Abstract

The small-angle approximation to the radiative transport equation is applied to particle suspensions that emulate ocean water. A particle size distribution is constructed from polystyrene and glass spheres with the best available data for particle size distributions in the ocean. A volume scattering function is calculated from the Mie theory for the particles in water and in oil. The refractive-index ratios of particles in water and particles in oil are 1.19 and 1.01, respectively. The ratio 1.19 is comparable to minerals and nonliving diatoms in ocean water, and the ratio 1.01 is comparable to the lower limit for microbes in water. The point-spread functions are measured as a function of optical thickness for both water and oil mixtures and compared with the point-spread functions generated from the small-angle approximation. Our results show that, under conditions that emulate ocean water, the small-angle approximation is valid only for small optical thicknesses. Specifically, the approximation is valid only for optical thicknesses less than 3.

© 2001 Optical Society of America

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References

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  1. N. L. Swanson, V. M. Gehman, B. D. Billard, T. L. Gennaro, “Limits of the small-angle approximation to the radiative transport equation,” J. Opt. Soc. Am. A 18, 385–391 (2001).
    [CrossRef]
  2. W. H. Wells, “Loss of resolution in water as a result of multiple small-angle scattering,” J. Opt. Soc. Am. 59, 686–691 (1969).
    [CrossRef]
  3. J. W. McLean, K. J. Voss, “Point-spread function in ocean water: comparison between theory and experiment,” Appl. Opt. 30, 2027–2030 (1991).
    [CrossRef] [PubMed]
  4. J. W. McLean, D. R. Crawford, C. L. Hindman, “Limits of small-angle scattering theory,” Appl. Opt. 26, 2053–2054 (1987).
    [CrossRef] [PubMed]
  5. W. H. Wells, “Theory of small-angle scattering,” in Optics of the Sea, AGARD Lect. Ser. 61, 3.3-1–3.3-19 (1973).
  6. In 1994A. R. Davenport, M. E. Stefanov (Dahlgren Division, Naval Surface Warfare Center, Va.; and Coastal Systems Station, Panama City, Fla.) observed that application of the small-angle approximation, using the Wells phase function as input, did not correspond with their observed data, as shown in Fig. 1.
  7. J. W. McLean, J. D. Freeman, “Effects of ocean waves on airborne lidar imaging,” Appl. Opt. 35, 3261–3269 (1996).
    [CrossRef] [PubMed]
  8. D. Arnush, “Underwater light beam propagation in the small-angle-scattering approximation,” J. Opt. Soc. Am. 62, 1109–1111 (1972).
    [CrossRef]
  9. T. J. Petzold, Volume Scattering Functions for Selected Ocean WatersSIO Ref. 72–78 (Visibility Laboratory, Scripps Institution of Oceanography, San Diego, Calif., 1972).
  10. O. B. Brown, H. R. Gordon, “Size–refractive index distribution of clear coastal water particulates from light scattering,” Appl. Opt. 13, 2874–2881 (1974).
    [CrossRef] [PubMed]
  11. G. Kullenberg, “Observed and computed scattering functions,” in Optical Aspects of Oceanography, N. G. Jerlov, N. E. Steemann, eds. (Academic, New York, 1974), pp. 25–49.
  12. R. W. Spinrad, J. R. V. Zaneveld, H. Pak, “Volume scattering function of suspended particulate matter at near-forward angles: a comparison of experimental and theoretical values,” Appl. Opt. 17, 1125–1130 (1978).
    [CrossRef] [PubMed]
  13. Y. Kuga, A. Ishimaru, H. W. Chang, L. Tsang, “Comparisons between the small-angle approximation and the numerical solution for radiative transfer theory,” Appl. Opt. 25, 3803–3805 (1986).
    [CrossRef] [PubMed]

2001

1996

1991

1987

1986

1978

1974

1973

W. H. Wells, “Theory of small-angle scattering,” in Optics of the Sea, AGARD Lect. Ser. 61, 3.3-1–3.3-19 (1973).

1972

1969

Arnush, D.

Billard, B. D.

Brown, O. B.

Chang, H. W.

Crawford, D. R.

Freeman, J. D.

Gehman, V. M.

Gennaro, T. L.

Gordon, H. R.

Hindman, C. L.

Ishimaru, A.

Kuga, Y.

Kullenberg, G.

G. Kullenberg, “Observed and computed scattering functions,” in Optical Aspects of Oceanography, N. G. Jerlov, N. E. Steemann, eds. (Academic, New York, 1974), pp. 25–49.

McLean, J. W.

Pak, H.

Petzold, T. J.

T. J. Petzold, Volume Scattering Functions for Selected Ocean WatersSIO Ref. 72–78 (Visibility Laboratory, Scripps Institution of Oceanography, San Diego, Calif., 1972).

Spinrad, R. W.

Swanson, N. L.

Tsang, L.

Voss, K. J.

Wells, W. H.

W. H. Wells, “Theory of small-angle scattering,” in Optics of the Sea, AGARD Lect. Ser. 61, 3.3-1–3.3-19 (1973).

W. H. Wells, “Loss of resolution in water as a result of multiple small-angle scattering,” J. Opt. Soc. Am. 59, 686–691 (1969).
[CrossRef]

Zaneveld, J. R. V.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Optics of the Sea

W. H. Wells, “Theory of small-angle scattering,” in Optics of the Sea, AGARD Lect. Ser. 61, 3.3-1–3.3-19 (1973).

Other

In 1994A. R. Davenport, M. E. Stefanov (Dahlgren Division, Naval Surface Warfare Center, Va.; and Coastal Systems Station, Panama City, Fla.) observed that application of the small-angle approximation, using the Wells phase function as input, did not correspond with their observed data, as shown in Fig. 1.

T. J. Petzold, Volume Scattering Functions for Selected Ocean WatersSIO Ref. 72–78 (Visibility Laboratory, Scripps Institution of Oceanography, San Diego, Calif., 1972).

G. Kullenberg, “Observed and computed scattering functions,” in Optical Aspects of Oceanography, N. G. Jerlov, N. E. Steemann, eds. (Academic, New York, 1974), pp. 25–49.

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Figures (8)

Fig. 1
Fig. 1

Images of standard bar chart panels. (a) The image taken during the tank test with τ = 3.05, a range of 6.1 m, and an attenuation of 0.5 m-1. The medium was water with 1–8-µm polystyrene spheres added. The refractive-index ratio was 1.19. (b) The image generated from an image of the bar chart panel in air convolved with the measured PSF at τ = 3.05. (c) The image generated from an image taken in air convolved with a PSF generated from the SAA with Wells’s algebraic VSF as input. Figure1 reprinted by permission, courtesy of Anne Davenport, Naval Surface Warfare Center, Coastal Systems Station, Panama City, Florida.

Fig. 2
Fig. 2

Comparison of Wells’s algebraic VSF with the VSF calculated from Mie theory for 1–8-µm spheres.

Fig. 3
Fig. 3

Comparison of VSF calculated from the particle size distribution with measured TOTO data, along with the modified Wells VSF.

Fig. 4
Fig. 4

Experimental setup for measuring the VSF. The components are f1, a fixed neutral-density filter; sf, a spatial filter and collimator; AP, an aperture; f2, removable neutral-density filters; s, the sample cell; l, a lens; pda, a photodiode array detector; m, a small mirror on a translation stage to block and unblock the main beam.

Fig. 5
Fig. 5

Measured VSF for particle distribution in water, m = 1.59, compared with calculated VSF and TOTO data.

Fig. 6
Fig. 6

Measured VSF for particle distribution in oil, m = 1.01, compared with calculated VSF and TOTO data.

Fig. 7
Fig. 7

Measured PSF compared with PSF from the small-angle theory for particles in water. Each set of curves corresponds to a value of τ shown at the right.

Fig. 8
Fig. 8

Measured PSF compared with PSF from the small-angle theory for particles in oil. Each set of curves corresponds to a value of τ shown at the right.

Tables (1)

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Table 1 Particle Sizes and Weights (percentage by number) used to Simulate Ocean Water

Equations (11)

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fθ, r=2π 0J02πθψFψ, rψdψ, Fψ, r=2π 0θmaxJ02πθψfθ, rθdθ,
σθ=2π 0J02πθψΣψψdψ, Σψ=2π 0θmaxJ02πθψσθθdθ.
4πσθdΩ=1.
Fψ, r=exp-ξr+sfrΣψ,
η=2π 0θmax σθsinθdθ.
Fψ, r=exp-τ1-ηΣψ.
σθ=θ02πθ02+θ23/2,
σθ=1θ3/2θ02+θ21/2.
σθ=E2θ/E20-E1θ/E10τ2-τ1aR2cos3arctany/R.
θmax=arcsinn1n2sinarctandR
fθ, L=Pθ, LP0Ω=Eθ, LE0aR2cos3arctany/R.

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