Abstract

An algorithm is described and evaluated for determining the absorption and backscattering coefficients a (z) and b b (z) from measurements of the nadir-viewing radiance L u (z) and downward irradiance E d (z). The method, derived from radiative transfer theory, is similar to a previously proposed one for E u (z) and E d (z), and both methods are demonstrated with numerical simulations and field data. Numerical simulations and a sensitivity analysis show that good estimates of a (z) and b b (z) can be obtained if the assumed scattering phase function is approximately correct. In an experiment in Long Island Sound, estimates of a (z) derived with these methods agreed well with those obtained from an in situ reflecting tube instrument.

© 1999 Optical Society of America

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References

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  1. J. R. V. Zaneveld, “A reflecting tube absorption meter,” in Ocean Optics X, R. W. Spinrad, ed., Proc. SPIE1302, 124–136 (1990).
    [CrossRef]
  2. R. M. Pope, E. S. Fry, “Absorption spectrum (380–700 nm) of pure water. II. Integrating cavity measurements,” Appl. Opt. 36, 8710–8723 (1997).
    [CrossRef]
  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: homogeneous waters,” Appl. Opt. 36, 2636–2641 (1997).
    [CrossRef] [PubMed]
  5. 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]
  6. N. J. McCormick, “Analytical transport theory applications in optical oceanography,” Ann. Nucl. Energy 23, 381–395 (1996).
    [CrossRef]
  7. S. Chandrasekhar, Radiative Transfer (Oxford U. Press, New York, 1950).
  8. M. Benassi, R. D. M. Garcia, A. H. J. Karp, C. E. Siewert, “A high-order spherical harmonics solution to the standard problem in radiative transfer,” Astrophys. J. 280, 853–864 (1984).
    [CrossRef]
  9. C. D. Mobley, B. Gentili, H. R. Gordon, Z. Jin, G. W. Kattawar, A. Morel, P. Reinersman, K. Stamnes, R. H. Stavn, “Comparison of numerical models for computing underwater light fields,” Appl. Opt. 32, 7484–7504 (1993).
    [CrossRef] [PubMed]
  10. Z. Tao, N. J. McCormick, R. Sanchez, “Ocean source and optical property estimation using explicit and implicit algorithms,” Appl. Opt. 33, 3265–3275 (1994).
    [CrossRef] [PubMed]
  11. T. J. Petzold, “Volume scattering functions for selected ocean waters,” (Scripps Institution of Oceanography, La Jolla, Calif., 1972).
  12. Z. Jin, K. Stamnes, “Radiative transfer in nonuniformly refracting layered media: atmosphere–ocean system,” Appl. Opt. 33, 431–442 (1994).
    [CrossRef] [PubMed]
  13. C. D. Mobley, “Hydrolight 3.1 Users’ Guide, 1996,” (SRI International, Menlo Park, Calif., 1996).
  14. W. S. Pegau, D. Gray, J. R. V. Zaneveld, “Absorption and attenuation of visible and near-infrared light in water: dependence on temperature and salinity,” Appl. Opt. 36, 6035–6046 (1997).
    [CrossRef] [PubMed]
  15. C. S. Roesler, J. R. V. Zaneveld, “High resolution vertical profiles of spectral absorption, attenuation, and scattering coefficients in highly stratified waters,” in Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE2258, 309–319 (1994).
    [CrossRef]
  16. A. C. Tam, C. K. N. Patel, “Optical absorptions of light and heavy water by laser optoacoustic spectroscopy,” Appl. Opt. 18, 3348–3357 (1979).
    [CrossRef] [PubMed]

1998 (1)

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]

1997 (4)

1996 (1)

N. J. McCormick, “Analytical transport theory applications in optical oceanography,” Ann. Nucl. Energy 23, 381–395 (1996).
[CrossRef]

1994 (2)

Z. Tao, N. J. McCormick, R. Sanchez, “Ocean source and optical property estimation using explicit and implicit algorithms,” Appl. Opt. 33, 3265–3275 (1994).
[CrossRef] [PubMed]

Z. Jin, K. Stamnes, “Radiative transfer in nonuniformly refracting layered media: atmosphere–ocean system,” Appl. Opt. 33, 431–442 (1994).
[CrossRef] [PubMed]

1993 (1)

1984 (1)

M. Benassi, R. D. M. Garcia, A. H. J. Karp, C. E. Siewert, “A high-order spherical harmonics solution to the standard problem in radiative transfer,” Astrophys. J. 280, 853–864 (1984).
[CrossRef]

1979 (1)

A. C. Tam, C. K. N. Patel, “Optical absorptions of light and heavy water by laser optoacoustic spectroscopy,” Appl. Opt. 18, 3348–3357 (1979).
[CrossRef] [PubMed]

Benassi, M.

M. Benassi, R. D. M. Garcia, A. H. J. Karp, C. E. Siewert, “A high-order spherical harmonics solution to the standard problem in radiative transfer,” Astrophys. J. 280, 853–864 (1984).
[CrossRef]

Boynton, G. C.

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]

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]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Oxford U. Press, New York, 1950).

Fry, E. S.

Garcia, R. D. M.

M. Benassi, R. D. M. Garcia, A. H. J. Karp, C. E. Siewert, “A high-order spherical harmonics solution to the standard problem in radiative transfer,” Astrophys. J. 280, 853–864 (1984).
[CrossRef]

Gentili, B.

Gordon, H. R.

Gray, D.

Jin, Z.

Karp, A. H. J.

M. Benassi, R. D. M. Garcia, A. H. J. Karp, C. E. Siewert, “A high-order spherical harmonics solution to the standard problem in radiative transfer,” Astrophys. J. 280, 853–864 (1984).
[CrossRef]

Kattawar, G. W.

Leathers, R. A.

McCormick, N. J.

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

N. J. McCormick, “Analytical transport theory applications in optical oceanography,” Ann. Nucl. Energy 23, 381–395 (1996).
[CrossRef]

Z. Tao, N. J. McCormick, R. Sanchez, “Ocean source and optical property estimation using explicit and implicit algorithms,” Appl. Opt. 33, 3265–3275 (1994).
[CrossRef] [PubMed]

Mobley, C. D.

Morel, A.

Patel, C. K. N.

A. C. Tam, C. K. N. Patel, “Optical absorptions of light and heavy water by laser optoacoustic spectroscopy,” Appl. Opt. 18, 3348–3357 (1979).
[CrossRef] [PubMed]

Pegau, W. S.

Petzold, T. J.

T. J. Petzold, “Volume scattering functions for selected ocean waters,” (Scripps Institution of Oceanography, La Jolla, Calif., 1972).

Pope, R. M.

Reinersman, P.

Roesler, C. S.

C. S. Roesler, J. R. V. Zaneveld, “High resolution vertical profiles of spectral absorption, attenuation, and scattering coefficients in highly stratified waters,” in Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE2258, 309–319 (1994).
[CrossRef]

Sanchez, R.

Z. Tao, N. J. McCormick, R. Sanchez, “Ocean source and optical property estimation using explicit and implicit algorithms,” Appl. Opt. 33, 3265–3275 (1994).
[CrossRef] [PubMed]

Siewert, C. E.

M. Benassi, R. D. M. Garcia, A. H. J. Karp, C. E. Siewert, “A high-order spherical harmonics solution to the standard problem in radiative transfer,” Astrophys. J. 280, 853–864 (1984).
[CrossRef]

Stamnes, K.

Stavn, R. H.

Tam, A. C.

A. C. Tam, C. K. N. Patel, “Optical absorptions of light and heavy water by laser optoacoustic spectroscopy,” Appl. Opt. 18, 3348–3357 (1979).
[CrossRef] [PubMed]

Tao, Z.

Z. Tao, N. J. McCormick, R. Sanchez, “Ocean source and optical property estimation using explicit and implicit algorithms,” Appl. Opt. 33, 3265–3275 (1994).
[CrossRef] [PubMed]

Zaneveld, J. R. V.

W. S. Pegau, D. Gray, J. R. V. Zaneveld, “Absorption and attenuation of visible and near-infrared light in water: dependence on temperature and salinity,” Appl. Opt. 36, 6035–6046 (1997).
[CrossRef] [PubMed]

C. S. Roesler, J. R. V. Zaneveld, “High resolution vertical profiles of spectral absorption, attenuation, and scattering coefficients in highly stratified waters,” in Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE2258, 309–319 (1994).
[CrossRef]

J. R. V. Zaneveld, “A reflecting tube absorption meter,” in Ocean Optics X, R. W. Spinrad, ed., Proc. SPIE1302, 124–136 (1990).
[CrossRef]

Ann. Nucl. Energy (1)

N. J. McCormick, “Analytical transport theory applications in optical oceanography,” Ann. Nucl. Energy 23, 381–395 (1996).
[CrossRef]

Appl. Opt. (3)

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]

Z. Tao, N. J. McCormick, R. Sanchez, “Ocean source and optical property estimation using explicit and implicit algorithms,” Appl. Opt. 33, 3265–3275 (1994).
[CrossRef] [PubMed]

A. C. Tam, C. K. N. Patel, “Optical absorptions of light and heavy water by laser optoacoustic spectroscopy,” Appl. Opt. 18, 3348–3357 (1979).
[CrossRef] [PubMed]

Appl. Opt. (6)

Astrophys. J. (1)

M. Benassi, R. D. M. Garcia, A. H. J. Karp, C. E. Siewert, “A high-order spherical harmonics solution to the standard problem in radiative transfer,” Astrophys. J. 280, 853–864 (1984).
[CrossRef]

Other (5)

T. J. Petzold, “Volume scattering functions for selected ocean waters,” (Scripps Institution of Oceanography, La Jolla, Calif., 1972).

C. D. Mobley, “Hydrolight 3.1 Users’ Guide, 1996,” (SRI International, Menlo Park, Calif., 1996).

C. S. Roesler, J. R. V. Zaneveld, “High resolution vertical profiles of spectral absorption, attenuation, and scattering coefficients in highly stratified waters,” in Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE2258, 309–319 (1994).
[CrossRef]

J. R. V. Zaneveld, “A reflecting tube absorption meter,” in Ocean Optics X, R. W. Spinrad, ed., Proc. SPIE1302, 124–136 (1990).
[CrossRef]

S. Chandrasekhar, Radiative Transfer (Oxford U. Press, New York, 1950).

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Figures (8)

Fig. 1
Fig. 1

Single-scattering albedo ω 0 , the largest eigenvalue ν1 , and the ratio of scattering to absorption versus the asymptotic radiance–irradiance ratio R L for the Petzold scattering phase function.

Fig. 2
Fig. 2

Normalized sensitivity coefficients (R / a)(d a /d R ), (R L / a)(d a /d R L), (R / b b)(d b b /d R ), and (R L / b b)(d b b/d R L) for the Petzold scattering phase function.

Fig. 3
Fig. 3

Simulated profiles of the irradiance and radiance–irradiance ratios and of the diffuse attenuation coefficients for the case of homogeneous optically deep water with a = 0.30, b = 0.70, a flat surface, and an illumination that is 30% diffuse skylight and 70% direct sunlight at 30° from the zenith.

Fig. 4
Fig. 4

Estimates of the absorption and backscattering coefficients for the simulation of Fig. 3 obtained with the E uE d and L uE d methods.

Fig. 5
Fig. 5

Determination of the absorption and backscattering coefficients with the L uE d (solid curve) and E uE d (dashed curve) methods for a simulation (dotted curve) with constant a (z) = 0.3 and b (z) varying sinusoidally with depth. The surface was flat and the illumination diffuse.

Fig. 6
Fig. 6

Comparison at site 8 of estimates of the absorption coefficients at (from right to left) 411, 443, 490, and 555 nm from the ac-9 (solid curve) with those from the L uE d method (dashed curve).

Fig. 7
Fig. 7

Absorption coefficients determined with the L uE d method versus those obtained with the ac-9. Included are data at 2-m intervals from sites 1–4 and 7–9. Also shown is the line a Sat = a ac-9.

Fig. 8
Fig. 8

Absorption coefficients determined with the E uE d method versus those obtained with the ac-9 for sites 1, 4, and 8.

Tables (1)

Tables Icon

Table 1 Location and Depth of the Cruise Stations in Long Island Sound

Equations (22)

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μ L z ,   μ / z + c z L z ,   μ = b z   1 1 β ˜ z ,   μ ,   μ d μ ,
b b = b   1 0 β ˜ μ ,   1 d μ ,
R z = E u z / E d z ,       R L z = L u z / E d z ,
E u z = 2 π   1 0 | μ | L z ,   μ d μ , E d z = 2 π   0 1 μ L z ,   μ d μ .
K u z = - d ln E u z d z ,     K d z = - d ln E d z d z ,
K L z ,   - 1 = - d ln L u z d z .
R = g ˜ 1 - ν 1 / g ˜ 1 ν 1 ,
R L = ϕ - ν 1 ,   1 / g ˜ 1 ν 1 ,
K = c / ν 1 ,
g ˜ 1 ν = 0 1 ϕ ν ,   μ μ d μ
β ˜ μ ,   μ = 1 2 n = 0 M 2 n + 1 f n P n μ P n μ ,       f 0 = 1 ,
ϕ ±   ν j ,     μ = ω ο ν j 2 ν j     μ n = 0 M 2 n + 1 f n ×   g n ± ν j P n μ ,     ν j   >   1 .
n g n ν j = h n - 1 ν j g n - 1 ν j - n - 1 g n - 2 ν j ,
g N + 1 ν j = 0 .
b b = b / 2 n = 0 M 2 n + 1 f n - 1 0 P n μ d μ
=   ( b / 2 ) 1 - n   odd 2 n + 1 f n   0 1 P n μ d μ .
K a   a K = K b b   b b K = 1 .
R L a   a R L = R L ν 1     ν 1 R L - ω 0 1 - ω 0 R L ω 0   ω 0 R L ,
R L b b   b b R L = R L ν 1     ν 1 R L + R L ω 0   ω 0 R L .
p a   a p = p ν 1   ν 1 p - ω 0 1 - ω 0 p ω 0   ω 0 p ,
p b b   b b p = p ν 1 ν 1 p + p ω 0   ω 0 p + p b ˜ b   b ˜ b p ,
b ˜ b b b b b b ˜ b = b ˜ b ν 1 ν 1 b ˜ b + b ˜ b ω 0 ω 0 b ˜ b + 1 .

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