)

J. T. O. Kirk, “Volume scattering function, average cosines, and the underwater light field,” Limnol. Oceanogr. 36, 455–467 (1991).
[CrossRef]

1984 (1)

J. T. O. Kirk, “Dependence of relationship between inherent and apparent optical properties of water on solar altitude,” Limnol. Oceanogr. 29, 350–356 (1984).
[CrossRef]

1961 (1)

R. W. Preisendorfer, “Application of radiative transfer theory to light measurements in the sea,” Int. Union Geodesy Geophys. Monogr. 10, 11–30 (1961).

Berwald, J.

J. Berwald, D. Stramski, C. D. Mobley, D. A. Kiefer, “Influences of absorption and scattering on vertical changes in the average cosine of the underwater light field,” Limnol. Oceanogr. 40, 1347–1357 (1995).
[CrossRef]

Honey, R. C.

Jaffe, J. S.

R. A. Maffione, J. S. Jaffe, “The average cosine due to an isotropic light source in the ocean,” J. Geophys. Res. 100, 13,179–13,192 (1995).
[CrossRef]

Jerlov, N. G.

N. G. Jerlov, Marine Optics (Elsevier, Amsterdam, 1976).

Kiefer, D. A.

J. Berwald, D. Stramski, C. D. Mobley, D. A. Kiefer, “Influences of absorption and scattering on vertical changes in the average cosine of the underwater light field,” Limnol. Oceanogr. 40, 1347–1357 (1995).
[CrossRef]

Kirk, J. T. O.

J. T. O. Kirk, “Characteristics of the light field in highly turbid waters: a Monte Carlo study,” Limnol. Oceanogr. 39, 702–706 (1994).
[CrossRef]

J. T. O. Kirk, “Volume scattering function, average cosines, and the underwater light field,” Limnol. Oceanogr. 36, 455–467 (1991).
[CrossRef]

J. T. O. Kirk, “Dependence of relationship between inherent and apparent optical properties of water on solar altitude,” Limnol. Oceanogr. 29, 350–356 (1984).
[CrossRef]

Maffione, R. A.

R. A. Maffione, J. S. Jaffe, “The average cosine due to an isotropic light source in the ocean,” J. Geophys. Res. 100, 13,179–13,192 (1995).
[CrossRef]

R. A. Maffione, K. J. Voss, R. C. Honey, “Measurement of the spectral absorption coefficient in the ocean with an isotropic source,” Appl. Opt. 32, 3273–3279 (1993).
[CrossRef] [PubMed]

Mobley, C. D.

J. Berwald, D. Stramski, C. D. Mobley, D. A. Kiefer, “Influences of absorption and scattering on vertical changes in the average cosine of the underwater light field,” Limnol. Oceanogr. 40, 1347–1357 (1995).
[CrossRef]

Petzold, T. J.

T. J. Petzold, “Volume scattering functions for selected ocean waters,” (Scripps Institution of Oceanography, University of California, San Diego, La Jolla, Calif., 1972).

Preisendorfer, R. W.

R. W. Preisendorfer, “Application of radiative transfer theory to light measurements in the sea,” Int. Union Geodesy Geophys. Monogr. 10, 11–30 (1961).

Stramski, D.

J. Berwald, D. Stramski, C. D. Mobley, D. A. Kiefer, “Influences of absorption and scattering on vertical changes in the average cosine of the underwater light field,” Limnol. Oceanogr. 40, 1347–1357 (1995).
[CrossRef]

Voss, K. J.

Appl. Opt. (1)

Int. Union Geodesy Geophys. Monogr. (1)

R. W. Preisendorfer, “Application of radiative transfer theory to light measurements in the sea,” Int. Union Geodesy Geophys. Monogr. 10, 11–30 (1961).

J. Geophys. Res. (1)

R. A. Maffione, J. S. Jaffe, “The average cosine due to an isotropic light source in the ocean,” J. Geophys. Res. 100, 13,179–13,192 (1995).
[CrossRef]

Limnol. Oceanogr. (4)

J. T. O. Kirk, “Characteristics of the light field in highly turbid waters: a Monte Carlo study,” Limnol. Oceanogr. 39, 702–706 (1994).
[CrossRef]

J. T. O. Kirk, “Volume scattering function, average cosines, and the underwater light field,” Limnol. Oceanogr. 36, 455–467 (1991).
[CrossRef]

J. T. O. Kirk, “Dependence of relationship between inherent and apparent optical properties of water on solar altitude,” Limnol. Oceanogr. 29, 350–356 (1984).
[CrossRef]

J. Berwald, D. Stramski, C. D. Mobley, D. A. Kiefer, “Influences of absorption and scattering on vertical changes in the average cosine of the underwater light field,” Limnol. Oceanogr. 40, 1347–1357 (1995).
[CrossRef]

Other (2)

T. J. Petzold, “Volume scattering functions for selected ocean waters,” (Scripps Institution of Oceanography, University of California, San Diego, La Jolla, Calif., 1972).

N. G. Jerlov, Marine Optics (Elsevier, Amsterdam, 1976).

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Tables (2)

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Table 1 Effect of Multiple Scattering on the Average Cosine of a Flux of Photonsa

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Table 2 Comparison of Values of the Integral Average Cosinea

Equations (45)

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KEz=aμ¯z.
μ¯z=Edz-EuzE0z,
μ¯s=4π β˜θcos θdω
μ¯s=0π β˜θcos θ 2π sin θdθ.
cos θ=cos θ0 cos α+sin θ0 sin α cos ϕ.
μ¯α=12π02πcos θ0 cos α+sin θ0 sin α cos ϕdϕ=cos θ0 cos α.
μ¯α, 0  π=μ¯α1P1+μ¯α2P2++μ¯αiPi++μ¯αNPN,
Pi=β˜αi2π sin αiΔα,
μ¯α, 0  π=i=1Ncos θ0  cos αi β˜αi2π sin αiΔα=α=0πcos θ0 cos α β˜α2π sin αΔα.=cos θ00π β˜αcos α 2π sin αdα.
μ¯α, 0  π=cos θ0μ¯s.
μ¯n+1=cos θ1μ¯sp1+cos θ2μ¯sp2++cos θiμ¯spi++cos θMμ¯spM=μ¯scos θ1p1+cos θ2p2++cos θipi++cos θMpM.
μ¯n+1=μ¯nμ¯s.
μ¯n=μ¯0μ¯sn.
exp-bcwΔt-l.
P1, l  l+δl=exp-blbδl exp-bcwΔt-l=exp-bcwΔtbδl,
P1=0cwΔtexp-bcwΔtbdl=bcwΔt exp-bcwΔt.
P1, l=bl exp-bl.
P2, l  l+δl=bl exp-blbδl exp-bcwΔt-l=exp-bcwΔtb2lδl.
P2=b2 exp-bcwΔt0cwΔt ldl=bcwΔt22exp-bcwΔt.
P3=bcwΔt36exp-bcwΔt,
Pn=bcwΔtnn!exp-bcwΔt.
0cwΔt Nbdl=NbcwΔt
n¯=bcwΔt,
n¯=0P0+1P1+2P2++nPn+=n=1 nbcwΔtnn!exp-bcwΔt,
μ¯Δt=P0μ¯0+P1μ¯1+P2μ¯2++Pnμ¯n+.
μ¯Δt=exp-bcwΔtμ¯0+bcwΔt exp-bcwΔtμ¯0μ¯s+bcwΔt22!exp-bcwΔtμ¯0μ¯s2++bcwΔtnn!exp-bcwΔtμ¯0μ¯sn+,
μ¯Δt=μ¯0 exp-bcwΔt1+bcwΔtμ¯s+bcwΔtμ¯s22!++bcwΔtμ¯snn!+=μ¯0 exp-bcwΔtexpbcwΔtμ¯s=μ¯0 exp-bcwΔt1-μ¯s
μ¯d=μ¯0 exp-bd1-μ¯s,
μ¯n¯=μ¯0 exp-n¯1-μ¯s,
μ¯ibcwΔt=μ¯0 exp-ibcwΔt1-μ¯s.
μ¯c=1NcNμ¯0 exp-acwΔtexp-bcwΔt1-μ¯s+Nμ¯0 exp-2acwΔtexp-2bcwΔt1-μ¯s++Nμ¯0 exp-iacwΔtexp-ibcwΔt1-μ¯s+=Nμ¯0Ncexp-cwΔta+b1-μ¯s+exp-2cwΔta+b1-μ¯s++exp-icwΔta+b1-μ¯s+,
Nc=N exp-acwΔt+N exp-2acwΔt+N exp-3acwΔt++N exp-iacwΔt+=N exp-acwΔt1+exp-acwΔt+exp-acwΔt2++exp-acwΔti-1+.
Nc=N exp-acwΔt1-exp-acwΔt.
Nc=N1-acwΔt1-1-acwΔt=NΔt1-acwΔtacw.
Nc=Ed0acw1-acwΔt.
Nc=Ed0acw.
μ¯c=μ¯0acwNEd0exp-cwΔta+b1-μ¯s×1+exp-cwΔta+b1-μ¯s+exp-cwΔta+b1-μ¯s2++exp-cwΔta+b1-μ¯si-1+.
μ¯0=μ¯0acwNEd0exp-cwΔta+b1-μ¯s1-exp-cwΔta+b1-μ¯s.
μ¯c=μ¯0acwNEd01-cwΔta+b1-μ¯scwΔta+b1-μ¯s=μ¯0aEd0NΔt1-cwΔta+b1-μ¯sa+b1-μ¯s.
μ¯c=μ¯0a1-cwΔta+b1-μ¯sa+b1-μ¯s,
μ¯c=μ¯0aa+b1-μ¯s,
μ¯c=μ¯01+ba1-μ¯s.
μ¯c=0 μ¯zUzdz0 Uzdz,
μ¯c=0Edz-Euzdz0 E0zdz,
μ¯c=ΣEdz-ΣEuzΣE0z,

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