Abstract

Digital images of overcast skies as seen from the earth’s surface open new windows onto the angular details of overcast colors and visible-wavelength spectra. After calibration with a spectroradiometer, a commercial CCD camera equipped with a fisheye lens can produce colorimetrically accurate all-sky maps of overcast spectra. Histograms and azimuthally averaged curves of the resulting chromaticities show consistent, but unexpected, patterns in time-averaged overcast colors. Although widely used models such as LOWTRAN7 and MODTRAN4 cannot explain these characteristic patterns, a simple semiempirical model based on the radiative transfer equation does, and it provides insights into the visible consequences of absorption and scattering both within and beneath overcasts.

© 2008 Optical Society of America

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References

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  1. M. Minnaert, Light and Color in the Outdoors, translated and revised by L. Seymour (Springer-Verlag, 1993), p. 325. This edition's foreword indicates that the book was first published in 1937, so Minnaert's earliest research predates the mid-1930s.
  2. W. E. K. Middleton, Vision through the Atmosphere (U. Toronto Press, 1952), pp. 155-172.
  3. A pioneering work on irradiance-based overcast color is W. E. K. Middleton's “The color of the overcast sky,” J. Opt. Soc. Am. 44, 793-798 (1954).
    [CrossRef]
  4. S. Nann and C. Riordan, “Solar spectral irradiance under clear and cloudy skies: measurements and a semiempirical model,” J. Appl. Meteorol. 30, 447-462 (1991).
    [CrossRef]
  5. J. Hernández-Andrés, R. L. Lee, Jr., J. Romero, and J. L. Nieves, “Color and spectral analysis of daylight in southern Europe,” J. Opt. Soc. Am. A 18, 1325-1335 (2001).
    [CrossRef]
  6. R. L. Lee, Jr. and J. Hernández-Andrés, “Colors of the daytime overcast sky,” Appl. Opt. 44, 5712-5722 (2005).
    [CrossRef] [PubMed]
  7. One early example is L. T. Maloney and B. A. Wandell, “Color constancy: a method for recovering surface spectral reflectance,” J. Opt. Soc. Am. A 3, 29-33 (1986).
    [CrossRef] [PubMed]
  8. R. L. Lee, Jr., “Colorimetric calibration of a video digitizing system: algorithm and applications,” Color Res. Appl. 13, 180-186 (1988).
    [CrossRef]
  9. D. Connah, S. Westland, and M. G. A. Thomson, “Recovering spectral information using digital camera systems,” Coloration Technol. 117, 309-312 (2001).
    [CrossRef]
  10. J. L. Nieves, E. M. Valero, S. M. C. Nascimento, J. Hernández-Andrés, and J. Romero, “Multispectral synthesis of daylight using a commercial digital CCD camera,” Appl. Opt. 44, 5696-5703 (2005).
    [CrossRef] [PubMed]
  11. PR-650 spectroradiometer from Photo Research, Inc., 9731 Topanga Canyon Place, Chatsworth, Calif. 91311. According to Photo Research, at specified radiance levels a properly calibrated PR-650 measures luminance and radiance accurate to within ±4%, has a spectral accuracy of ±2 nm, and its CIE 1931 colorimetric errors are x<0.001, y<0.001 for a 2856 K blackbody (CIE standard illuminant A).
  12. J. Romero, A. García-Beltrán, and J. Hernández-Andrés, “Linear bases for representation of natural and artificial illuminants,” J. Opt. Soc. Am. A 14, 1007-1014 (1997).
    [CrossRef]
  13. J. Hernández-Andrés, J. L. Nieves, E. M. Valero, and J. Romero, “Spectral-daylight recovery by use of only a few sensors,” J. Opt. Soc. Am. A 21, 13-23 (2004).
    [CrossRef]
  14. For any pair of chromaticities separated by Δu′ and Δv′, calculate this error as the Euclidean distance Δu′v′=[(Δu′)2+(Δv′)2]½.
  15. G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley, 1982), pp. 306-310.
  16. All photographs are corrected for the effective integrated transmissivity of this fisheye lens as a function of angle from its optical axis, but tests show that no additional spectral corrections are needed to the transform matrix F.
  17. R. L. Lee, Jr. and J. Hernández-Andrés, “Short-term variability of overcast brightness,” Appl. Opt. 44, 5704-5711 (2005).
    [CrossRef] [PubMed]
  18. For example, see Spatial Distribution of Daylight--CIE Standard Overcast Sky and Clear Sky, CIE Standard S 003/E-1996 (Commission Internationale de l'Eclairage, 1996), p. 3.
  19. R. L. Lee, Jr. and D. E. Devan, “Observed brightness distributions in overcast skies,” Appl. Opt. 47, H116-H127 (2008).
    [CrossRef] [PubMed]
  20. For a definition of g^, see R. L. Lee, Jr., “Twilight and daytime colors of the clear sky,” Appl. Opt. 33, 4629-4638, 4959 (1994).
    [CrossRef] [PubMed]
  21. R. L. Lee, Jr., “Horizon brightness revisited: measurements and a model of clear-sky radiances,” Appl. Opt. 33, 4620-4628, 4959 (1994).
    [CrossRef] [PubMed]
  22. D. B. Judd, D. L. MacAdam, and G. Wyszecki, “Spectral distribution of typical daylight as a function of correlated color temperature,” J. Opt. Soc. Am. 54, 1031-1040 (1964).
    [CrossRef]
  23. J. Hernández-Andrés, R. L. Lee, Jr., and J. Romero, “Calculating correlated color temperatures across the entire gamut of daylight and skylight chromaticities,” Appl. Opt. 38, 5703-5709 (1999).
    [CrossRef]
  24. , pp. 224-225.
  25. C. F. Bohren and E. E. Clothiaux, Fundamentals of Atmospheric Radiation (Wiley-VCH, 2006), p. 294.
  26. C. F. Bohren, Clouds in a Glass of Beer: Simple Experiments in Atmospheric Physics (Wiley, 1987), p. 149.

2008 (1)

2005 (3)

2004 (1)

2001 (2)

D. Connah, S. Westland, and M. G. A. Thomson, “Recovering spectral information using digital camera systems,” Coloration Technol. 117, 309-312 (2001).
[CrossRef]

J. Hernández-Andrés, R. L. Lee, Jr., J. Romero, and J. L. Nieves, “Color and spectral analysis of daylight in southern Europe,” J. Opt. Soc. Am. A 18, 1325-1335 (2001).
[CrossRef]

1999 (1)

1997 (1)

1994 (2)

1991 (1)

S. Nann and C. Riordan, “Solar spectral irradiance under clear and cloudy skies: measurements and a semiempirical model,” J. Appl. Meteorol. 30, 447-462 (1991).
[CrossRef]

1988 (1)

R. L. Lee, Jr., “Colorimetric calibration of a video digitizing system: algorithm and applications,” Color Res. Appl. 13, 180-186 (1988).
[CrossRef]

1986 (1)

1964 (1)

1954 (1)

Bohren, C. F.

C. F. Bohren and E. E. Clothiaux, Fundamentals of Atmospheric Radiation (Wiley-VCH, 2006), p. 294.

C. F. Bohren, Clouds in a Glass of Beer: Simple Experiments in Atmospheric Physics (Wiley, 1987), p. 149.

Clothiaux, E. E.

C. F. Bohren and E. E. Clothiaux, Fundamentals of Atmospheric Radiation (Wiley-VCH, 2006), p. 294.

Connah, D.

D. Connah, S. Westland, and M. G. A. Thomson, “Recovering spectral information using digital camera systems,” Coloration Technol. 117, 309-312 (2001).
[CrossRef]

Devan, D. E.

García-Beltrán, A.

Hernández-Andrés, J.

Judd, D. B.

Lee, R. L.

MacAdam, D. L.

Maloney, L. T.

Middleton, W. E. K.

W. E. K. Middleton, Vision through the Atmosphere (U. Toronto Press, 1952), pp. 155-172.

Middleton's, W. E. K.

Minnaert, M.

M. Minnaert, Light and Color in the Outdoors, translated and revised by L. Seymour (Springer-Verlag, 1993), p. 325. This edition's foreword indicates that the book was first published in 1937, so Minnaert's earliest research predates the mid-1930s.

Nann, S.

S. Nann and C. Riordan, “Solar spectral irradiance under clear and cloudy skies: measurements and a semiempirical model,” J. Appl. Meteorol. 30, 447-462 (1991).
[CrossRef]

Nascimento, S. M. C.

Nieves, J. L.

Riordan, C.

S. Nann and C. Riordan, “Solar spectral irradiance under clear and cloudy skies: measurements and a semiempirical model,” J. Appl. Meteorol. 30, 447-462 (1991).
[CrossRef]

Romero, J.

Stiles, W. S.

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley, 1982), pp. 306-310.

Thomson, M. G. A.

D. Connah, S. Westland, and M. G. A. Thomson, “Recovering spectral information using digital camera systems,” Coloration Technol. 117, 309-312 (2001).
[CrossRef]

Valero, E. M.

Wandell, B. A.

Westland, S.

D. Connah, S. Westland, and M. G. A. Thomson, “Recovering spectral information using digital camera systems,” Coloration Technol. 117, 309-312 (2001).
[CrossRef]

Wyszecki, G.

D. B. Judd, D. L. MacAdam, and G. Wyszecki, “Spectral distribution of typical daylight as a function of correlated color temperature,” J. Opt. Soc. Am. 54, 1031-1040 (1964).
[CrossRef]

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley, 1982), pp. 306-310.

Appl. Opt. (7)

Color Res. Appl. (1)

R. L. Lee, Jr., “Colorimetric calibration of a video digitizing system: algorithm and applications,” Color Res. Appl. 13, 180-186 (1988).
[CrossRef]

Coloration Technol. (1)

D. Connah, S. Westland, and M. G. A. Thomson, “Recovering spectral information using digital camera systems,” Coloration Technol. 117, 309-312 (2001).
[CrossRef]

J. Appl. Meteorol. (1)

S. Nann and C. Riordan, “Solar spectral irradiance under clear and cloudy skies: measurements and a semiempirical model,” J. Appl. Meteorol. 30, 447-462 (1991).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (4)

Other (10)

M. Minnaert, Light and Color in the Outdoors, translated and revised by L. Seymour (Springer-Verlag, 1993), p. 325. This edition's foreword indicates that the book was first published in 1937, so Minnaert's earliest research predates the mid-1930s.

W. E. K. Middleton, Vision through the Atmosphere (U. Toronto Press, 1952), pp. 155-172.

PR-650 spectroradiometer from Photo Research, Inc., 9731 Topanga Canyon Place, Chatsworth, Calif. 91311. According to Photo Research, at specified radiance levels a properly calibrated PR-650 measures luminance and radiance accurate to within ±4%, has a spectral accuracy of ±2 nm, and its CIE 1931 colorimetric errors are x<0.001, y<0.001 for a 2856 K blackbody (CIE standard illuminant A).

For any pair of chromaticities separated by Δu′ and Δv′, calculate this error as the Euclidean distance Δu′v′=[(Δu′)2+(Δv′)2]½.

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley, 1982), pp. 306-310.

All photographs are corrected for the effective integrated transmissivity of this fisheye lens as a function of angle from its optical axis, but tests show that no additional spectral corrections are needed to the transform matrix F.

For example, see Spatial Distribution of Daylight--CIE Standard Overcast Sky and Clear Sky, CIE Standard S 003/E-1996 (Commission Internationale de l'Eclairage, 1996), p. 3.

, pp. 224-225.

C. F. Bohren and E. E. Clothiaux, Fundamentals of Atmospheric Radiation (Wiley-VCH, 2006), p. 294.

C. F. Bohren, Clouds in a Glass of Beer: Simple Experiments in Atmospheric Physics (Wiley, 1987), p. 149.

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Figures (17)

Fig. 1
Fig. 1

Spectral and colorimetric calibration equipment consists of two digital still cameras (a Nikon E5000 and an Olympus E300, labeled “b” and “c,” respectively) and a Photo Research PR-650 spectroradiometer (labeled “a”) affixed to a metal bracket, which is itself mounted on a tripod.

Fig. 2
Fig. 2

Normalized radiances for the 95th percentile RMSE spectrum reconstructed from Nikon E5000 camera data compared with the original radiometer spectrum of an overcast feature. Each curve is normalized so that its spectral radiances L λ satisfy ( L λ ) 2 = 1 .

Fig. 3
Fig. 3

Portion of the CIE 1976 UCS diagram, showing Sc overcast chromaticities u , v measured at USNA in Annapolis, Maryland, on 30 August 2006 and 12 January 2007. Each curve plots temporally and azimuthally averaged chromaticities versus view-elevation angle h above the astronomical horizon; the mean unrefracted sun elevation h 0 during each curve’s photography session (here, 38   min ) is also listed.

Fig. 4
Fig. 4

Time-averaged fisheye images of nonprecipitating Sc overcasts photographed at USNA on (a) 30 August 2006 and (b) 12 January 2007. The mean image in (a) is calculated from 53 individual photographs and has a measured cloud base z 365 m , whereas (b) is calculated from 78 photographs and has 2400 m < z < 2700 m .

Fig. 5
Fig. 5

Temporally and azimuthally averaged chromaticity curves for Sc overcasts measured at USNA on 20 February and 7 March 2007.

Fig. 6
Fig. 6

Temporally and azimuthally averaged chromaticity curves for Sc overcasts measured at USNA on 29 November 2006 and 9 November 2007. Light drizzle fell throughout measurements on the latter date. To show more detail, the u , v scaling is slightly anisotropic here.

Fig. 7
Fig. 7

Temporally and azimuthally averaged chromaticity curves for a clear (CLR) sky and a Sc overcast (OVC) measured at USNA on 17 January 2007 and 5 October 2006, respectively. The CLR chromaticities are averaged within two 90 ° wide sectors that are symmetric about the clear-sky principal plane (relative azimuths of 45 ° 135 ° and 225 ° 315 ° ), so these colors exclude the solar and antisolar regions. To show as much detail as possible, the u , v scaling is anisotropic here.

Fig. 8
Fig. 8

Histogram of u , v chromaticities for Sc overcast measured at USNA on the morning of 12 January 2007 when 27.6 ° < h 0 < 29.1 ° . The absolute pixel frequency in any u , v bin is the product of its gray-scale relative frequency f rel and the total number of image pixels n.

Fig. 9
Fig. 9

Histogram of u , v chromaticities for Sc overcast measured at USNA on the morning of 7 March 2007 when 39.6 ° < h 0 < 43.2 ° . Throughout measurements, precipitation varied from light snow to flurries.

Fig. 10
Fig. 10

Time-averaged fisheye image calculated from 61 individual photographs of a Sc overcast photographed at USNA on 14 September 2007; the measured cloud base is 2400 m . For consistency in printing, no image processing is used to correct the small, but constant, color bias of this and other RAW-format images (Figs. 4a, 4b).

Fig. 11
Fig. 11

Histogram of u , v chromaticities for nonprecipitating Sc overcast measured at USNA on the afternoon of 14 September 2007 when 53.0 ° > h 0 > 50.6 ° .

Fig. 12
Fig. 12

Histograms of inverse CCT for overcasts measured at USNA on 7 March 2007 (solid bars) and 14 September 2007 (striped bars). Each bar’s height is proportional to log 10 ( n ) , where n is the number of pixels in the corresponding interval of inverse CCT. Each juxtaposed pair of bars for the two dates has a common abscissa value (e.g., the leftmost pair is for 144 MK 1 ).

Fig. 13
Fig. 13

Histograms of inverse CCT for overcasts measured at USNA on 7 March 2007 (solid bars) and 12 January 2007 (striped bars).

Fig. 14
Fig. 14

Temporally and azimuthally averaged chromaticity curve for the 20 February 2007 Sc overcast compared with Sc simulations by LOWTRAN7 and MODTRAN4. For a wide range of input parameters, neither model predicts overcast chromaticity patterns well and both greatly underestimate typical overcast color gamuts.

Fig. 15
Fig. 15

Overcast chromaticity regimes typical of many overcasts and the changing balance of radiative transfer factors that explains them: (1) bluing caused by increasing spectral absorption in clouds from the zenith to h 30 ° 40 ° , (2) reddening caused by increasing extinction beneath the overcast over ever-longer slant optical paths for 30 ° 40 ° > h > 3 ° , and (3) bluing caused by increasing airlight for h < 3 ° .

Fig. 16
Fig. 16

Scattering geometry for an object of radiance L 0 seen by an observer through a multiple-scattering medium along a path of total optical depth τ f . The path consists of differential air volumes d V , each having an angular scattering phase function J ( τ ) . Here L 0 is the spectral radiance emerging from the cloud base and τ f is for the nominally clear air beneath it.

Fig. 17
Fig. 17

Temporally and azimuthally averaged chromaticity curve for the 20 February 2007 Sc overcast compared with a simulation from the semiempirical model. Like MODTRAN and LOWTRAN, this model is based on the radiative transfer equation and, unlike them, it correctly reproduces the chromaticity gamuts and turnings observed in real overcasts.

Tables (1)

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Table 1 Mean Skylight Chromaticities and Chromaticity Gamuts

Equations (3)

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F = L ρ T ( ρ ρ T ) 1 ,
L r = F ρ r ,
L f = L 0 exp ( τ f ) A + 0 τ f ( J ( τ ) exp ( τ τ f ) ) d τ B ,

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