Abstract

We propose and test an inverse ocean optics procedure with numerically simulated data for the determination of inherent optical properties using in-water radiance measurements. If data are available at only one depth within a deep homogeneous water layer, then the single-scattering albedo and the single parameter that characterizes the Henyey-Greenstein phase function can be estimated. If data are available at two depths, then these two parameters can be determined along with the optical thickness so that the absorption and scattering coefficients, and also the backscattering coefficient, can be estimated. With a knowledge of these parameters, the albedo and Lambertian fraction of reflected radiance of the bottom can be determined if measurements are made close to the bottom. A simplified method for determining the optical properties of the water also is developed for only three irradiance-type measurements if the radiance is approximately in the asymptotic regime.

© 2003 Optical Society of America

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References

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  1. C. D. Mobley, Light and Water. Radiative Transfer in Natural Waters (Academic, New York, 1994), pp. 157–165 and 281 ff.
  2. H. R. Gordon, G. C. Boynton, “Radiance-irradiance inversion algorithm for estimating the absorption and backscattering coefficients of natural waters: homogeneous waters,” Appl. Opt. 36, 2636–2641 (1997).
    [CrossRef] [PubMed]
  3. R. A. Leathers, N. J. McCormick, “Ocean inherent optical property estimation from irradiances,” Appl. Opt. 36, 8685–8698 (1997).
    [CrossRef]
  4. H. R. Gordon, G. C. Boynton, “Radiance-irradiance inversion algorithm for estimating the absorption and backscattering coefficients of natural waters: vertically stratified water bodies,” Appl. Opt. 37, 3886–3896 (1998).
    [CrossRef]
  5. R. A. Leathers, C. S. Roesler, N. J. McCormick, “Ocean inherent optical property determination from in-water light field measurements,” Appl. Opt. 38, 5096–5103 (1999).
    [CrossRef]
  6. E. S. Chalhoub, H. F. Campos Velho, “Simultaneous estimation of radiation phase function and albedo in natural waters,” J. Quant. Spectrosc. Radiat. Transfer 69, 137–149 (2001).
    [CrossRef]
  7. L. C. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
    [CrossRef]
  8. C. D. Mobley, Hydrolight software program, available from Sequoia Scientific, Inc., Westpark Technical Center, 15317 NE 90th St., Redmond, Wash. 98052.
  9. C. D. Mobley, L. K. Sundman, E. Boss, “Phase function effects on oceanic light fields,” Appl. Opt. 41, 1035–1050 (2002).
    [CrossRef] [PubMed]
  10. K. J. Voss, G. Zibordi, “Radiometric and geometric calibration of a visible spectral electrooptic ‘fisheye’ camera radiance distribution system,” J. Atmos. Oceanic Technol. 6, 652–662 (1989).
    [CrossRef]
  11. K. J. Voss, “Use of the radiance distribution to measure the optical absorption coefficient in the ocean,” Limnol. Oceanogr. 34, 1614–1622 (1989).
    [CrossRef]
  12. K. J. Voss, A. L. Chapin, “Measurement of the point spread function in the ocean,” Appl. Opt. 29, 3638–3642 (1990).
    [CrossRef] [PubMed]
  13. K. J. Voss, Y. Liu, “Polarized radiance distribution measurements of skylight. I. System description and characterization,” Appl. Opt. 36, 6083–6094 (1997).
    [CrossRef] [PubMed]
  14. R. A. Leathers, N. J. McCormick, “Algorithms for ocean-bottom albedo determination from in-water natural-light measurements,” Appl. Opt. 38, 3199–3205 (1999).
    [CrossRef]
  15. C. E. Siewert, “Inverse solutions to radiative-transfer problems based on the binomial or the Henyey-Greenstein scattering law,” J. Quant. Spectrosc. Radiat. Transfer 72, 827–835 (2002).
    [CrossRef]
  16. W. Doss, W. H. Wells, “Undersea compound radiometer,” Appl. Opt. 22, 2313–2321 (1983).
  17. N. J. McCormick, “Transport scattering coefficients from reflection and transmission measurements,” J. Math. Phys. 20, 1504–1507 (1979).
    [CrossRef]
  18. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960), Sect. 48.3.
  19. H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980), Vol. 2, p. 307.
  20. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, Cambridge, UK, 1997), Chap. 9.
  21. R. D. M. Garcia, C. E. Siewert, “On computing the Chandrasekhar polynomials in high order and high degree,” J. Quant. Spectrosc. Radiat. Transfer 43, 201–205 (1990).
    [CrossRef]
  22. T. J. Petzold, “Volume scattering functions for selected ocean waters,” SIO Ref. 71–78 (Scripps Institution of Oceanography, San Diego, Calif., 1972).
  23. N. J. McCormick, “Asymptotic optical attenuation,” Limnol. Oceanogr. 37, 1570–1578 (1992).
    [CrossRef]
  24. N. J. McCormick, “Methods for estimating the similarity parameter of clouds from internal measurements of the scattered radiation field,” J. Quant. Spectrosc. Radiat. Transfer 33, 63–70 (1985).
    [CrossRef]

2002 (2)

C. E. Siewert, “Inverse solutions to radiative-transfer problems based on the binomial or the Henyey-Greenstein scattering law,” J. Quant. Spectrosc. Radiat. Transfer 72, 827–835 (2002).
[CrossRef]

C. D. Mobley, L. K. Sundman, E. Boss, “Phase function effects on oceanic light fields,” Appl. Opt. 41, 1035–1050 (2002).
[CrossRef] [PubMed]

2001 (1)

E. S. Chalhoub, H. F. Campos Velho, “Simultaneous estimation of radiation phase function and albedo in natural waters,” J. Quant. Spectrosc. Radiat. Transfer 69, 137–149 (2001).
[CrossRef]

1999 (2)

1998 (1)

1997 (3)

1992 (1)

N. J. McCormick, “Asymptotic optical attenuation,” Limnol. Oceanogr. 37, 1570–1578 (1992).
[CrossRef]

1990 (2)

R. D. M. Garcia, C. E. Siewert, “On computing the Chandrasekhar polynomials in high order and high degree,” J. Quant. Spectrosc. Radiat. Transfer 43, 201–205 (1990).
[CrossRef]

K. J. Voss, A. L. Chapin, “Measurement of the point spread function in the ocean,” Appl. Opt. 29, 3638–3642 (1990).
[CrossRef] [PubMed]

1989 (2)

K. J. Voss, G. Zibordi, “Radiometric and geometric calibration of a visible spectral electrooptic ‘fisheye’ camera radiance distribution system,” J. Atmos. Oceanic Technol. 6, 652–662 (1989).
[CrossRef]

K. J. Voss, “Use of the radiance distribution to measure the optical absorption coefficient in the ocean,” Limnol. Oceanogr. 34, 1614–1622 (1989).
[CrossRef]

1985 (1)

N. J. McCormick, “Methods for estimating the similarity parameter of clouds from internal measurements of the scattered radiation field,” J. Quant. Spectrosc. Radiat. Transfer 33, 63–70 (1985).
[CrossRef]

1983 (1)

1979 (1)

N. J. McCormick, “Transport scattering coefficients from reflection and transmission measurements,” J. Math. Phys. 20, 1504–1507 (1979).
[CrossRef]

1941 (1)

L. C. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Boss, E.

Boynton, G. C.

Campos Velho, H. F.

E. S. Chalhoub, H. F. Campos Velho, “Simultaneous estimation of radiation phase function and albedo in natural waters,” J. Quant. Spectrosc. Radiat. Transfer 69, 137–149 (2001).
[CrossRef]

Chalhoub, E. S.

E. S. Chalhoub, H. F. Campos Velho, “Simultaneous estimation of radiation phase function and albedo in natural waters,” J. Quant. Spectrosc. Radiat. Transfer 69, 137–149 (2001).
[CrossRef]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960), Sect. 48.3.

Chapin, A. L.

Doss, W.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, Cambridge, UK, 1997), Chap. 9.

Garcia, R. D. M.

R. D. M. Garcia, C. E. Siewert, “On computing the Chandrasekhar polynomials in high order and high degree,” J. Quant. Spectrosc. Radiat. Transfer 43, 201–205 (1990).
[CrossRef]

Gordon, H. R.

Greenstein, J. L.

L. C. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Henyey, L. C.

L. C. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Leathers, R. A.

Liu, Y.

McCormick, N. J.

R. A. Leathers, N. J. McCormick, “Algorithms for ocean-bottom albedo determination from in-water natural-light measurements,” Appl. Opt. 38, 3199–3205 (1999).
[CrossRef]

R. A. Leathers, C. S. Roesler, N. J. McCormick, “Ocean inherent optical property determination from in-water light field measurements,” Appl. Opt. 38, 5096–5103 (1999).
[CrossRef]

R. A. Leathers, N. J. McCormick, “Ocean inherent optical property estimation from irradiances,” Appl. Opt. 36, 8685–8698 (1997).
[CrossRef]

N. J. McCormick, “Asymptotic optical attenuation,” Limnol. Oceanogr. 37, 1570–1578 (1992).
[CrossRef]

N. J. McCormick, “Methods for estimating the similarity parameter of clouds from internal measurements of the scattered radiation field,” J. Quant. Spectrosc. Radiat. Transfer 33, 63–70 (1985).
[CrossRef]

N. J. McCormick, “Transport scattering coefficients from reflection and transmission measurements,” J. Math. Phys. 20, 1504–1507 (1979).
[CrossRef]

Mobley, C. D.

C. D. Mobley, L. K. Sundman, E. Boss, “Phase function effects on oceanic light fields,” Appl. Opt. 41, 1035–1050 (2002).
[CrossRef] [PubMed]

C. D. Mobley, Hydrolight software program, available from Sequoia Scientific, Inc., Westpark Technical Center, 15317 NE 90th St., Redmond, Wash. 98052.

C. D. Mobley, Light and Water. Radiative Transfer in Natural Waters (Academic, New York, 1994), pp. 157–165 and 281 ff.

Petzold, T. J.

T. J. Petzold, “Volume scattering functions for selected ocean waters,” SIO Ref. 71–78 (Scripps Institution of Oceanography, San Diego, Calif., 1972).

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, Cambridge, UK, 1997), Chap. 9.

Roesler, C. S.

Siewert, C. E.

C. E. Siewert, “Inverse solutions to radiative-transfer problems based on the binomial or the Henyey-Greenstein scattering law,” J. Quant. Spectrosc. Radiat. Transfer 72, 827–835 (2002).
[CrossRef]

R. D. M. Garcia, C. E. Siewert, “On computing the Chandrasekhar polynomials in high order and high degree,” J. Quant. Spectrosc. Radiat. Transfer 43, 201–205 (1990).
[CrossRef]

Sundman, L. K.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, Cambridge, UK, 1997), Chap. 9.

van de Hulst, H. C.

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980), Vol. 2, p. 307.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, Cambridge, UK, 1997), Chap. 9.

Voss, K. J.

K. J. Voss, Y. Liu, “Polarized radiance distribution measurements of skylight. I. System description and characterization,” Appl. Opt. 36, 6083–6094 (1997).
[CrossRef] [PubMed]

K. J. Voss, A. L. Chapin, “Measurement of the point spread function in the ocean,” Appl. Opt. 29, 3638–3642 (1990).
[CrossRef] [PubMed]

K. J. Voss, “Use of the radiance distribution to measure the optical absorption coefficient in the ocean,” Limnol. Oceanogr. 34, 1614–1622 (1989).
[CrossRef]

K. J. Voss, G. Zibordi, “Radiometric and geometric calibration of a visible spectral electrooptic ‘fisheye’ camera radiance distribution system,” J. Atmos. Oceanic Technol. 6, 652–662 (1989).
[CrossRef]

Wells, W. H.

Zibordi, G.

K. J. Voss, G. Zibordi, “Radiometric and geometric calibration of a visible spectral electrooptic ‘fisheye’ camera radiance distribution system,” J. Atmos. Oceanic Technol. 6, 652–662 (1989).
[CrossRef]

Appl. Opt. (9)

W. Doss, W. H. Wells, “Undersea compound radiometer,” Appl. Opt. 22, 2313–2321 (1983).

K. J. Voss, A. L. Chapin, “Measurement of the point spread function in the ocean,” Appl. Opt. 29, 3638–3642 (1990).
[CrossRef] [PubMed]

H. R. Gordon, G. C. Boynton, “Radiance-irradiance inversion algorithm for estimating the absorption and backscattering coefficients of natural waters: homogeneous waters,” Appl. Opt. 36, 2636–2641 (1997).
[CrossRef] [PubMed]

K. J. Voss, Y. Liu, “Polarized radiance distribution measurements of skylight. I. System description and characterization,” Appl. Opt. 36, 6083–6094 (1997).
[CrossRef] [PubMed]

H. R. Gordon, G. C. Boynton, “Radiance-irradiance inversion algorithm for estimating the absorption and backscattering coefficients of natural waters: vertically stratified water bodies,” Appl. Opt. 37, 3886–3896 (1998).
[CrossRef]

R. A. Leathers, N. J. McCormick, “Algorithms for ocean-bottom albedo determination from in-water natural-light measurements,” Appl. Opt. 38, 3199–3205 (1999).
[CrossRef]

R. A. Leathers, N. J. McCormick, “Ocean inherent optical property estimation from irradiances,” Appl. Opt. 36, 8685–8698 (1997).
[CrossRef]

C. D. Mobley, L. K. Sundman, E. Boss, “Phase function effects on oceanic light fields,” Appl. Opt. 41, 1035–1050 (2002).
[CrossRef] [PubMed]

R. A. Leathers, C. S. Roesler, N. J. McCormick, “Ocean inherent optical property determination from in-water light field measurements,” Appl. Opt. 38, 5096–5103 (1999).
[CrossRef]

Astrophys. J. (1)

L. C. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

J. Atmos. Oceanic Technol. (1)

K. J. Voss, G. Zibordi, “Radiometric and geometric calibration of a visible spectral electrooptic ‘fisheye’ camera radiance distribution system,” J. Atmos. Oceanic Technol. 6, 652–662 (1989).
[CrossRef]

J. Math. Phys. (1)

N. J. McCormick, “Transport scattering coefficients from reflection and transmission measurements,” J. Math. Phys. 20, 1504–1507 (1979).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (4)

E. S. Chalhoub, H. F. Campos Velho, “Simultaneous estimation of radiation phase function and albedo in natural waters,” J. Quant. Spectrosc. Radiat. Transfer 69, 137–149 (2001).
[CrossRef]

C. E. Siewert, “Inverse solutions to radiative-transfer problems based on the binomial or the Henyey-Greenstein scattering law,” J. Quant. Spectrosc. Radiat. Transfer 72, 827–835 (2002).
[CrossRef]

R. D. M. Garcia, C. E. Siewert, “On computing the Chandrasekhar polynomials in high order and high degree,” J. Quant. Spectrosc. Radiat. Transfer 43, 201–205 (1990).
[CrossRef]

N. J. McCormick, “Methods for estimating the similarity parameter of clouds from internal measurements of the scattered radiation field,” J. Quant. Spectrosc. Radiat. Transfer 33, 63–70 (1985).
[CrossRef]

Limnol. Oceanogr. (2)

N. J. McCormick, “Asymptotic optical attenuation,” Limnol. Oceanogr. 37, 1570–1578 (1992).
[CrossRef]

K. J. Voss, “Use of the radiance distribution to measure the optical absorption coefficient in the ocean,” Limnol. Oceanogr. 34, 1614–1622 (1989).
[CrossRef]

Other (6)

T. J. Petzold, “Volume scattering functions for selected ocean waters,” SIO Ref. 71–78 (Scripps Institution of Oceanography, San Diego, Calif., 1972).

C. D. Mobley, Light and Water. Radiative Transfer in Natural Waters (Academic, New York, 1994), pp. 157–165 and 281 ff.

C. D. Mobley, Hydrolight software program, available from Sequoia Scientific, Inc., Westpark Technical Center, 15317 NE 90th St., Redmond, Wash. 98052.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960), Sect. 48.3.

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980), Vol. 2, p. 307.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, Cambridge, UK, 1997), Chap. 9.

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Figures (6)

Fig. 1
Fig. 1

Evaluation of g and ϖ for the forward problem run with the HG phase function with g = 0.9. The solid curves show the errors in g, and the dashed curves show the errors in ϖ. The numbers next to the curves show the wind speed in meters per second.

Fig. 2
Fig. 2

Same as Fig. 1 except for the forward problem run with the particle phase function.

Fig. 3
Fig. 3

Evaluation of ϖ, c, and b b for the forward problem run with the HG phase function with g = 0.9. A depth difference of Δz = 4 m was used in the inverse algorithm. The heavy solid curves show the errors in c, the thin solid curves show the errors in ϖ, and the dashed curves show the errors in b b . The numbers next to the curves show the wind speed in meters per second.

Fig. 4
Fig. 4

Same as Fig. 3 except for the forward problem run with the particle phase function.

Fig. 5
Fig. 5

Evaluation of bottom reflectivity ρ for a water column of 5 m with a Lambertian bottom. The numbers next to the curves show the wind speed in meters per second.

Fig. 6
Fig. 6

Evaluation of g and ϖ with the approximate algorithm for the forward problem run with the particle phase function and no wind. The solid curves show the errors in g, and the dashed curves show the errors in ϖ. The numbers next to the curves show ϖ used in the forward problem run.

Tables (1)

Tables Icon

Table 1 Benchmark Tests of Eqs. (5) and (6) for Radiance Measurements Computed from Hydrolight with g = 0.9, ϖ = 0.5, n = 1, and with the Sun in a Black Sky

Equations (41)

Equations on this page are rendered with MathJax. Learn more.

μLz, μ/z+cLz, μ=b -11 β˜μ, μLz, μdμ, 0zz*,
bbb=-10 β˜μ, 1dμ.
β˜μ, μ=12n=0N2n+1fnPnμPnμ, f0=1,
μ Lτ, μτ+Lτ, μ=ϖ2n=02n+1gnPnμ×-11 PnμLτ, μdμ, 0ττ*,
ϖ n=0-1n2n+1gnEnτ22-Enτ1]2=4Sτ2-Sτ1,
ϖ n=0-1n2n+1g˜nE˜nτ22-E˜nτ12=4S˜τ2-S˜τ1,
Enτ=-11 PnμLτ, μdμ,
Sτ=01 Lτ, μLτ, -μdμ,
E˜nτ=-11 μPnμLτ, μdμ,
S˜τ=01 μ2Lτ, μLτ, -μdμ,
g˜n=gn/1-ϖgn.
Lτ, μ=2-1n=02n+1EnτPnμ.
Sτ=n=0-1n2n+1En2τ,
n=0-1n2n+11-ϖgnEnτ22-Enτ1]2=0.
S˜τ=-n=0-1n2n+1E˜n2τ,
n=0-1n2n+11-ϖgn-1×E˜nτ22-E˜nτ12=0.
c=τ2-τ1/z2-z1.
Δτ=τ2-τ1=νj lnMτ1, νj/Mτ2, νj, j=1 to J,
Mτ, νj=-11 μϕνj, μLτ, μdμ,
ϕνj, μ=ϖνj2n=02n+1gngnνjPnμνj-μ,
ϖ2n=0-112n+1gngnμPnμν-μdμ-1=0.
gnν=-11 Pnμϕν, μdμ
2n+11-ϖgnνgnν=n+1gn+1ν+ngn-1ν
a=1-ϖc,
b=ϖc.
bbb=1-g2g1-g1+g1/2-1.
Lτ*, -μ=ρ1-γLτ*, μ+2γ×01 μLτ*, μdμ, 0μ1,
Enτ2=-1nρ1-γEnhτ2+2ργαnE1hτ2+Enhτ2,
Sτ2=ρ1-γ01Lτ2, μ2dμ+2ργE0hτ2E1hτ2,
E˜nτ2=-1n+1ρ1-γE˜nhτ2+2ργα˜nE˜1hτ2+E˜nhτ2,
S˜τ2=ρ1-γ01μLτ2, μ2dμ+23 ργE1hτ2×2E2hτ2+E0hτ2,
Enhτ2=01 PnμLτ2, μdμ,
E˜nhτ2=01 μPnμLτ2, μdμ,
αn=01 Pnμdμ,
α˜n=01 μPnμdμ.
Lτ, μAν1ϕν1, μexp-τ/ν1+A-ν1ϕ-ν1, μexpτ/ν1,
Enτgnν1Aν1exp-τ/ν1+-1nA-ν1expτ/ν1.
E˜nτν11-ϖgngnν1Aν1exp-τ/ν1+-1n+1A-ν1expτ/ν1.
E2τ/E0τ=g2ν1=2-131-ϖ1-ϖgν12-1.
ν11-ϖ2E02τ2-E02τ1=E12τ2-E12τ1,
Δτ=ν1 lnE1τ1/E1τ2,

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