Abstract

The radiative transfer equation is solved by Monte Carlo methods for natural waters in which the optical properties are distributed with depth. It is demonstrated that interpreting the reflectance of a continuously stratified ocean in terms of an equivalent homogeneous ocean yields the average of a particular combination of the water's optical properties over the dimensionless penetration depth τ90. Although in general the dimensionless penetration depth cannot be remotely measured, a method is presented for estimating the actual penetration depth z90 from the remote observations if the medium's absorption coefficient is known, independent of depth, and sufficiently large. The application of this to the remote measurement of the vertical distribution of suspended sediments is discussed in detail.

© 1978 Optical Society of America

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References

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  1. H. R. Gordon, W. R. McCluney, Appl. Opt. 14, 413 (1975).
    [CrossRef] [PubMed]
  2. H. R. Gordon, O. B. Brown, Appl. Opt. 14, 2892 (1975).
    [CrossRef] [PubMed]
  3. H. R. Gordon, Appl. Opt. 15, 1974 (1976).
    [CrossRef] [PubMed]
  4. H. R. Gordon, O. B. Brown, M. M. Jacobs, Appl. Opt. 14, 417 (1975).
    [CrossRef] [PubMed]
  5. H. R. Gordon, Appl. Opt. 15, 2611 (1976).
    [CrossRef] [PubMed]
  6. N. G. Jerlov, in Optical Aspects of Oceanography, N. G. Jerlov, E. Steemen Nielsen, Eds. (Academic, New York, 1974), pp. 77– 94.

1976 (2)

1975 (3)

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Figures (4)

Fig. 1
Fig. 1

Variation in b/a with optical depth τ resulting from Eq. (3) with ω = 0.4. In the left panel n = 0, ζ = 1.24, and = 1.5,1.0, and 0.5, respectively, from top to bottom, while in the right panel n = 1 and (ζ,∊) = (5.1, 1.5), (3.4, 1.0), and (1.7, 0.5), respectively, for the curves with maximum b/a at τ = 0.67, 1.0, and 2.0.

Fig. 2
Fig. 2

Relationship between R and k B ¯ for stratified waters. Solid line is for a homogeneous ocean with k B = k B ¯.

Fig. 3
Fig. 3

Relationship between R and ( k B ¯ ) z for stratified waters. Solid line is for a homogeneous ocean with k B = ( k B ¯ ) z.

Fig. 4
Fig. 4

Demonstration of the validity of using Eq. (5) to find z90.

Tables (2)

Tables Icon

Table I Dependence of D0(τ) on τ for θ0 = 0

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Table II Percent Error In Using Eq. (4) to Find K(τ)

Equations (21)

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I = I 1 + I 2 R ( 1 r R ) ,
R = 0.0001 + 0.3244 x + 0.1425 x 2 + 0.1308 x 3 ,
x = k B / ( 1 + k B ) , k = b / a , B = π / 2 π P ( θ ) sin θ d θ / 0 π P ( θ ) sin θ d θ .
a = a w + i a i , b B = b w / 2 + i ( b B ) i ,
ω 0 ( τ ) = ω [ 1 + ζ τ n exp ( τ ) ] ,
τ = 0 z ( a + b ) d z .
k B ¯ = 1 τ 90 0 τ 90 k B d τ ,
τ 90 0 z 90 ( a + b ) d z
K ( τ ) ( a + b ) D 0 ( τ ) [ 1 ω 0 ( 1 B ) ] ,
K ( τ ) = D 0 ( τ ) a ( 1 + k B ) ,
H ( z , ) H ( 0 , ) = exp [ 0 z K ( z ) d z ] ,
0 z 90 K ( z ) d z = 1.
D 0 a z 90 [ 1 + ( k B ¯ ) z ] = 1 ,
( k B ¯ ) z = 1 z 90 0 z 90 k B d z .
z 90 a D 0 [ 1 + ( k B ¯ ) z ] = 0.86 + 0.072 log 10 ( k B ¯ ) z
0 z 90 k B d z = z 90 ( k B ¯ ) z
0 z 90 kBdz = z 90 a ( k B ¯ ) z .
b B = γ C ,
0 z 90 Cdz = z 90 a ( k B ) z γ ,
0 z 90 ( λ 1 ) Cdz and 0 z 90 ( λ 2 ) Cdz
z 90 ( λ 1 ) z 90 ( λ 2 ) Cdz ,

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