Abstract

Clear daytime skies persistently display a subtle local maximum of radiance near the astronomical horizon. Spectroradiometry and digital image analysis confirm this maximum's reality, and they show that its angular width and elevation vary with solar elevation, azimuth relative to the Sun, and aerosol optical depth. Many existing models of atmospheric scattering do not generate this near-horizon radiance maximum, but a simple second-order scattering model does, and it reproduces many of the maximum's details.

© 1994 Optical Society of America

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References

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  1. We use radiance and brightness as synonyms in this paper, while recognizing that they are not linearly related. For example, see G. Wyszecki, W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae (Wiley, New York, 1982), pp. 259 and 495.
  2. R. Perez, J. Michalsky, R. Seals, “Modeling sky luminance angular distribution for real sky conditions: experimental evaluation of existing algorithms,” J. Ilium. Eng. Soc. 21, 84–92 (1992).
  3. F. M. F. Siala, M. A. Rosen, F. C. Hooper, “Models for the directional distribution of the diffuse sky radiance,” J. Sol. Energy Eng. 112, 102–109 (1990).
    [CrossRef]
  4. C. R. Prasad, A. K. Inamdar, P. Venkatesh, “Computation of diffuse solar radiation,” Sol. Energy 39, 521–532 (1987).
    [CrossRef]
  5. J. V. Dave, “A direct solution of the spherical harmonics approximation to the radiative transfer equation for an arbitrary solar elevation. Part I: theory,” J. Atmos. Sci. 32, 790–798 (1975).
    [CrossRef]
  6. For basic discussions of optical path length's effect on sky radiance (in a purely molecular atmosphere) see C. F. Bohren, A. B. Fraser, “Colors of the sky,” Phys. Teach. 23, 267–272 (1985) and Ref. 7.
    [CrossRef]
  7. C. F. Bohren, “Multiple scattering of light and some of its observable consequences,” Am. J. Phys. 55, 524–533 (1987).
    [CrossRef]
  8. G. Zibordi, K. J. Voss, “Geometrical and spectral distribution of sky radiance: comparison between simulations and field measurements,” Remote Sensing Environ. 27, 343–358 (1989).
    [CrossRef]
  9. R. L. Lee, “What are ‘all the colors of the rainbow’?” Appl. Opt. 30, 3401–3407, 3545 (1991).
    [CrossRef] [PubMed]
  10. D. K. Lynch, P. Schwartz, “Intensity profile of the 22° halo,” J. Opt. Soc. Am. A 2, 584–589 (1985).
    [CrossRef]
  11. A. Deepak, R. R. Adams, “Photography and photographic-photometry of the solar aureole,” Appl. Opt. 22, 1646–1654 (1983).
    [CrossRef] [PubMed]
  12. L. J. B. McArthur, J. E. Hay, “A technique for mapping the distribution of diffuse solar radiation over the sky hemisphere,” J. Appl. Meteorol. 20, 421–429 (1981).
    [CrossRef]
  13. M. A. Rosen, “The angular distribution of diffuse sky radiance: an assessment of the effects of haze,” J. Sol. Energy Eng. 113, 200–205 (1991).
    [CrossRef]
  14. A. W. Harrison, “Directional sky luminance versus cloud cover and solar position,” Sol. Energy 46, 13–19 (1991).
    [CrossRef]
  15. F. C. Hooper, A. P. Brunger, C. S. Chan, “A clear sky model of diffuse sky radiance,” J. Sol. Energy Eng. 109, 9–14 (1987).
    [CrossRef]
  16. C. G. Justus, M. V. Paris, “A model for solar spectral irradiance and radiance at the bottom and top of a cloudless atmosphere,” J. Cli. Appl. Meteorol. 24, 193–205 (1985).
    [CrossRef]
  17. J. V. Dave, “Extensive datasets of the diffuse radiation in realistic atmospheric models with aerosols and common absorbing gases,” Sol. Energy 21, 361–369 (1978).
    [CrossRef]
  18. R. L. Lee, “Colorimetric calibration of a video digitizing system: algorithm and applications,” Col. Res. Appl. 13, 180–186 (1988).
    [CrossRef]
  19. A Photo Research PR-704 spectroradiometer with a nominal 0.5° FOV was used.
  20. D. K. Lynch, “Step brightness changes of distant mountain ridges and their perception,” Appl. Opt. 30, 3508–3513 (1991).
    [CrossRef] [PubMed]
  21. C. F. Bohren, A. B. Fraser, “At what altitude does the horizon cease to be visible?” Am. J. Phys. 54, 222–227 (1986).
    [CrossRef]
  22. E. J. McCartney, Optics of the Atmosphere: Scattering by Molecules and Particles (Wiley, New York, 1976), pp. 136–138.
  23. As in the other models' simulations, all our radiance profiles are monochromatic. The wavelength λ used here is 475 nm, a typical dominant wavelength for the clear sky. This wavelength determines both the solar spectral irradiance and the angular scattering phase function for aerosols.
  24. R. L. Lee, “Twilight and daytime colors of the clear sky,” Appl. Opt. 33, 4629–4638 (1994).
    [CrossRef] [PubMed]
  25. G. E. Shaw, “Sun photometry,” Bull. Am. Meteorol. Soc. 64, 4–10 (1983).
    [CrossRef]

1994 (1)

1992 (1)

R. Perez, J. Michalsky, R. Seals, “Modeling sky luminance angular distribution for real sky conditions: experimental evaluation of existing algorithms,” J. Ilium. Eng. Soc. 21, 84–92 (1992).

1991 (4)

R. L. Lee, “What are ‘all the colors of the rainbow’?” Appl. Opt. 30, 3401–3407, 3545 (1991).
[CrossRef] [PubMed]

M. A. Rosen, “The angular distribution of diffuse sky radiance: an assessment of the effects of haze,” J. Sol. Energy Eng. 113, 200–205 (1991).
[CrossRef]

A. W. Harrison, “Directional sky luminance versus cloud cover and solar position,” Sol. Energy 46, 13–19 (1991).
[CrossRef]

D. K. Lynch, “Step brightness changes of distant mountain ridges and their perception,” Appl. Opt. 30, 3508–3513 (1991).
[CrossRef] [PubMed]

1990 (1)

F. M. F. Siala, M. A. Rosen, F. C. Hooper, “Models for the directional distribution of the diffuse sky radiance,” J. Sol. Energy Eng. 112, 102–109 (1990).
[CrossRef]

1989 (1)

G. Zibordi, K. J. Voss, “Geometrical and spectral distribution of sky radiance: comparison between simulations and field measurements,” Remote Sensing Environ. 27, 343–358 (1989).
[CrossRef]

1988 (1)

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

1987 (3)

F. C. Hooper, A. P. Brunger, C. S. Chan, “A clear sky model of diffuse sky radiance,” J. Sol. Energy Eng. 109, 9–14 (1987).
[CrossRef]

C. R. Prasad, A. K. Inamdar, P. Venkatesh, “Computation of diffuse solar radiation,” Sol. Energy 39, 521–532 (1987).
[CrossRef]

C. F. Bohren, “Multiple scattering of light and some of its observable consequences,” Am. J. Phys. 55, 524–533 (1987).
[CrossRef]

1986 (1)

C. F. Bohren, A. B. Fraser, “At what altitude does the horizon cease to be visible?” Am. J. Phys. 54, 222–227 (1986).
[CrossRef]

1985 (3)

C. G. Justus, M. V. Paris, “A model for solar spectral irradiance and radiance at the bottom and top of a cloudless atmosphere,” J. Cli. Appl. Meteorol. 24, 193–205 (1985).
[CrossRef]

For basic discussions of optical path length's effect on sky radiance (in a purely molecular atmosphere) see C. F. Bohren, A. B. Fraser, “Colors of the sky,” Phys. Teach. 23, 267–272 (1985) and Ref. 7.
[CrossRef]

D. K. Lynch, P. Schwartz, “Intensity profile of the 22° halo,” J. Opt. Soc. Am. A 2, 584–589 (1985).
[CrossRef]

1983 (2)

1981 (1)

L. J. B. McArthur, J. E. Hay, “A technique for mapping the distribution of diffuse solar radiation over the sky hemisphere,” J. Appl. Meteorol. 20, 421–429 (1981).
[CrossRef]

1978 (1)

J. V. Dave, “Extensive datasets of the diffuse radiation in realistic atmospheric models with aerosols and common absorbing gases,” Sol. Energy 21, 361–369 (1978).
[CrossRef]

1975 (1)

J. V. Dave, “A direct solution of the spherical harmonics approximation to the radiative transfer equation for an arbitrary solar elevation. Part I: theory,” J. Atmos. Sci. 32, 790–798 (1975).
[CrossRef]

Adams, R. R.

Bohren, C. F.

C. F. Bohren, “Multiple scattering of light and some of its observable consequences,” Am. J. Phys. 55, 524–533 (1987).
[CrossRef]

C. F. Bohren, A. B. Fraser, “At what altitude does the horizon cease to be visible?” Am. J. Phys. 54, 222–227 (1986).
[CrossRef]

For basic discussions of optical path length's effect on sky radiance (in a purely molecular atmosphere) see C. F. Bohren, A. B. Fraser, “Colors of the sky,” Phys. Teach. 23, 267–272 (1985) and Ref. 7.
[CrossRef]

Brunger, A. P.

F. C. Hooper, A. P. Brunger, C. S. Chan, “A clear sky model of diffuse sky radiance,” J. Sol. Energy Eng. 109, 9–14 (1987).
[CrossRef]

Chan, C. S.

F. C. Hooper, A. P. Brunger, C. S. Chan, “A clear sky model of diffuse sky radiance,” J. Sol. Energy Eng. 109, 9–14 (1987).
[CrossRef]

Dave, J. V.

J. V. Dave, “Extensive datasets of the diffuse radiation in realistic atmospheric models with aerosols and common absorbing gases,” Sol. Energy 21, 361–369 (1978).
[CrossRef]

J. V. Dave, “A direct solution of the spherical harmonics approximation to the radiative transfer equation for an arbitrary solar elevation. Part I: theory,” J. Atmos. Sci. 32, 790–798 (1975).
[CrossRef]

Deepak, A.

Fraser, A. B.

C. F. Bohren, A. B. Fraser, “At what altitude does the horizon cease to be visible?” Am. J. Phys. 54, 222–227 (1986).
[CrossRef]

For basic discussions of optical path length's effect on sky radiance (in a purely molecular atmosphere) see C. F. Bohren, A. B. Fraser, “Colors of the sky,” Phys. Teach. 23, 267–272 (1985) and Ref. 7.
[CrossRef]

Harrison, A. W.

A. W. Harrison, “Directional sky luminance versus cloud cover and solar position,” Sol. Energy 46, 13–19 (1991).
[CrossRef]

Hay, J. E.

L. J. B. McArthur, J. E. Hay, “A technique for mapping the distribution of diffuse solar radiation over the sky hemisphere,” J. Appl. Meteorol. 20, 421–429 (1981).
[CrossRef]

Hooper, F. C.

F. M. F. Siala, M. A. Rosen, F. C. Hooper, “Models for the directional distribution of the diffuse sky radiance,” J. Sol. Energy Eng. 112, 102–109 (1990).
[CrossRef]

F. C. Hooper, A. P. Brunger, C. S. Chan, “A clear sky model of diffuse sky radiance,” J. Sol. Energy Eng. 109, 9–14 (1987).
[CrossRef]

Inamdar, A. K.

C. R. Prasad, A. K. Inamdar, P. Venkatesh, “Computation of diffuse solar radiation,” Sol. Energy 39, 521–532 (1987).
[CrossRef]

Justus, C. G.

C. G. Justus, M. V. Paris, “A model for solar spectral irradiance and radiance at the bottom and top of a cloudless atmosphere,” J. Cli. Appl. Meteorol. 24, 193–205 (1985).
[CrossRef]

Lee, R. L.

R. L. Lee, “Twilight and daytime colors of the clear sky,” Appl. Opt. 33, 4629–4638 (1994).
[CrossRef] [PubMed]

R. L. Lee, “What are ‘all the colors of the rainbow’?” Appl. Opt. 30, 3401–3407, 3545 (1991).
[CrossRef] [PubMed]

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

Lynch, D. K.

McArthur, L. J. B.

L. J. B. McArthur, J. E. Hay, “A technique for mapping the distribution of diffuse solar radiation over the sky hemisphere,” J. Appl. Meteorol. 20, 421–429 (1981).
[CrossRef]

McCartney, E. J.

E. J. McCartney, Optics of the Atmosphere: Scattering by Molecules and Particles (Wiley, New York, 1976), pp. 136–138.

Michalsky, J.

R. Perez, J. Michalsky, R. Seals, “Modeling sky luminance angular distribution for real sky conditions: experimental evaluation of existing algorithms,” J. Ilium. Eng. Soc. 21, 84–92 (1992).

Paris, M. V.

C. G. Justus, M. V. Paris, “A model for solar spectral irradiance and radiance at the bottom and top of a cloudless atmosphere,” J. Cli. Appl. Meteorol. 24, 193–205 (1985).
[CrossRef]

Perez, R.

R. Perez, J. Michalsky, R. Seals, “Modeling sky luminance angular distribution for real sky conditions: experimental evaluation of existing algorithms,” J. Ilium. Eng. Soc. 21, 84–92 (1992).

Prasad, C. R.

C. R. Prasad, A. K. Inamdar, P. Venkatesh, “Computation of diffuse solar radiation,” Sol. Energy 39, 521–532 (1987).
[CrossRef]

Rosen, M. A.

M. A. Rosen, “The angular distribution of diffuse sky radiance: an assessment of the effects of haze,” J. Sol. Energy Eng. 113, 200–205 (1991).
[CrossRef]

F. M. F. Siala, M. A. Rosen, F. C. Hooper, “Models for the directional distribution of the diffuse sky radiance,” J. Sol. Energy Eng. 112, 102–109 (1990).
[CrossRef]

Schwartz, P.

Seals, R.

R. Perez, J. Michalsky, R. Seals, “Modeling sky luminance angular distribution for real sky conditions: experimental evaluation of existing algorithms,” J. Ilium. Eng. Soc. 21, 84–92 (1992).

Shaw, G. E.

G. E. Shaw, “Sun photometry,” Bull. Am. Meteorol. Soc. 64, 4–10 (1983).
[CrossRef]

Siala, F. M. F.

F. M. F. Siala, M. A. Rosen, F. C. Hooper, “Models for the directional distribution of the diffuse sky radiance,” J. Sol. Energy Eng. 112, 102–109 (1990).
[CrossRef]

Stiles, W. S.

We use radiance and brightness as synonyms in this paper, while recognizing that they are not linearly related. For example, see G. Wyszecki, W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae (Wiley, New York, 1982), pp. 259 and 495.

Venkatesh, P.

C. R. Prasad, A. K. Inamdar, P. Venkatesh, “Computation of diffuse solar radiation,” Sol. Energy 39, 521–532 (1987).
[CrossRef]

Voss, K. J.

G. Zibordi, K. J. Voss, “Geometrical and spectral distribution of sky radiance: comparison between simulations and field measurements,” Remote Sensing Environ. 27, 343–358 (1989).
[CrossRef]

Wyszecki, G.

We use radiance and brightness as synonyms in this paper, while recognizing that they are not linearly related. For example, see G. Wyszecki, W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae (Wiley, New York, 1982), pp. 259 and 495.

Zibordi, G.

G. Zibordi, K. J. Voss, “Geometrical and spectral distribution of sky radiance: comparison between simulations and field measurements,” Remote Sensing Environ. 27, 343–358 (1989).
[CrossRef]

Am. J. Phys. (2)

C. F. Bohren, “Multiple scattering of light and some of its observable consequences,” Am. J. Phys. 55, 524–533 (1987).
[CrossRef]

C. F. Bohren, A. B. Fraser, “At what altitude does the horizon cease to be visible?” Am. J. Phys. 54, 222–227 (1986).
[CrossRef]

Appl. Opt. (4)

Bull. Am. Meteorol. Soc. (1)

G. E. Shaw, “Sun photometry,” Bull. Am. Meteorol. Soc. 64, 4–10 (1983).
[CrossRef]

Col. Res. Appl. (1)

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

J. Appl. Meteorol. (1)

L. J. B. McArthur, J. E. Hay, “A technique for mapping the distribution of diffuse solar radiation over the sky hemisphere,” J. Appl. Meteorol. 20, 421–429 (1981).
[CrossRef]

J. Atmos. Sci. (1)

J. V. Dave, “A direct solution of the spherical harmonics approximation to the radiative transfer equation for an arbitrary solar elevation. Part I: theory,” J. Atmos. Sci. 32, 790–798 (1975).
[CrossRef]

J. Cli. Appl. Meteorol. (1)

C. G. Justus, M. V. Paris, “A model for solar spectral irradiance and radiance at the bottom and top of a cloudless atmosphere,” J. Cli. Appl. Meteorol. 24, 193–205 (1985).
[CrossRef]

J. Ilium. Eng. Soc. (1)

R. Perez, J. Michalsky, R. Seals, “Modeling sky luminance angular distribution for real sky conditions: experimental evaluation of existing algorithms,” J. Ilium. Eng. Soc. 21, 84–92 (1992).

J. Opt. Soc. Am. A (1)

J. Sol. Energy Eng. (3)

F. M. F. Siala, M. A. Rosen, F. C. Hooper, “Models for the directional distribution of the diffuse sky radiance,” J. Sol. Energy Eng. 112, 102–109 (1990).
[CrossRef]

M. A. Rosen, “The angular distribution of diffuse sky radiance: an assessment of the effects of haze,” J. Sol. Energy Eng. 113, 200–205 (1991).
[CrossRef]

F. C. Hooper, A. P. Brunger, C. S. Chan, “A clear sky model of diffuse sky radiance,” J. Sol. Energy Eng. 109, 9–14 (1987).
[CrossRef]

Phys. Teach. (1)

For basic discussions of optical path length's effect on sky radiance (in a purely molecular atmosphere) see C. F. Bohren, A. B. Fraser, “Colors of the sky,” Phys. Teach. 23, 267–272 (1985) and Ref. 7.
[CrossRef]

Remote Sensing Environ. (1)

G. Zibordi, K. J. Voss, “Geometrical and spectral distribution of sky radiance: comparison between simulations and field measurements,” Remote Sensing Environ. 27, 343–358 (1989).
[CrossRef]

Sol. Energy (3)

J. V. Dave, “Extensive datasets of the diffuse radiation in realistic atmospheric models with aerosols and common absorbing gases,” Sol. Energy 21, 361–369 (1978).
[CrossRef]

C. R. Prasad, A. K. Inamdar, P. Venkatesh, “Computation of diffuse solar radiation,” Sol. Energy 39, 521–532 (1987).
[CrossRef]

A. W. Harrison, “Directional sky luminance versus cloud cover and solar position,” Sol. Energy 46, 13–19 (1991).
[CrossRef]

Other (4)

A Photo Research PR-704 spectroradiometer with a nominal 0.5° FOV was used.

We use radiance and brightness as synonyms in this paper, while recognizing that they are not linearly related. For example, see G. Wyszecki, W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae (Wiley, New York, 1982), pp. 259 and 495.

E. J. McCartney, Optics of the Atmosphere: Scattering by Molecules and Particles (Wiley, New York, 1976), pp. 136–138.

As in the other models' simulations, all our radiance profiles are monochromatic. The wavelength λ used here is 475 nm, a typical dominant wavelength for the clear sky. This wavelength determines both the solar spectral irradiance and the angular scattering phase function for aerosols.

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Figures (14)

Fig. 1
Fig. 1

Clear-sky scattering geometry for an observer at point X whose view zenith and azimuth angles are θυ and ϕυ, respectively. ϕrel is the difference between the sun's azimuth and ϕυ. Clear-sky radiances reaching the observer include contributions from directly scattered sunlight, Ls, and from multiply scattered surface light and skylight, Ldiff. At each elemental scattering volume dV along the observer's line of sight, Ls and Ldiff are scattered through various angles Ψ (Ψdir for Ls is shown above).

Fig. 2
Fig. 2

Comparison of photographic and spectroradiometric measures of clear-sky radiances at University Park, Pa., at 1605 GMT on 6 October 1992 (see Plate 37). A running average has smoothed the detailed photographic data of Fig. 3. The solar zenith angle θ0 = 48°, the instruments' azimuth with respect to the sun, ϕrel, is 118°, and the equivalent Lambertian surface reflectance rsfc = 0.25.

Fig. 3
Fig. 3

Normalized radiance versus view elevation angle for a clear-sky scene at University Park, Pa., at 1605 GMT on 6 October 1992 (see Plate 37). The astronomical horizon corresponds to a view elevation of 0°. These photographically derived data span a 0.5°-wide meridional swath near the scene's center that matches the FOV of spectroradiometer data taken simultaneously (see Fig. 2). Error bars span two standard deviations σn of the azimuthal radiances at selected elevation angles.

Fig. 4
Fig. 4

Second-order scattering model's meridional radiance profiles for a combined molecular and aerosol atmosphere. Model parameters were chosen to closely match those of Fig. 8 in Ref. 4. Multiplying the scaled radiances [which include a factor of 1/(π sr)] by the solar spectral irradiance Escale at wavelength λ converts them into absolute radiances.

Fig. 5
Fig. 5

Comparison of the azimuthal variation of clear-sky radiances across Plate 38 with those predicted by the second-order model, with the same model parameters that generate the best-fit meridional radiance profile of Fig. 6. τmol is the molecular normal optical depth, τaer is the aerosol normal optical depth, and Haer is the aerosol scale height. A constant molecular scale height of 8.4 km and a single-scattering albedo ϖ0 of 0.97 are used throughout this paper.

Fig. 6
Fig. 6

Comparison of measured and modeled clear-sky radiances for the Bald Eagle Mountain scene at University Park, Pa., ∼1530 GMT on 5 February 1987 (see Plate 38).

Fig. 7
Fig. 7

Comparison of measured and modeled radiances for an Antarctic clear-sky scene (see Plate 39). Although the snow is brighter than the sky, radiances are still normalized here by a local maximum occurring above the horizon.

Fig. 8
Fig. 8

Comparison of measured and modeled radiances for a clear-sky scene off Hamilton, Bermuda, at ∼1530 GMT on 2 June 1988 (see Plate 40).

Fig. 9
Fig. 9

Comparison of measured and modeled radiances for a clear-sky scene on the Chesapeake Bay (North Beach, Md.) at 2300 GMT on 24 March 1992 (see Plate 41, Ref. 24).

Fig. 10
Fig. 10

Comparison of measured and modeled radiances for a clear-sky scene at University Park, Pa., at 1605 GMT on 6 October 1992 (see Plate 37).

Fig. 11
Fig. 11

Comparison of measured clear-sky radiance profiles in atmospheres with different aerosol normal optical depths τaer. See Plate 37 (University Park), Plate 38 (Bald Eagle), and Plate 40 (Bermuda) for the original photographs.

Fig. 12
Fig. 12

Contours of the clear-sky radiance maximum's elevation as a function of solar elevation and aerosol normal optical depth. The observer is looking opposite the Sun (ϕrel = 180°), the surface Lambertian reflectance rsfc = 0.1, and the aerosol scale height Haer = 1 km.

Fig. 13
Fig. 13

Second-order model radiances integrated over slant optical path τslant at several view zenith angles θυ. The total optical path lengths τfυ) are measured from the observer to the atmosphere's top and are paired with the corresponding clear-sky radiances Lfυ).

Fig. 14
Fig. 14

Second-order model's meridional radiance profiles at ϕrel = 180° in a nonabsorbing, purely molecular atmosphere with different normal optical depths. In this atmosphere, the near-horizon radiance maximum does not exist for the two smaller optical depths, but it does for the two larger ones.

Equations (4)

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L sky = τ = 0 τ f J ( Ψ , τ ) exp ( τ τ f ) d τ ,
J dir ϖ 0 exp ( τ ) L s ¯ ω s P dir ( Ψ ) 4 π = ϖ 0 E s exp ( τ ) P dir ( Ψ ) 4 π ,
J diff , 1 = ϖ 0 4 π ϕ = 0 2 π θ = 0 π P diff ( θ , ϕ ) L diff ( θ , ϕ ) sin ( θ ) d θ d ϕ ,
L diff ( θ , ϕ ) = L sfc exp ( τ sfc ) + ϖ 0 4 π 0 τ diff E s exp ( τ θ ϕ ) P θ ϕ exp ( τ τ diff ) d τ .

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