Abstract

We have developed a Monte Carlo program that is capable of calculating both the scalar and the Stokes vector radiances in an atmosphere–ocean system in a single computer run. The correlated sampling technique is used to compute radiance distributions for both the scalar and the Stokes vector formulations simultaneously, thus permitting a direct comparison of the errors induced. We show the effect of the volume-scattering phase function on the errors in radiance calculations when one neglects polarization effects. The model used in this study assumes a conservative Rayleigh-scattering atmosphere above a flat ocean. Within the ocean, the volume-scattering function (the first element in the Mueller matrix) is varied according to both a Henyey–Greenstein phase function, with asymmetry factors G = 0.0, 0.5, and 0.9, and also to a Rayleigh-scattering phase function. The remainder of the reduced Mueller matrix for the ocean is taken to be that for Rayleigh scattering, which is consistent with ocean water measurements.

© 1993 Optical Society of America

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Corrections

Charles N. Adams and George W. Kattawar, "Effect of volume-scattering function on the errors induced when polarization is neglected in radiance calculations in an atmosphere–ocean system: errata," Appl. Opt. 33, 453-453 (1994)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-33-3-453

References

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  1. C. N. Adams, G. W. Kattawar, “Solutions of the equations of radiative transfer by an invariant imbedding approach,” J. Quant. Spectrosc. Radiat. Transfer 10, 341–366 (1970).
    [CrossRef]
  2. G. W. Kattawar, C. N. Adams, “Stokes vector calculations of the submarine light field in an atmosphere-ocean with scattering according to a Rayleigh phase matrix: effect of interface refractive index on the radiance and polarization,” Limnol. Oceanogr. 34, 1463–1472 (1989).
    [CrossRef]
  3. G. W. Kattawar, C. N. Adams, “Errors in radiance calculations induced by using scalar rather than Stokes vector theory in a realistic atmosphere-ocean system,” in Ocean Optics X, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1302, 2–12 (1990).
  4. A. T. Young, “Rayleigh scattering,” Appl. Opt. 4, 633–636 (1981).
  5. A. T. Young, “Rayleigh scattering,” Phys. Today 35(1), 2–8 (1982).
    [CrossRef]
  6. K. J. Voss, E. S. Fry, “Measurement of the Mueller matrix for ocean water,” Appl. Opt. 23, 4427–4439 (1984).
    [CrossRef] [PubMed]

1989 (1)

G. W. Kattawar, C. N. Adams, “Stokes vector calculations of the submarine light field in an atmosphere-ocean with scattering according to a Rayleigh phase matrix: effect of interface refractive index on the radiance and polarization,” Limnol. Oceanogr. 34, 1463–1472 (1989).
[CrossRef]

1984 (1)

1982 (1)

A. T. Young, “Rayleigh scattering,” Phys. Today 35(1), 2–8 (1982).
[CrossRef]

1981 (1)

A. T. Young, “Rayleigh scattering,” Appl. Opt. 4, 633–636 (1981).

1970 (1)

C. N. Adams, G. W. Kattawar, “Solutions of the equations of radiative transfer by an invariant imbedding approach,” J. Quant. Spectrosc. Radiat. Transfer 10, 341–366 (1970).
[CrossRef]

Adams, C. N.

G. W. Kattawar, C. N. Adams, “Stokes vector calculations of the submarine light field in an atmosphere-ocean with scattering according to a Rayleigh phase matrix: effect of interface refractive index on the radiance and polarization,” Limnol. Oceanogr. 34, 1463–1472 (1989).
[CrossRef]

C. N. Adams, G. W. Kattawar, “Solutions of the equations of radiative transfer by an invariant imbedding approach,” J. Quant. Spectrosc. Radiat. Transfer 10, 341–366 (1970).
[CrossRef]

G. W. Kattawar, C. N. Adams, “Errors in radiance calculations induced by using scalar rather than Stokes vector theory in a realistic atmosphere-ocean system,” in Ocean Optics X, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1302, 2–12 (1990).

Fry, E. S.

Kattawar, G. W.

G. W. Kattawar, C. N. Adams, “Stokes vector calculations of the submarine light field in an atmosphere-ocean with scattering according to a Rayleigh phase matrix: effect of interface refractive index on the radiance and polarization,” Limnol. Oceanogr. 34, 1463–1472 (1989).
[CrossRef]

C. N. Adams, G. W. Kattawar, “Solutions of the equations of radiative transfer by an invariant imbedding approach,” J. Quant. Spectrosc. Radiat. Transfer 10, 341–366 (1970).
[CrossRef]

G. W. Kattawar, C. N. Adams, “Errors in radiance calculations induced by using scalar rather than Stokes vector theory in a realistic atmosphere-ocean system,” in Ocean Optics X, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1302, 2–12 (1990).

Voss, K. J.

Young, A. T.

A. T. Young, “Rayleigh scattering,” Phys. Today 35(1), 2–8 (1982).
[CrossRef]

A. T. Young, “Rayleigh scattering,” Appl. Opt. 4, 633–636 (1981).

Appl. Opt. (2)

J. Quant. Spectrosc. Radiat. Transfer (1)

C. N. Adams, G. W. Kattawar, “Solutions of the equations of radiative transfer by an invariant imbedding approach,” J. Quant. Spectrosc. Radiat. Transfer 10, 341–366 (1970).
[CrossRef]

Limnol. Oceanogr. (1)

G. W. Kattawar, C. N. Adams, “Stokes vector calculations of the submarine light field in an atmosphere-ocean with scattering according to a Rayleigh phase matrix: effect of interface refractive index on the radiance and polarization,” Limnol. Oceanogr. 34, 1463–1472 (1989).
[CrossRef]

Phys. Today (1)

A. T. Young, “Rayleigh scattering,” Phys. Today 35(1), 2–8 (1982).
[CrossRef]

Other (1)

G. W. Kattawar, C. N. Adams, “Errors in radiance calculations induced by using scalar rather than Stokes vector theory in a realistic atmosphere-ocean system,” in Ocean Optics X, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1302, 2–12 (1990).

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Figures (14)

Fig. 1
Fig. 1

Comparison of Monte Carlo sampling scheme with analytic results for three HG volume-scattering functions with asymmetry factors G = 0.0, 0.60, and 0.90.

Fig. 2
Fig. 2

Comparison of the calculated errors in reflected radiance between the scalar and the vector approaches computed by both a highly accurate invariant imbedding scheme (solid curve) compared with the Monte Carlo calculation (symbols) by the method of correlated sampling. The calculations are for a conservative medium of optical depths 0.10, 0.50, 1.0, and 5.0 scattering according to a Rayleigh phase matrix for a solar zenith angle of 10.24° (upper graph) and for a solar zenith angle of 78.85° (lower graph).

Fig. 3
Fig. 3

Quantities plotted are as follows: I is the Stokes vector diffuse radiance, i.e., it does not contain the direct solar beam or the direct specularly reflected beam; P is the degree of polarization defined in the standard way, i.e., P = (Q2 + U2 + V2)1/2/I; R is the ratio of the single-scattered radiance to the total radiance; and E is the error, in percent, between the scalar and the Stokes vector approach, i.e., E = 100 × (IsIυ)/Iυ, where the subscripts s and υ refer to the scalar and vector approaches, respectively. The detector is placed at the top of the atmosphere (τ = 0) and the solar zenith angle is 3°; the quantities are averaged over an azimuthal range of 0° ≤ ϕ ≤ 30° and the left-hand portion of each curve (zenith angles 0°−90°) denotes upward radiation whereas the right-hand portion (zenith angles 90°−0°) denotes downward radiation.

Fig. 4
Fig. 4

Same as Fig. 3, except the single-scattering albedo for the ocean has been set to ω0 = 0.2.

Fig. 5
Fig. 5

Four error curves for the model used in Fig. 3 for the following situations: a, single scattering with no interface with and without the ocean (G = 0.9); b, single scattering with an interface with and without the ocean (G = 0.9); c, all orders of scattering with no interface with and without the ocean (G = 0.9); d, all orders of scattering with an interface with and without the ocean (G = 0.9).

Fig. 6
Fig. 6

Same as Fig. 3, except the detector is placed just above the ocean interface, τD = 0.1499. The interface is at τ = 0.1500.

Fig. 7
Fig. 7

Same as Fig. 5, except the detector is placed just above the ocean interface, τD = 0.1499.

Fig. 8
Fig. 8

Same as Fig. 3, except the detector is placed just below the interface, τD = 0.1501.

Fig. 9
Fig. 9

Same as Fig. 3, except the detector is placed at an optical depth of τD = 1.00 from the top of the atmosphere.

Fig. 10
Fig. 10

Same as Fig. 3, except the solar zenith angle has been moved to 53°, which is almost at the Brewster angle, and the azimuthal range has been changed to 60° ≤ ϕ ≤ 90°.

Fig. 11
Fig. 11

Same as Fig. 10, except the detector is placed just above the ocean interface, τD = 0.1499.

Fig. 12
Fig. 12

Same as Fig. 5, except the solar zenith angle has been moved to 53° and the detector is just above the ocean surface, τD = 0.1499, and the azimuthal range has been changed to 60° ≤ ϕ ≤ 90°.

Fig. 13
Fig. 13

Same as Fig. 10, except the detector is placed just below the ocean interface, τD = 0.1501.

Fig. 14
Fig. 14

Same as Fig. 10, except the detector is placed at an optical depth of τD = 1.0 from the top of the atmosphere.

Equations (10)

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P ( μ ) = 3 16 π [ μ 2 + 1 μ 2 1 0 0 μ 2 1 μ 2 + 1 0 0 0 0 μ 0 0 0 0 μ ] ,
Ω P 11 ( Ω ) d Ω = 1 .
R ( μ ) = [ 1 μ 2 1 μ 2 + 1 0 0 μ 2 μ 2 + 1 1 0 0 0 0 μ μ 2 + 1 0 0 0 0 μ μ 2 + 1 ] .
P 11 ( G , μ ) = 1 G 2 4 π ( 1 2 G μ + G 2 ) 3 / 2
G = μ = Ω μ P 11 ( G , Ω ) d Ω .
μ = 1 + G 2 [ 1 G 2 G ( 2 R 1 ) + 1 ] 2 2 G ,
f = f ( x ) p ( x ) d x .
f = f ( x ) p ( x ) p ( x ) p ( x ) d x = f ( x ) w ( x ) p ( x ) d x ,
σ 2 [ f ( x ) w ( x ) ] = [ f ( x ) w ( x ) f ] 2 p ( x ) d x .
B = 0 2 π 1 0 P 11 ( μ ) d μ d ϕ

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