Abstract

A model of a clear atmosphere is presented based upon two assumptions: (1) the point-function equilibrium radiance for a given path of sight does not change with altitude; (2) there is no absorption. As a result of these assumptions, the equation of transfer can be integrated. The path radiance for any slant path becomes a function of the equilibrium radiance and the beam transmittance of that path. In addition, the equilibrium radiance is a function of the scalar irradiance from the sun, sky, and earth and the proportional directional scattering coefficient for ground level. Sky radiances, and path radiances through the atmosphere for both upward and downward paths are determined by four parameters; the proportional directional scattering function for ground level, the total vertical beam transmittance of the atmosphere, the scalar albedo, and the solar zenith angle.

There is evidence that the real atmosphere does on some days conform to the above two assumptions to a useful extent for the visible portion of the spectrum.

© 1969 Optical Society of America

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References

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  1. S. Q. Duntley, J. Opt. Soc. Am. 38, 181 (1948).
  2. S. Q. Duntley, Visibility Studies and Some Applications in the Field of Camouflage. Summary Tech. Rept. of Div. 16, NDRC (Columbia University Press, New York, 1946), Vol. 2, p. 20.
  3. W. E. K. Middleton, Vision Through the Atmosphere (University of Toronto Press, Canada, 1952), p. 64–68.
  4. S. Q. Duntley, A. R. Boileau, and R. W. Preisendorfer, J. Opt. Soc. Am. 47, 501 (1957), Eq. (10).
    [Crossref]
  5. The path radiance for the upward path between altitude zt and z is found from the experimental data by use of the relation Nr*(z,θ,ϕ) = N∞*(z,θ,ϕ) − N∞*(zt,θ,ϕ)Tr(z,θ). For upward-looking paths of sight N∞*(z,θ,ϕ) is the sky radiance at altitude z.
  6. E. O. Hulburt, J. Opt. Soc. Am. 31, 474 (1941).
    [Crossref]
  7. G. I. Pokrowski, Z. Physik 34, 49, 496 (1925); Z. Physik 53, 67 (1926); Z. Physik. 30, 697 (1929).
    [Crossref]
  8. R. W. Kittler, in Proceedings of the CIE International Conference on Sunlighting, April 1965 at Newcastle-upon-Tyne (Bouwcentrum, Rotterdam, 1967).
  9. R. Anthony, J. Meteorol. 10, 60 (1953).
    [Crossref]
  10. F. E. Volz and K. Bullrich, J. Meteorol. 18, 306 (1961).
    [Crossref]
  11. R. W. Fenn, Beitr. Physik Atmosphäre 37, 71 (1964).
  12. R. Tousey and E. O. Hulburt, J. Opt. Soc. Am. 37, 78 (1947).
    [Crossref]
  13. S. Chandrasekhar, Radiative Transfer (Clarendon Press, Oxford, England, 1950).
  14. L. Goldberg, in The Earth as a Planet, G. P. Kuiper, Ed. (University of Chicago Press, Chicago, 1954), Ch. 5.

1964 (1)

R. W. Fenn, Beitr. Physik Atmosphäre 37, 71 (1964).

1961 (1)

F. E. Volz and K. Bullrich, J. Meteorol. 18, 306 (1961).
[Crossref]

1957 (1)

S. Q. Duntley, A. R. Boileau, and R. W. Preisendorfer, J. Opt. Soc. Am. 47, 501 (1957), Eq. (10).
[Crossref]

1953 (1)

R. Anthony, J. Meteorol. 10, 60 (1953).
[Crossref]

1948 (1)

S. Q. Duntley, J. Opt. Soc. Am. 38, 181 (1948).

1947 (1)

1941 (1)

E. O. Hulburt, J. Opt. Soc. Am. 31, 474 (1941).
[Crossref]

1925 (1)

G. I. Pokrowski, Z. Physik 34, 49, 496 (1925); Z. Physik 53, 67 (1926); Z. Physik. 30, 697 (1929).
[Crossref]

Anthony, R.

R. Anthony, J. Meteorol. 10, 60 (1953).
[Crossref]

Boileau, A. R.

S. Q. Duntley, A. R. Boileau, and R. W. Preisendorfer, J. Opt. Soc. Am. 47, 501 (1957), Eq. (10).
[Crossref]

Bullrich, K.

F. E. Volz and K. Bullrich, J. Meteorol. 18, 306 (1961).
[Crossref]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Clarendon Press, Oxford, England, 1950).

Duntley, S. Q.

S. Q. Duntley, A. R. Boileau, and R. W. Preisendorfer, J. Opt. Soc. Am. 47, 501 (1957), Eq. (10).
[Crossref]

S. Q. Duntley, J. Opt. Soc. Am. 38, 181 (1948).

S. Q. Duntley, Visibility Studies and Some Applications in the Field of Camouflage. Summary Tech. Rept. of Div. 16, NDRC (Columbia University Press, New York, 1946), Vol. 2, p. 20.

Fenn, R. W.

R. W. Fenn, Beitr. Physik Atmosphäre 37, 71 (1964).

Goldberg, L.

L. Goldberg, in The Earth as a Planet, G. P. Kuiper, Ed. (University of Chicago Press, Chicago, 1954), Ch. 5.

Hulburt, E. O.

Kittler, R. W.

R. W. Kittler, in Proceedings of the CIE International Conference on Sunlighting, April 1965 at Newcastle-upon-Tyne (Bouwcentrum, Rotterdam, 1967).

Middleton, W. E. K.

W. E. K. Middleton, Vision Through the Atmosphere (University of Toronto Press, Canada, 1952), p. 64–68.

Pokrowski, G. I.

G. I. Pokrowski, Z. Physik 34, 49, 496 (1925); Z. Physik 53, 67 (1926); Z. Physik. 30, 697 (1929).
[Crossref]

Preisendorfer, R. W.

S. Q. Duntley, A. R. Boileau, and R. W. Preisendorfer, J. Opt. Soc. Am. 47, 501 (1957), Eq. (10).
[Crossref]

Tousey, R.

Volz, F. E.

F. E. Volz and K. Bullrich, J. Meteorol. 18, 306 (1961).
[Crossref]

Beitr. Physik Atmosphäre (1)

R. W. Fenn, Beitr. Physik Atmosphäre 37, 71 (1964).

J. Meteorol. (2)

R. Anthony, J. Meteorol. 10, 60 (1953).
[Crossref]

F. E. Volz and K. Bullrich, J. Meteorol. 18, 306 (1961).
[Crossref]

J. Opt. Soc. Am. (4)

E. O. Hulburt, J. Opt. Soc. Am. 31, 474 (1941).
[Crossref]

S. Q. Duntley, J. Opt. Soc. Am. 38, 181 (1948).

S. Q. Duntley, A. R. Boileau, and R. W. Preisendorfer, J. Opt. Soc. Am. 47, 501 (1957), Eq. (10).
[Crossref]

R. Tousey and E. O. Hulburt, J. Opt. Soc. Am. 37, 78 (1947).
[Crossref]

Z. Physik (1)

G. I. Pokrowski, Z. Physik 34, 49, 496 (1925); Z. Physik 53, 67 (1926); Z. Physik. 30, 697 (1929).
[Crossref]

Other (6)

R. W. Kittler, in Proceedings of the CIE International Conference on Sunlighting, April 1965 at Newcastle-upon-Tyne (Bouwcentrum, Rotterdam, 1967).

The path radiance for the upward path between altitude zt and z is found from the experimental data by use of the relation Nr*(z,θ,ϕ) = N∞*(z,θ,ϕ) − N∞*(zt,θ,ϕ)Tr(z,θ). For upward-looking paths of sight N∞*(z,θ,ϕ) is the sky radiance at altitude z.

S. Q. Duntley, Visibility Studies and Some Applications in the Field of Camouflage. Summary Tech. Rept. of Div. 16, NDRC (Columbia University Press, New York, 1946), Vol. 2, p. 20.

W. E. K. Middleton, Vision Through the Atmosphere (University of Toronto Press, Canada, 1952), p. 64–68.

S. Chandrasekhar, Radiative Transfer (Clarendon Press, Oxford, England, 1950).

L. Goldberg, in The Earth as a Planet, G. P. Kuiper, Ed. (University of Chicago Press, Chicago, 1954), Ch. 5.

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Figures (2)

Fig. 1
Fig. 1

Equilibrium luminance for many upward-looking paths of sight initiating at an altitude of 6100 m (20 000 ft) and terminating at an altitude of 305 m (1000 ft). Each point plotted in Fig. 1 represents the equilibrium luminance of a different path of sight; all azimuths and zenith angles from 0° to 85° are represented. The data were obtained by A. R. Boileau on Flight 112 of an instrumented B-29 USAF aircraft assigned to the Visibility Laboratory. The flight took place on 16 May 1957 near Eglin Air Force Base, Florida. The measured beam transmittance for the vertical path of sight between altitudes 6100 m and 305 m was 0.897. The solar zenith angle was 25°. These data illustrate that equilibrium luminance depends on angle from the sun but not appreciably upon zenith angle θ or azimuth angle ϕ.

Fig. 2
Fig. 2

Measured profiles of path function and horizontal equilibrium luminance over western Florida near Eglin AFB, 16 May 1957, Visibility Laboratory Flight 112, measured by A. R. Boileau. Sun zenith angle is 25°. Path function, N*(z,90,113), is at β = 100°. Sky condition: clear and blue. The profile of attenuation length was computed by means of Eq. (4). Note that the graph has three scales, with the equilibrium-luminance scale one half the size of the attenuation-length and path-function scales. Thus the graph is a nomographic expression of the relationship expressed in Eq. (4); that is, equilibrium luminance equals the product of the attenuation length and the path function. A: B*(z,90°,113°); B: Bq(z,90°,113°); C: L(z).

Equations (41)

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N q ( 0 , θ , ϕ ) = N q ( z , θ , ϕ ) .
Δ N ( z , θ , ϕ ) / Δ z sec θ = N * ( z , θ , ϕ ) - [ N ( z , θ , ϕ ) / L ( z ) ] .
Δ N ( z , θ , ϕ ) = [ N * ( z , θ , ϕ ) L ( z ) - N ( z , θ , ϕ ) ] × sec θ Δ z / L ( z ) .
N q ( z , θ , ϕ ) = N * ( z , θ , ϕ ) L ( z ) .
Δ N ( z , θ , ϕ ) = [ N q ( 0 , θ , ϕ ) - N ( z , θ , ϕ ) ] sec θ Δ z / L ( z ) .
Δ N ( z , θ , ϕ ) / [ N ( z , θ , ϕ ) - N q ( 0 , θ , ϕ ) ] = - sec θ Δ z / L ( z ) .
N 0 N r d N ( x , θ , ϕ ) N ( z , θ , ϕ ) - N q ( 0 , θ , ϕ ) = - 0 r sec θ L ( z ) d z .
ln [ N r ( z , θ , ϕ ) - N q ( 0 , θ , ϕ ) N 0 ( z t , θ , ϕ ) - N q ( 0 , θ , ϕ ) ] = ln T r ( z , θ ) ,
N r ( z , θ , ϕ ) - N q ( 0 , θ , ϕ ) N 0 ( z t , θ , ϕ ) - N q ( 0 , θ , ϕ ) = T r ( z , θ ) ,
N r ( z , θ , ϕ ) = N 0 ( z t , θ , ϕ ) T r ( z , θ ) + N q ( 0 , θ , ϕ ) [ 1 - T r ( z , θ ) ] .
N r * ( z , θ , ϕ ) = N q ( 0 , θ , ϕ ) [ 1 - T r ( z , θ ) ] .
N q ( 0 , θ , ϕ ) = N * ( 0 , θ , ϕ ) / [ 1 - T ( 0 , θ ) ] .
N * ( z , θ , ϕ ) = N s r ( z , θ s , 0 ) σ ( z , β ) d Ω s + 4 π N ( z , θ , ϕ ) σ ( z , β ) d Ω .
h s ( z ) = N s ( z , θ s , 0 ) d Ω s .
L ( z ) = 1 / s ( z ) ,
N q ( z , θ , ϕ ) = h s ( z ) σ ( z , β ) s ( z ) + 4 π N ( z , θ , ϕ ) σ ( z , β ) s ( z ) d Ω .
N q ( z , θ , ϕ ) = h s ( z ) σ ( z , β ) s ( z ) + N ¯ ( z ) 4 π σ ( z , β ) s ( z ) d Ω .
h k ( z ) + h u ( z ) = 4 π N ( z , θ , ϕ ) d Ω = N ¯ ( z ) 4 π d Ω .
N ¯ ( z ) = [ h k ( z ) + h u ( z ) ] / 4 π .
4 π σ ( z , β ) s ( z ) d Ω = 1.
N q ( z , θ , ϕ ) = h s ( z ) σ ( z , β ) s ( z ) + h k ( z ) + h u ( z ) 4 π .
4 π N q ( z , θ , ϕ ) d Ω = h s 4 π σ ( z , β ) s ( z ) d Ω + [ h k ( z ) + h u ( z ) 4 π ] 4 π d Ω = h s ( z ) + h k ( z ) + h u ( z ) = h ( z ) ,
σ ( z , β ) s ( z ) = { N q ( z , θ , ϕ ) - [ h k ( z ) + h u ( z ) 4 π ] } ÷ h s ( z ) .
A s = h u ( z ) / [ h s ( z ) + h k ( z ) ] .
h u ( z ) = h s ( z ) A s + h k ( z ) A s .
N q ( z , θ , ϕ ) = h s ( z ) [ σ ( z , β ) s ( z ) + A s 4 π ] + h k ( z ) [ 1 + A s 4 π ] .
h s ( z ) = h s ( ) T ( z , θ s ) ,
N q 1 ( z , θ , ϕ ) = h s ( z ) [ σ ( z , β ) s ( z ) + A s 4 π ] .
N r 1 * ( z , θ , ϕ ) = N q 1 ( z , θ , ϕ ) [ 1 - T ( z , θ ) ] ,
h k 1 ( z ) = 2 π N q 1 ( z , θ , ϕ ) [ 1 - T ( z , θ ) ] d Ω ;
h k 1 ( z ) = h s ( z ) 2 π [ σ ( z , β ) s ( z ) + A s 4 π ] [ 1 - T ( z , θ ) ] d Ω .
h k 2 ( z ) = 2 π N q 2 ( z , θ , ϕ ) [ 1 - T ( z , θ ) ] d Ω ,
h k 2 ( z ) = h k 1 + h k 1 ( 1 + A s 4 π ) 2 π [ 1 - T ( z , θ ) ] d Ω .
h k 3 ( z ) = 2 π N q 3 ( z , θ , ϕ ) [ 1 - T ( z , θ ) ] d Ω ,
h k 3 ( z ) = h k 1 + h k 1 ( 1 + A s 4 π ) 2 π [ 1 - T ( z , θ ) ] d Ω + h k 1 ( 1 + A s 4 π ) 2 { 2 π [ 1 - T ( z , θ ) ] d Ω } 2 .
B = ( 1 + A s ) / 4 π
X = 2 π [ 1 - T ( z , θ ) ] d Ω .
h k 3 ( z ) = h k 1 ( 1 + B X + B 2 X 2 ) .
h k n ( z ) = h k 1 ( 1 + B X + B 2 X 2 + B n - 1 X n - 1 ) .
h k ( z ) = h k ( z ) = h k 1 / ( 1 - B X ) .
N q ( z , θ , ϕ ) = h s ( z ) { σ ( z , β ) s ( z ) + A s 4 π + ( 1 + A s 4 π ) × [ 2 π [ σ ( z , β ) s ( z ) + A s 4 π ] [ 1 - T ( z , θ ) ] d Ω / 1 - ( 1 + A s 4 π ) 2 π [ 1 - T ( z , θ ) ] d Ω ] } .