Abstract

Monte Carlo simulations of the transfer of radiation in the ocean are used to compute the apparent optical properties of a flat homogeneous ocean as a function of the inherent optical properties. The data are used to find general relationships between the inherent and apparent optical properties for optical depths τ ≤ 4. The results indicate that the apparent optical properties depend on the phase function only through the back scattering probability. It is shown that these relations can be used with measurements of the upwelling and downwelling irradiance, the beam attenuation coefficient, and the incident radiance distribution to determine the absorption coefficient, the scattering coefficient, and the backward and forward scattering probabilities.

© 1975 Optical Society of America

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References

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  1. T. J. Petzold, Volume Scattering Functions for Selected Waters (Scripps Institution of Oceanography, University of California at San Diego, 1972), SIO Ref. 72–78.
  2. N. G. Jerlov, Optical Oceanography (Elsevier, Amsterdam, 1968).
  3. J. E. Tyler, R. C. Smith, Measurements of Spectral Irradiance Underwater (Gordon and Breach, New York, 1970).
  4. R. W. Preisendorfer, U.G.G.I., Monogr. 10, 11 (1961).
  5. J. E. Tyler, R. C. Smith, W. H. Wilson, J. Opt. Soc. Am. 62, 83 (1972).
    [CrossRef]
  6. H. R. Gordon, Appl. Opt. 12, 2803 (1973).
    [CrossRef] [PubMed]
  7. In this work μ is always cosθ, and unprimed μ’s refer to rays beneath the sea surface, while primed μ’s refer to rays above the sea surface.
  8. H. R. Gordon, O. B. Brown, Appl. Opt. 12, 1544 (1973).
    [CrossRef]
  9. G. Kullenberg, Deep Sea Res. 15, 423 (1968).
  10. G. N. Plass, G. W. Kattawar, J. Binstock, J. Quant. Spectrosc. Radiat. Transfer. 13, 1081 (1973).
    [CrossRef]
  11. G. N. Plass, G. W. Kattawar, F. E. Catchings, Appl. Opt. 12, 314 (1973).
    [CrossRef] [PubMed]
  12. L. Prieur, A. Morel, Cahiers Ocean 23, 35 (1971).
  13. G. F. Beardsley, J. R. V. Zaneveld, J. Opt. Soc. Am. 59, 373 (1969).
    [CrossRef]
  14. C. Cox, W. Munk, J. Opt. Soc. Am. 44, 838 (1954).
    [CrossRef]
  15. G. Maul, H. R. Gordon. Relationships Between ERTS Radiances and Gradients across Oceanic Fronts, presented at the Third ERTS-1 Principal Investigator’s Symposium.
  16. H. R. Gordon, W. R. McCluney, Appl. Opt. 14, 413 (1975).
    [CrossRef] [PubMed]
  17. M. Herman, J. Lenoble, J. Quant. Spectrosc. Radiat. Transfer 8, 355 (1968).

1975

1973

1972

1971

L. Prieur, A. Morel, Cahiers Ocean 23, 35 (1971).

1969

1968

G. Kullenberg, Deep Sea Res. 15, 423 (1968).

M. Herman, J. Lenoble, J. Quant. Spectrosc. Radiat. Transfer 8, 355 (1968).

1961

R. W. Preisendorfer, U.G.G.I., Monogr. 10, 11 (1961).

1954

Beardsley, G. F.

Binstock, J.

G. N. Plass, G. W. Kattawar, J. Binstock, J. Quant. Spectrosc. Radiat. Transfer. 13, 1081 (1973).
[CrossRef]

Brown, O. B.

Catchings, F. E.

Cox, C.

Gordon, H. R.

H. R. Gordon, W. R. McCluney, Appl. Opt. 14, 413 (1975).
[CrossRef] [PubMed]

H. R. Gordon, Appl. Opt. 12, 2803 (1973).
[CrossRef] [PubMed]

H. R. Gordon, O. B. Brown, Appl. Opt. 12, 1544 (1973).
[CrossRef]

G. Maul, H. R. Gordon. Relationships Between ERTS Radiances and Gradients across Oceanic Fronts, presented at the Third ERTS-1 Principal Investigator’s Symposium.

Herman, M.

M. Herman, J. Lenoble, J. Quant. Spectrosc. Radiat. Transfer 8, 355 (1968).

Jerlov, N. G.

N. G. Jerlov, Optical Oceanography (Elsevier, Amsterdam, 1968).

Kattawar, G. W.

G. N. Plass, G. W. Kattawar, J. Binstock, J. Quant. Spectrosc. Radiat. Transfer. 13, 1081 (1973).
[CrossRef]

G. N. Plass, G. W. Kattawar, F. E. Catchings, Appl. Opt. 12, 314 (1973).
[CrossRef] [PubMed]

Kullenberg, G.

G. Kullenberg, Deep Sea Res. 15, 423 (1968).

Lenoble, J.

M. Herman, J. Lenoble, J. Quant. Spectrosc. Radiat. Transfer 8, 355 (1968).

Maul, G.

G. Maul, H. R. Gordon. Relationships Between ERTS Radiances and Gradients across Oceanic Fronts, presented at the Third ERTS-1 Principal Investigator’s Symposium.

McCluney, W. R.

Morel, A.

L. Prieur, A. Morel, Cahiers Ocean 23, 35 (1971).

Munk, W.

Petzold, T. J.

T. J. Petzold, Volume Scattering Functions for Selected Waters (Scripps Institution of Oceanography, University of California at San Diego, 1972), SIO Ref. 72–78.

Plass, G. N.

G. N. Plass, G. W. Kattawar, J. Binstock, J. Quant. Spectrosc. Radiat. Transfer. 13, 1081 (1973).
[CrossRef]

G. N. Plass, G. W. Kattawar, F. E. Catchings, Appl. Opt. 12, 314 (1973).
[CrossRef] [PubMed]

Preisendorfer, R. W.

R. W. Preisendorfer, U.G.G.I., Monogr. 10, 11 (1961).

Prieur, L.

L. Prieur, A. Morel, Cahiers Ocean 23, 35 (1971).

Smith, R. C.

J. E. Tyler, R. C. Smith, W. H. Wilson, J. Opt. Soc. Am. 62, 83 (1972).
[CrossRef]

J. E. Tyler, R. C. Smith, Measurements of Spectral Irradiance Underwater (Gordon and Breach, New York, 1970).

Tyler, J. E.

J. E. Tyler, R. C. Smith, W. H. Wilson, J. Opt. Soc. Am. 62, 83 (1972).
[CrossRef]

J. E. Tyler, R. C. Smith, Measurements of Spectral Irradiance Underwater (Gordon and Breach, New York, 1970).

Wilson, W. H.

Zaneveld, J. R. V.

Appl. Opt.

Cahiers Ocean

L. Prieur, A. Morel, Cahiers Ocean 23, 35 (1971).

Deep Sea Res.

G. Kullenberg, Deep Sea Res. 15, 423 (1968).

J. Opt. Soc. Am.

J. Quant. Spectrosc. Radiat. Transfer

M. Herman, J. Lenoble, J. Quant. Spectrosc. Radiat. Transfer 8, 355 (1968).

J. Quant. Spectrosc. Radiat. Transfer.

G. N. Plass, G. W. Kattawar, J. Binstock, J. Quant. Spectrosc. Radiat. Transfer. 13, 1081 (1973).
[CrossRef]

U.G.G.I., Monogr.

R. W. Preisendorfer, U.G.G.I., Monogr. 10, 11 (1961).

Other

T. J. Petzold, Volume Scattering Functions for Selected Waters (Scripps Institution of Oceanography, University of California at San Diego, 1972), SIO Ref. 72–78.

N. G. Jerlov, Optical Oceanography (Elsevier, Amsterdam, 1968).

J. E. Tyler, R. C. Smith, Measurements of Spectral Irradiance Underwater (Gordon and Breach, New York, 1970).

In this work μ is always cosθ, and unprimed μ’s refer to rays beneath the sea surface, while primed μ’s refer to rays above the sea surface.

G. Maul, H. R. Gordon. Relationships Between ERTS Radiances and Gradients across Oceanic Fronts, presented at the Third ERTS-1 Principal Investigator’s Symposium.

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Figures (10)

Fig. 1
Fig. 1

Comparison between the direct Monte Carlo simulation results for R(τ, −) and D(τ, +) (data points) with those determined from irradiances fit to Eq. (17) (smooth curves.)

Fig. 2
Fig. 2

Dependence of K(0, )/c on ω0 and F for phase functions A, B, and C in the sun case.

Fig. 3
Fig. 3

Dependence of K(0, )/c on ω0 and F for phase functions A, B, and C in the sky case.

Fig. 4
Fig. 4

Dependence of K(τ, −)/c on ω0F and τ for phase functions A, B, and C in the sun case. Each successive τ curve has been shifted upward one unit to facilitate presentation.

Fig. 5
Fig. 5

Dependence of the diffuse reflectance of the ocean on ω0 and the parameter X = ω0B/(1 − ω0F) for phase functions A, B, and C in the sun case.

Fig. 6
Fig. 6

Dependence of R (τ, −) on X and τ for phase functions A, B, and C in the sun case. Each successive τ curve has been shifted upward 1 order of magnitude to facilitate presentation.

Fig. 7
Fig. 7

Comparison of if K (0, )/c and R (0, ) computed from Eq. (12) and (13) (lines) with Monte Carlo computations (points) for phase function B in the sun case.

Fig. 8
Fig. 8

Comparison of D (0, ±) computed from Eqs. (14) and (15) (lines) with Monte Carlo computations (points) for phase function B in the sun case.

Fig. 9
Fig. 9

Comparison of computations of K (τ, −) for Rayleigh scattering τ and phase functions A (most forward scattering) and PB (least forward scattering) used in this study.

Fig. 10
Fig. 10

Comparison of computations of R (τ, −) for Rayleigh scattering τ and phase functions A and PB used in this study.

Tables (9)

Tables Icon

Table I Phase Functions (×102)

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Table II Comparison Between Matrix Operator (MO) and Monte Carlo (MC) Calculations of R(τ,−) for a Rayleigh Scattering Atmosphere

Tables Icon

Table III Variation in K(τ,−)/cD0 with the Incident Radiance Distribution for τ = 1.50 and Phase Function B

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Table IV Expansion Coefficients for Eqs. (19) and (19′)

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Table V Variation in R(τ,−) with the Incident Radiance Distribution for τ = 1.50 and Phase Function B

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Table VI Expansion Coefficients for Eqs. (20) and (20′) for the Sun and Sky Cases

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Table VII Variation in D(τ,−) − D0(τ,−) with the Incident Radiance Distribution for τ = 1.50 and Phase Function B

Tables Icon

Table VIII Expansion Coefficients in Eqs. (21) for the Sun and Sky Cases

Tables Icon

Table IX Test of Scheme for Finding ω0 and B

Equations (31)

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H ( z , ± ) = ± N ( z , θ , ϕ ) | cos θ | d Ω ,
h ( z , ± ) = ± N ( z , θ , ϕ ) d Ω , h ( z ) = h ( z , + ) + h ( z , ) ,
D ( z , ± ) = h ( z , ± ) / H ( z , ± ) ;
R ( z , ) = H ( z , + ) / H ( z , ) = 1 / R ( z , + ) ;
K ( z , ± ) = [ 1 / H ( z , ± ) ] ( d / d z ) H ( z , ± ) .
β ( θ ) [ Δ J ( θ ) ] / ( H 0 A Δ r ) .
b = 2 π 0 π β ( θ ) sin θ d θ .
[ μ d d z + c ( z ) ] N ( z , θ , ϕ ) = b ( z ) Ω P ( z ; θ , ϕ ; θ , ϕ ) N ( z , θ , ϕ ) d Ω ,
P ( z ; θ , ϕ ; θ , ϕ ) = β ( z , θ , ϕ , θ , ϕ ) / b ( z )
τ = 0 z c ( z ) d z
( μ d d τ + 1 ) N ( τ , θ , ϕ ) = ω 0 ( τ ) Ω P ( τ , θ , ϕ ; θ , ϕ ) N ( τ , θ , ϕ ) d Ω ,
K ( τ , ± ) c = 1 H ( τ , ± ) d d τ H ( τ , ± ) .
K ( τ , ) = { a + b [ 1 F ( μ 0 ) ] } / μ 0 = c [ 1 ω 0 F ( μ 0 ) ] / μ 0 ,
F ( μ 0 ) = 1 [ 0 2 π 0 1 P ( δ ) d μ d ϕ / 2 π 1 1 P ( μ ) d μ ] ,
cos δ = μ μ 0 + ( 1 μ 0 2 ) 1 / 2 ( 1 μ 2 ) 1 / 2 cos ( ϕ ) .
R ( 0 , ) = ω 0 1 ω 0 F 0 2 π 0 1 P ( δ ) μ d μ d ϕ μ + μ 0 .
D ( 0 , ) = 1 μ 0 + 1 μ 0 2 ω 0 1 ω 0 F 0 2 π 0 1 r ( μ , μ ) × ( μ 0 μ μ 0 + μ ) P ( δ ) d μ d ϕ + ,
D ( 0 , + ) = 0 1 P ( μ ) μ 0 + μ d μ / 0 1 P ( μ ) μ μ 0 + μ d μ .
1 ω 0 ( τ ) = 1 h ( τ ) d d τ [ H ( τ , + ) H ( τ , ) ] .
I = exp [ ( c 0 + c 1 τ + c 2 τ 2 ) ] ,
K ( τ , ) / [ c D 0 ( τ , ) ] = 1 ω 0 F ,
D 0 ( τ , ) = 0 2 π 0 1 N ( μ , ϕ ) T ( μ , μ ) × exp ( τ / μ ) d μ d ϕ / 0 2 π 0 1 N ( μ , ϕ ) T ( μ , μ ) μ × exp ( τ / μ ) d μ d ϕ ,
K ( τ , ) / c D 0 ( τ , ) = n = 0 N k n ( τ ) ( ω 0 F ) n
ω 0 F = n = 0 N k n ( τ ) [ K ( τ , ) / c D 0 ( τ , ) ] n .
X = ω 0 B / ( 1 ω 0 F ) .
R ( τ , ) = n = 0 N r n ( τ ) X n ,
X = n = 0 N r n ( τ ) [ R ( τ , ) ] n .
D ( τ , ) D 0 ( τ , ) = n = 0 N d n ( τ ) X n .
a = K ( z , ) R ( z , ) K ( z , + ) D ( z , ) + R ( z , ) D ( z , + ) .
a = 0.1446 m 1 / D ( τ , ) ,
P ( θ ) = 3 ( 1 + cos 2 θ ) / 16 π .

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