Abstract

Upwelling radiance measurements made with instruments designed to float at the sea surface are shaded both by the instrument housing and by the buoy that holds the instrument. The amount of shading is wavelength dependent and is affected by the local marine and atmospheric conditions. Radiance measurements made with such instruments should be corrected for this self-shading error before being applied to remote sensing calibrations or remote sensing algorithm validation. Here we use Monte Carlo simulations to compute the self-shading error of a commercially available buoyed radiometer so that measurements made with this instrument can be improved. This approach can be easily adapted to the dimensions of other instruments.

©2001 Optical Society of America

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References

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  • Publication
  1. H. R. Gordon and K. Ding, “Self-shading of in-water optical instruments,” Limnol. Oceanogr. 37, 491–500 (1992).
    [Crossref]
  2. D. R. Lyzenga, “Remote sensing of bottom reflectance and water attenuation parameters in shallow water using aircraft and Landsat data,” Int. J. Remote Sensing 2, 71–82 (1981).
    [Crossref]
  3. H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10909–10924 (1988).
    [Crossref]
  4. J. E. O’Reilley, S. Maritorena, B. G. Mitchell, D. A. Siegel, K. L. Carder, S. A. Garver, M. Kahru, and C. McClain, “Ocean color chlorophyll algorithms for SeaWiFS,” J. Geophys. Res. 103, 24937–24953 (1998).
    [Crossref]
  5. D. R. Lyzenga, “Passive remote sensing techniques for mapping water depth and bottom features,” Appl. Opt. 17, 379–383 (1978).
    [Crossref] [PubMed]
  6. G. Zibordi and G. M. Ferrari, “Instrument self-shading in underwater optical measurements: experimental data,” Appl. Opt. 34, 2750–2754 (1995).
    [Crossref] [PubMed]
  7. E. Aas and B. Korsbø, “Self-shading effect by radiance meters on upward radiance observed in coastal waters,” Limnol. Oceanogr. 42, 968–974 (1997).
    [Crossref]
  8. J. Piskozub, A. R. Weeks, J. N. Schwarz, and I. S. Robinson, “Self-shading of upwelling irradiance for an instrument with sensors on a sidearm,” Appl. Opt. 39, 1872–1878 (2000).
    [Crossref]
  9. G. S. Fargion and J. L. Mueller, Ocean optics protocols for satellite ocean color sensor validation, revision 2 (NASA Tech. Memo. 209966, NASA Goddard Space Flight Center, Greenbelt, Maryland, 2000).
  10. Coastal Benthic Optical Properties, http://www.psicorp.com/cobop/cobop.html.
  11. H. R. Gordon, “Ship perturbation of irradiance measurements at sea. 1. Monte-Carlo simulations,” Appl. Opt. 24, 4172–4182 (1985).
    [Crossref] [PubMed]
  12. C. D. Mobley, Light and Water. Radiative Transfer in Natural Waters, (Academic Press, New York, 1994).
  13. G. R. Fournier and J. L. Forand, “Analytic phase function for ocean water,” in Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE2258, 194–201 (1994).
  14. C. D. Mobley, B. Gentili, H. R. Gordon, Z. Jin, G. W. Kattawar, A. Morel, P. Reinersman, K. Stamnes, and R. H. Stavn, “Comparison of numerical models for computing underwater light fields,” Appl. Opt. 32, 7484–7504 (1993).
    [Crossref] [PubMed]
  15. C. Cox and W. Munk, “Statistics of the sea surface derived from sun glitter,” J. Mar. Res. 13, 198–227 (1954).
  16. S. McLean, Satlantic Incorporated, Halifax, Nova Scotia, Canada. Personal correspondence.
  17. A. Ariew, “Are probabilities necessary for evolutionary explanations?,” Biol. Phil. 13, 245–253 (1998).
    [Crossref]
  18. A. W. Harrison and C. A. Coombes, “An opaque cloud cover model of sky short wavelength radiance,” Solar Energy 41, 387–392 (1988).
    [Crossref]
  19. C. S. Roesler, “Theoretical and experimental approaches to improve the accuracy of particulate absorption coefficients derived from the quantitative filter technique,” Limnol. Oceanogr. 43, 1649–1660 (1998).
    [Crossref]
  20. R. A. Leathers, T. V. Downes, and C. O. Davis, “Analysis of a point-source integrating-cavity absorption meter,” Appl. Opt. 39, 6118–6127 (2000).
    [Crossref]
  21. L. Prieur and S. Sathyendranath, “An optical classification of coastal and oceanic waters based on the specific spectral absorption curves of phytoplankton pigments, dissolved organic matter, and other particulate materials,” Limnol. Oceanogr. 26, 671–689 (1981).
    [Crossref]
  22. R. M. Pope and E. S. Fry, “Absorption spectrum (380–700nm) of pure water. II. Integrating cavity measurements,” Appl. Opt. 368710–8723, (1997).
    [Crossref]
  23. R. C. Smith and K. S. Baker, “Optical properties of the clearest natural waters (200–800 nm),” Appl. Opt. 20, 177–184 (1981).
    [Crossref] [PubMed]
  24. W. W. Gregg and K. L. Carder, “A simple spectral solar irradiance model for cloudless maritime atmospheres,” Limnol. Oceanogr. 351657–1675, (1990).
    [Crossref]
  25. H. R. Gordon, “Dependence of the diffuse reflectance of natural waters on the sun angle,” Limnol. Oceanogr. 34, 1484–1489 (1989).
    [Crossref]
  26. K. S. Baker and R. C. Smith, “Irradiance transmittance through the air-water interface,” in Ocean Optics X, R. W. Spinrad, ed., Proc. SPIE1302, 556–565 (1990).

2000 (2)

1998 (3)

J. E. O’Reilley, S. Maritorena, B. G. Mitchell, D. A. Siegel, K. L. Carder, S. A. Garver, M. Kahru, and C. McClain, “Ocean color chlorophyll algorithms for SeaWiFS,” J. Geophys. Res. 103, 24937–24953 (1998).
[Crossref]

A. Ariew, “Are probabilities necessary for evolutionary explanations?,” Biol. Phil. 13, 245–253 (1998).
[Crossref]

C. S. Roesler, “Theoretical and experimental approaches to improve the accuracy of particulate absorption coefficients derived from the quantitative filter technique,” Limnol. Oceanogr. 43, 1649–1660 (1998).
[Crossref]

1997 (2)

E. Aas and B. Korsbø, “Self-shading effect by radiance meters on upward radiance observed in coastal waters,” Limnol. Oceanogr. 42, 968–974 (1997).
[Crossref]

R. M. Pope and E. S. Fry, “Absorption spectrum (380–700nm) of pure water. II. Integrating cavity measurements,” Appl. Opt. 368710–8723, (1997).
[Crossref]

1995 (1)

1993 (1)

1992 (1)

H. R. Gordon and K. Ding, “Self-shading of in-water optical instruments,” Limnol. Oceanogr. 37, 491–500 (1992).
[Crossref]

1990 (1)

W. W. Gregg and K. L. Carder, “A simple spectral solar irradiance model for cloudless maritime atmospheres,” Limnol. Oceanogr. 351657–1675, (1990).
[Crossref]

1989 (1)

H. R. Gordon, “Dependence of the diffuse reflectance of natural waters on the sun angle,” Limnol. Oceanogr. 34, 1484–1489 (1989).
[Crossref]

1988 (2)

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10909–10924 (1988).
[Crossref]

A. W. Harrison and C. A. Coombes, “An opaque cloud cover model of sky short wavelength radiance,” Solar Energy 41, 387–392 (1988).
[Crossref]

1985 (1)

1981 (3)

L. Prieur and S. Sathyendranath, “An optical classification of coastal and oceanic waters based on the specific spectral absorption curves of phytoplankton pigments, dissolved organic matter, and other particulate materials,” Limnol. Oceanogr. 26, 671–689 (1981).
[Crossref]

D. R. Lyzenga, “Remote sensing of bottom reflectance and water attenuation parameters in shallow water using aircraft and Landsat data,” Int. J. Remote Sensing 2, 71–82 (1981).
[Crossref]

R. C. Smith and K. S. Baker, “Optical properties of the clearest natural waters (200–800 nm),” Appl. Opt. 20, 177–184 (1981).
[Crossref] [PubMed]

1978 (1)

1954 (1)

C. Cox and W. Munk, “Statistics of the sea surface derived from sun glitter,” J. Mar. Res. 13, 198–227 (1954).

Aas, E.

E. Aas and B. Korsbø, “Self-shading effect by radiance meters on upward radiance observed in coastal waters,” Limnol. Oceanogr. 42, 968–974 (1997).
[Crossref]

Ariew, A.

A. Ariew, “Are probabilities necessary for evolutionary explanations?,” Biol. Phil. 13, 245–253 (1998).
[Crossref]

Baker, K. S.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10909–10924 (1988).
[Crossref]

R. C. Smith and K. S. Baker, “Optical properties of the clearest natural waters (200–800 nm),” Appl. Opt. 20, 177–184 (1981).
[Crossref] [PubMed]

K. S. Baker and R. C. Smith, “Irradiance transmittance through the air-water interface,” in Ocean Optics X, R. W. Spinrad, ed., Proc. SPIE1302, 556–565 (1990).

Brown, J. W.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10909–10924 (1988).
[Crossref]

Brown, O. B.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10909–10924 (1988).
[Crossref]

Carder, K. L.

J. E. O’Reilley, S. Maritorena, B. G. Mitchell, D. A. Siegel, K. L. Carder, S. A. Garver, M. Kahru, and C. McClain, “Ocean color chlorophyll algorithms for SeaWiFS,” J. Geophys. Res. 103, 24937–24953 (1998).
[Crossref]

W. W. Gregg and K. L. Carder, “A simple spectral solar irradiance model for cloudless maritime atmospheres,” Limnol. Oceanogr. 351657–1675, (1990).
[Crossref]

Clark, D. K.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10909–10924 (1988).
[Crossref]

Coombes, C. A.

A. W. Harrison and C. A. Coombes, “An opaque cloud cover model of sky short wavelength radiance,” Solar Energy 41, 387–392 (1988).
[Crossref]

Cox, C.

C. Cox and W. Munk, “Statistics of the sea surface derived from sun glitter,” J. Mar. Res. 13, 198–227 (1954).

Davis, C. O.

Ding, K.

H. R. Gordon and K. Ding, “Self-shading of in-water optical instruments,” Limnol. Oceanogr. 37, 491–500 (1992).
[Crossref]

Downes, T. V.

Evans, R. H.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10909–10924 (1988).
[Crossref]

Fargion, G. S.

G. S. Fargion and J. L. Mueller, Ocean optics protocols for satellite ocean color sensor validation, revision 2 (NASA Tech. Memo. 209966, NASA Goddard Space Flight Center, Greenbelt, Maryland, 2000).

Ferrari, G. M.

Forand, J. L.

G. R. Fournier and J. L. Forand, “Analytic phase function for ocean water,” in Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE2258, 194–201 (1994).

Fournier, G. R.

G. R. Fournier and J. L. Forand, “Analytic phase function for ocean water,” in Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE2258, 194–201 (1994).

Fry, E. S.

Garver, S. A.

J. E. O’Reilley, S. Maritorena, B. G. Mitchell, D. A. Siegel, K. L. Carder, S. A. Garver, M. Kahru, and C. McClain, “Ocean color chlorophyll algorithms for SeaWiFS,” J. Geophys. Res. 103, 24937–24953 (1998).
[Crossref]

Gentili, B.

Gordon, H. R.

C. D. Mobley, B. Gentili, H. R. Gordon, Z. Jin, G. W. Kattawar, A. Morel, P. Reinersman, K. Stamnes, and R. H. Stavn, “Comparison of numerical models for computing underwater light fields,” Appl. Opt. 32, 7484–7504 (1993).
[Crossref] [PubMed]

H. R. Gordon and K. Ding, “Self-shading of in-water optical instruments,” Limnol. Oceanogr. 37, 491–500 (1992).
[Crossref]

H. R. Gordon, “Dependence of the diffuse reflectance of natural waters on the sun angle,” Limnol. Oceanogr. 34, 1484–1489 (1989).
[Crossref]

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10909–10924 (1988).
[Crossref]

H. R. Gordon, “Ship perturbation of irradiance measurements at sea. 1. Monte-Carlo simulations,” Appl. Opt. 24, 4172–4182 (1985).
[Crossref] [PubMed]

Gregg, W. W.

W. W. Gregg and K. L. Carder, “A simple spectral solar irradiance model for cloudless maritime atmospheres,” Limnol. Oceanogr. 351657–1675, (1990).
[Crossref]

Harrison, A. W.

A. W. Harrison and C. A. Coombes, “An opaque cloud cover model of sky short wavelength radiance,” Solar Energy 41, 387–392 (1988).
[Crossref]

Jin, Z.

Kahru, M.

J. E. O’Reilley, S. Maritorena, B. G. Mitchell, D. A. Siegel, K. L. Carder, S. A. Garver, M. Kahru, and C. McClain, “Ocean color chlorophyll algorithms for SeaWiFS,” J. Geophys. Res. 103, 24937–24953 (1998).
[Crossref]

Kattawar, G. W.

Korsbø, B.

E. Aas and B. Korsbø, “Self-shading effect by radiance meters on upward radiance observed in coastal waters,” Limnol. Oceanogr. 42, 968–974 (1997).
[Crossref]

Leathers, R. A.

Lyzenga, D. R.

D. R. Lyzenga, “Remote sensing of bottom reflectance and water attenuation parameters in shallow water using aircraft and Landsat data,” Int. J. Remote Sensing 2, 71–82 (1981).
[Crossref]

D. R. Lyzenga, “Passive remote sensing techniques for mapping water depth and bottom features,” Appl. Opt. 17, 379–383 (1978).
[Crossref] [PubMed]

Maritorena, S.

J. E. O’Reilley, S. Maritorena, B. G. Mitchell, D. A. Siegel, K. L. Carder, S. A. Garver, M. Kahru, and C. McClain, “Ocean color chlorophyll algorithms for SeaWiFS,” J. Geophys. Res. 103, 24937–24953 (1998).
[Crossref]

McClain, C.

J. E. O’Reilley, S. Maritorena, B. G. Mitchell, D. A. Siegel, K. L. Carder, S. A. Garver, M. Kahru, and C. McClain, “Ocean color chlorophyll algorithms for SeaWiFS,” J. Geophys. Res. 103, 24937–24953 (1998).
[Crossref]

McLean, S.

S. McLean, Satlantic Incorporated, Halifax, Nova Scotia, Canada. Personal correspondence.

Mitchell, B. G.

J. E. O’Reilley, S. Maritorena, B. G. Mitchell, D. A. Siegel, K. L. Carder, S. A. Garver, M. Kahru, and C. McClain, “Ocean color chlorophyll algorithms for SeaWiFS,” J. Geophys. Res. 103, 24937–24953 (1998).
[Crossref]

Mobley, C. D.

Morel, A.

Mueller, J. L.

G. S. Fargion and J. L. Mueller, Ocean optics protocols for satellite ocean color sensor validation, revision 2 (NASA Tech. Memo. 209966, NASA Goddard Space Flight Center, Greenbelt, Maryland, 2000).

Munk, W.

C. Cox and W. Munk, “Statistics of the sea surface derived from sun glitter,” J. Mar. Res. 13, 198–227 (1954).

O’Reilley, J. E.

J. E. O’Reilley, S. Maritorena, B. G. Mitchell, D. A. Siegel, K. L. Carder, S. A. Garver, M. Kahru, and C. McClain, “Ocean color chlorophyll algorithms for SeaWiFS,” J. Geophys. Res. 103, 24937–24953 (1998).
[Crossref]

Piskozub, J.

Pope, R. M.

Prieur, L.

L. Prieur and S. Sathyendranath, “An optical classification of coastal and oceanic waters based on the specific spectral absorption curves of phytoplankton pigments, dissolved organic matter, and other particulate materials,” Limnol. Oceanogr. 26, 671–689 (1981).
[Crossref]

Reinersman, P.

Robinson, I. S.

Roesler, C. S.

C. S. Roesler, “Theoretical and experimental approaches to improve the accuracy of particulate absorption coefficients derived from the quantitative filter technique,” Limnol. Oceanogr. 43, 1649–1660 (1998).
[Crossref]

Sathyendranath, S.

L. Prieur and S. Sathyendranath, “An optical classification of coastal and oceanic waters based on the specific spectral absorption curves of phytoplankton pigments, dissolved organic matter, and other particulate materials,” Limnol. Oceanogr. 26, 671–689 (1981).
[Crossref]

Schwarz, J. N.

Siegel, D. A.

J. E. O’Reilley, S. Maritorena, B. G. Mitchell, D. A. Siegel, K. L. Carder, S. A. Garver, M. Kahru, and C. McClain, “Ocean color chlorophyll algorithms for SeaWiFS,” J. Geophys. Res. 103, 24937–24953 (1998).
[Crossref]

Smith, R. C.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10909–10924 (1988).
[Crossref]

R. C. Smith and K. S. Baker, “Optical properties of the clearest natural waters (200–800 nm),” Appl. Opt. 20, 177–184 (1981).
[Crossref] [PubMed]

K. S. Baker and R. C. Smith, “Irradiance transmittance through the air-water interface,” in Ocean Optics X, R. W. Spinrad, ed., Proc. SPIE1302, 556–565 (1990).

Stamnes, K.

Stavn, R. H.

Weeks, A. R.

Zibordi, G.

Appl. Opt. (8)

Biol. Phil. (1)

A. Ariew, “Are probabilities necessary for evolutionary explanations?,” Biol. Phil. 13, 245–253 (1998).
[Crossref]

Int. J. Remote Sensing (1)

D. R. Lyzenga, “Remote sensing of bottom reflectance and water attenuation parameters in shallow water using aircraft and Landsat data,” Int. J. Remote Sensing 2, 71–82 (1981).
[Crossref]

J. Geophys. Res. (2)

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10909–10924 (1988).
[Crossref]

J. E. O’Reilley, S. Maritorena, B. G. Mitchell, D. A. Siegel, K. L. Carder, S. A. Garver, M. Kahru, and C. McClain, “Ocean color chlorophyll algorithms for SeaWiFS,” J. Geophys. Res. 103, 24937–24953 (1998).
[Crossref]

J. Mar. Res. (1)

C. Cox and W. Munk, “Statistics of the sea surface derived from sun glitter,” J. Mar. Res. 13, 198–227 (1954).

Limnol. Oceanogr. (6)

E. Aas and B. Korsbø, “Self-shading effect by radiance meters on upward radiance observed in coastal waters,” Limnol. Oceanogr. 42, 968–974 (1997).
[Crossref]

L. Prieur and S. Sathyendranath, “An optical classification of coastal and oceanic waters based on the specific spectral absorption curves of phytoplankton pigments, dissolved organic matter, and other particulate materials,” Limnol. Oceanogr. 26, 671–689 (1981).
[Crossref]

W. W. Gregg and K. L. Carder, “A simple spectral solar irradiance model for cloudless maritime atmospheres,” Limnol. Oceanogr. 351657–1675, (1990).
[Crossref]

H. R. Gordon, “Dependence of the diffuse reflectance of natural waters on the sun angle,” Limnol. Oceanogr. 34, 1484–1489 (1989).
[Crossref]

C. S. Roesler, “Theoretical and experimental approaches to improve the accuracy of particulate absorption coefficients derived from the quantitative filter technique,” Limnol. Oceanogr. 43, 1649–1660 (1998).
[Crossref]

H. R. Gordon and K. Ding, “Self-shading of in-water optical instruments,” Limnol. Oceanogr. 37, 491–500 (1992).
[Crossref]

Solar Energy (1)

A. W. Harrison and C. A. Coombes, “An opaque cloud cover model of sky short wavelength radiance,” Solar Energy 41, 387–392 (1988).
[Crossref]

Other (6)

S. McLean, Satlantic Incorporated, Halifax, Nova Scotia, Canada. Personal correspondence.

G. S. Fargion and J. L. Mueller, Ocean optics protocols for satellite ocean color sensor validation, revision 2 (NASA Tech. Memo. 209966, NASA Goddard Space Flight Center, Greenbelt, Maryland, 2000).

Coastal Benthic Optical Properties, http://www.psicorp.com/cobop/cobop.html.

C. D. Mobley, Light and Water. Radiative Transfer in Natural Waters, (Academic Press, New York, 1994).

G. R. Fournier and J. L. Forand, “Analytic phase function for ocean water,” in Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE2258, 194–201 (1994).

K. S. Baker and R. C. Smith, “Irradiance transmittance through the air-water interface,” in Ocean Optics X, R. W. Spinrad, ed., Proc. SPIE1302, 556–565 (1990).

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Figures (7)

Fig. 1.
Fig. 1. Self-shading error of the TSRB (solid line) in deep water for b/a=2 compared with that computed with Eqs. (1) and (2) for radii 0.044 m and 0.15 m. The TSRB error was computed for an empirical sky model at 480 nm with the sun zenith angle θ 0=30 degrees.

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Fig. 2.
Fig. 2. Self-shading error of the TSRB in deep water for b/a=2, five solar zenith angles, and no skylight.

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Fig. 3.
Fig. 3. Shaded and unshaded normalized radiance and percent shading error of the TSRB and of the TSRB cylindrical body without the buoy compared with Eq. (1) for radii 0.044 m and 0.15 m. The computations are for a sun in a black sky, deep water, and a=b=0.2 m-1.

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Fig. 4.
Fig. 4. Percent shading error of the TSRB in deep water versus solar zenith angle θ 0 for a sun in a black sky, six values of a, and b/a=2.

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Fig. 5.
Fig. 5. Percent shading error of the TSRB in deep water for a=0.02 m-1 (dashed), 0.05 m-1 (solid), and 1.0 m-1 (dotted) and θ 0=0° (top, blue), 10° (middle, green), and 20° (bottom, red).

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Fig. 6.
Fig. 6. TSRB self-shading error as a function of water depth for absorption coefficient a=0.2 m-1, scattering coefficient b=0.4 m-1, and bottom albedo Rb =0.2. The dashed lines show the shading error in optically deep waters for solar zenith angles θ 0=0°, 10°, and 20°.

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Fig. 7.
Fig. 7. Example TSRB shading correction. The upper plot shows the water absorption spectrum a(λ), the ratio f(λ) of skylight to sunlight, and the corresponding shading error ε(λ). The lower plot shows the measured and corrected upwelling radiance spectra.

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Tables (1)

Tables Icon

Table 1. Percent shading error (100×ε) of a TSRB for given values of absorption coefficient a, scattering coefficient b, and solar zenith angle θ 0.

View Table

Equations (10)

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

ε = [ 1 − exp ( − kar ) ] ,
ε = ε sun + ε sky f 1 + f ,
ε = ( L u true − L u m ) ⁄ L u true ,
[ x y z ] = [ x 1 y 1 z 1 ] + [ α β γ ] s , 0 ≤ s ≤ L ,
[ x y z ] = [ x 0 y 0 z 0 ] + [ ρ cos ( θ ) ρ sin ( θ ) u ] , 0 ≤ ρ ≤ r 0 ≤ θ ≤ 2 π , 0 ≤ u ≤ h ,
( x 1 − x 0 + α s ) 2 + ( y 1 − y 0 + β s ) 2 = r 2 .
s = − B ± B 2 − 4 AC 2 A , A = α 2 + β 2 , B = 2 α ( x 1 − x 0 ) + 2 β ( y 1 − y 0 ) , C = ( x 1 − x 0 ) 2 + ( y 1 − y 0 ) 2 − r 2 .
u = z 1 − z 0 + γ s .
a ( λ ) = K d ( λ ) [ 1 − R ( λ ) ] cos θ 0 0.6 + [ 0.47 + 2.5 R ( λ ) ] cos θ 0 ,
L u true ( λ ) = L u m ( λ ) ⁄ ( 1 − ε ( λ ) ) .

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