Abstract

A solution is presented to the radiative transfer of the solar irradiation through a turbid atmosphere, bounded by a Lambert surface, based on the single-scattering approximation, i.e., an assumption that a photon that underwent scattering either leaves the top of the atmosphere or strikes the surface. The solution depends on idealization of the scattering phase function of the aerosols. The equations developed here are subsequently applied to analyze quantitatively (1) the enhancement of the surface irradiation and (2) the enhancement of the scattered radiant emittance as seen from above the atmosphere, caused by the surface reflectance and subsequent atmospheric scattering. An order-of-magnitude error analysis is presented.

© 1978 Optical Society of America

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References

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  1. J. Otterman, R. S. Fraser, “Adjacency Effects on Imaging by Surface Reflection and Atmospheric Scattering: Cross-Radiance to Zenith”, submitted, Applied Optics (1978).
  2. J. E. Hansen, L. D. Travis, Space Sci. Rev. 16, 527 (1974).
    [CrossRef]
  3. D. Deirmendjian, L. Sekera, Tellus 6, 382 (1954).
    [CrossRef]
  4. K. Ya. Kondratyev, Radiation in the Atmosphere (Academic, New York, 1969).
  5. K. L. Coulson, J. Atmos. Sci. 25, 759 (1968).
    [CrossRef]
  6. J. Otterman, R. S. Fraser, Remote Sensing Environ. 5, 247 (1976).
    [CrossRef]
  7. J. E. Colwell, Remote Sensing Environ. 3, 175 (1974).
    [CrossRef]
  8. J. Otterman, Geophys. Res. Lett. 4, 441 (1977).
    [CrossRef]
  9. M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (U.S. Department of Commerce, Applied Mathematics Series 55, National Bureau of Standards, Washington, D.C., 1964).

1977 (1)

J. Otterman, Geophys. Res. Lett. 4, 441 (1977).
[CrossRef]

1976 (1)

J. Otterman, R. S. Fraser, Remote Sensing Environ. 5, 247 (1976).
[CrossRef]

1974 (2)

J. E. Colwell, Remote Sensing Environ. 3, 175 (1974).
[CrossRef]

J. E. Hansen, L. D. Travis, Space Sci. Rev. 16, 527 (1974).
[CrossRef]

1968 (1)

K. L. Coulson, J. Atmos. Sci. 25, 759 (1968).
[CrossRef]

1954 (1)

D. Deirmendjian, L. Sekera, Tellus 6, 382 (1954).
[CrossRef]

Colwell, J. E.

J. E. Colwell, Remote Sensing Environ. 3, 175 (1974).
[CrossRef]

Coulson, K. L.

K. L. Coulson, J. Atmos. Sci. 25, 759 (1968).
[CrossRef]

Deirmendjian, D.

D. Deirmendjian, L. Sekera, Tellus 6, 382 (1954).
[CrossRef]

Fraser, R. S.

J. Otterman, R. S. Fraser, Remote Sensing Environ. 5, 247 (1976).
[CrossRef]

J. Otterman, R. S. Fraser, “Adjacency Effects on Imaging by Surface Reflection and Atmospheric Scattering: Cross-Radiance to Zenith”, submitted, Applied Optics (1978).

Hansen, J. E.

J. E. Hansen, L. D. Travis, Space Sci. Rev. 16, 527 (1974).
[CrossRef]

Kondratyev, K. Ya.

K. Ya. Kondratyev, Radiation in the Atmosphere (Academic, New York, 1969).

Otterman, J.

J. Otterman, Geophys. Res. Lett. 4, 441 (1977).
[CrossRef]

J. Otterman, R. S. Fraser, Remote Sensing Environ. 5, 247 (1976).
[CrossRef]

J. Otterman, R. S. Fraser, “Adjacency Effects on Imaging by Surface Reflection and Atmospheric Scattering: Cross-Radiance to Zenith”, submitted, Applied Optics (1978).

Sekera, L.

D. Deirmendjian, L. Sekera, Tellus 6, 382 (1954).
[CrossRef]

Travis, L. D.

J. E. Hansen, L. D. Travis, Space Sci. Rev. 16, 527 (1974).
[CrossRef]

Geophys. Res. Lett. (1)

J. Otterman, Geophys. Res. Lett. 4, 441 (1977).
[CrossRef]

J. Atmos. Sci. (1)

K. L. Coulson, J. Atmos. Sci. 25, 759 (1968).
[CrossRef]

Remote Sensing Environ. (2)

J. Otterman, R. S. Fraser, Remote Sensing Environ. 5, 247 (1976).
[CrossRef]

J. E. Colwell, Remote Sensing Environ. 3, 175 (1974).
[CrossRef]

Space Sci. Rev. (1)

J. E. Hansen, L. D. Travis, Space Sci. Rev. 16, 527 (1974).
[CrossRef]

Tellus (1)

D. Deirmendjian, L. Sekera, Tellus 6, 382 (1954).
[CrossRef]

Other (3)

K. Ya. Kondratyev, Radiation in the Atmosphere (Academic, New York, 1969).

M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (U.S. Department of Commerce, Applied Mathematics Series 55, National Bureau of Standards, Washington, D.C., 1964).

J. Otterman, R. S. Fraser, “Adjacency Effects on Imaging by Surface Reflection and Atmospheric Scattering: Cross-Radiance to Zenith”, submitted, Applied Optics (1978).

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Figures (5)

Fig. 1
Fig. 1

The geometry of scattering into the hemisphere below vs above the horizontal plane: (A) for Rayleigh scattering; (B) for Mie scattering.

Fig. 2
Fig. 2

The increase in the surface irradiance by surface reflection and subsequent backscattering Srb for (i) Q = 0.1, f/b = 1.0; (ii) Q = 0.3, f/b = 3.0; (iii) Q = 0.3, f/b = 5.0. θ = 0°.

Fig. 3
Fig. 3

The increase in the surface irradiance by surface reflection and subsequent backscattering Srb for (i) Q = 0.1, f/b = 1.0; (ii) Q = 0.3, f/b = 3.0; (iii) Q = 0.3, f/b = 5.0. θ = 60°.

Fig. 4
Fig. 4

The increase in the atmospheric radiant emittance by surface reflection and subsequent upward scattering Srf for (i) Q = 0.1, f/b = 1.0; (ii) Q = 0.3, f/b = 3.0; (iii) Q = 0.3, f/b = 5.0. θ = 0°.

Fig. 5
Fig. 5

The increase in the atmospheric radiant emittance by surface reflection and subsequent upward scattering Srf for (i) Q = 0.1, f/b = 1.0; (ii) Q = 0.3, f/b = 3.0; (iii) Q = 0.3, f/b = 5.0. θ = 60°.

Tables (2)

Tables Icon

Table I Total (Global) Surface Irradiance for a Rayleigh Atmosphere, Sun at the Zenith, Surface Reflectivity a0 = 0.8, from Deirmendjian and Sekera,3 and Calculated Using Eq. (15)

Tables Icon

Table II Tabulation of Function C1(Q)

Equations (42)

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P R ( ρ ) = [ 3 / ( 16 π ) ] ( 1 + cos 2 ρ ) .
1 + α = 2 2 π forward hemisphere P M d Ω 4 π P M d Ω = 2 0 π / 2 P M ( ρ ) sin ρ d ρ 0 π P M ( ρ ) sin ρ d ρ ,
p = ( 1 + α ) / ( 1 α ) ,
P M ( ϕ ) = 1 4 π ( i 2 α i ( n 1 + 1 ) cos n i ρ + j β j ( m j + 1 ) cos m j ρ )
= 1 4 π ( j β j ( m j + 1 ) cos m j ρ ) 90 ° < ρ 180 ° ,
i α i + j β j = 1
Q = R + M + B .
G d = cos θ 0 exp ( Q / cos θ 0 ) .
G s d = cos θ 0 [ 1 exp ( Q / cos θ 0 ) ] [ ( R + ( 1 + α ) M ) / 2 Q ] .
G o = G d + G s d = cos θ 0 { exp ( Q / cos θ 0 ) + [ 1 exp ( Q / cos θ 0 ) ] R + ( 1 + α ) M 2 Q } .
G t = G o + k G o + k 2 G o + = G o / ( 1 k ) ,
k = a 0 2 π π 0 π / 2 cos ϕ sin ϕ × [ 1 exp ( Q / cos ϕ ) ] R + ( 1 α ) M 2 Q d ϕ ,
C 1 ( Q ) = 0 π / 2 cos ϕ sin ϕ [ 1 exp ( Q / cos ϕ ) ] d ϕ = 1 2 { 1 ( 1 Q ) exp ( Q ) + Q 2 [ + 0.5772 + ln Q + n = 1 ( 1 ) n Q n n n ! ] } .
k = a 0 C 1 ( Q ) [ R + ( 1 + α ) M ] / Q .
G t = cos θ 0 × exp ( Q / cos θ 0 ) + [ 1 exp ( Q / cos θ 0 ) ] R + ( 1 + α ) M 2 Q 1 a 0 R + ( 1 α ) M Q C 1 ( Q ) ,
G t = cos θ 0 1 + exp ( R / cos θ 0 ) 2 [ 1 a 0 C 1 ( R ) ] .
E o = ( 1 a 0 ) G t ,
E a = 2 π a 0 G t π 0 π / 2 cos ϕ sin ϕ [ 1 exp ( Q / cos ϕ ) ] d ϕ B Q + cos θ 0 [ 1 exp ( Q / cos θ 0 ) ] B Q = 2 G t a 0 B Q C 1 ( Q ) + cos θ 0 [ 1 exp ( Q / cos θ 0 ) ] B Q .
A t cos θ 0 = cos θ 0 ( E o + E a ) = cos θ 0 { 1 [ 1 exp ( Q / cos θ 0 ) ] B Q [ 1 a 0 + 2 a 0 B Q C 1 ( Q ) ] × exp ( Q / cos θ 0 ) + [ 1 exp ( Q / cos θ 0 ) ] R + ( 1 + α ) M 2 Q 1 a 0 R + ( 1 α ) M Q C 1 ( Q ) } .
A p = 2 π 0 π / 2 sin θ 0 A t cos θ 0 d θ 0 2 π 0 π / 2 sin θ 0 cos θ 0 d θ 0 = 1 [ 1 a 0 + 2 a 0 B Q C 1 ( Q ) ] × 2 2 C 1 ( Q ) + C 1 ( Q ) R + ( 1 + α ) M Q 1 a 0 R + ( 1 α ) M Q C 1 ( Q ) 2 B Q C 1 ( Q ) ,
L n r = r G t exp ( Q ) / π .
L n d = cos θ 0 [ 1 exp ( Q / cos θ 0 ) ] × [ β ( m + 1 ) cos m θ 0 M 4 π Q + ( cos 2 θ 0 + 1 ) 3 R 16 π Q ] .
L n a = G t a π ( 2 π 0 π / 2 [ 1 exp ( Q / cos ϕ ) ] sin ϕ cos ϕ × { [ 2 α ( n + 1 ) cos n ϕ + β ( m + 1 ) cos m ϕ ] M 4 π Q + ( cos 2 ϕ + 1 ) 3 R 16 π Q } d ϕ ) = G t a π { [ 2 α ( n + 1 ) C n + 1 ( Q ) + β ( m + 1 ) C m + 1 ( Q ) ] M 2 Q + [ C 3 ( Q ) + C 1 ( Q ) ] 3 R 8 Q } ,
C m + 1 ( Q ) = 0 π / 2 [ 1 exp ( Q / cos ϕ ) ] sin ϕ cos ( m + 1 ) ϕ d ϕ .
L n = L n r + L n d + L n a .
f = [ R + ( 1 + α ) M ] / 2 Q
b = [ R + ( 1 α ) M ] / 2 Q .
G s d = cos θ 0 [ 1 exp ( Q / cos θ 0 ) ] R + ( 1 + α ) M 2 Q = cos θ 0 [ 1 exp ( Q / cos θ 0 ] f .
G r b = cos θ 0 { exp ( Q / cos θ 0 ) + [ 1 exp ( Q / cos θ 0 ) ] f } × [ 1 1 2 a 0 b C 1 ( Q ) 1 ] = cos θ 0 { exp ( Q / cos θ 0 ) + [ 1 exp ( Q / cos θ 0 ) ] f } 2 a 0 b C 1 ( Q ) 1 2 a 0 b C 1 ( Q ) .
S r b = G r b / G s d = a 0 b [ exp ( Q / cos θ 0 ) + [ 1 exp ( Q / cos θ 0 ) ] f ] 2 C 1 ( Q ) f [ 1 exp ( Q / cos θ 0 ) ] [ 1 2 a 0 b C 1 ( Q ) ] = a 0 b { 1 + exp ( Q / cos θ 0 ) f [ 1 exp ( Q / cos θ 0 ) ] } 2 C 1 ( Q ) 1 2 a 0 b C 1 ( Q ) .
exp ( Q / cos θ 0 ) 1 exp ( Q / cos θ 0 ) cos θ 0 Q 1.
S r b 2 a 0 b cos θ 0 / f
W s d = cos θ 0 [ 1 exp ( Q / cos θ 0 ) ] R + ( 1 α ) M 2 Q = cos θ 0 [ 1 exp ( Q / cos θ 0 ) ] b ,
W r f = a 0 G t R + ( 1 + α ) M Q C 1 ( Q ) = a 0 cos θ 0 { exp ( Q / cos θ 0 ) + [ 1 exp ( Q / cos θ 0 ) ] f } 2 f C 1 ( Q ) 1 2 a 0 b C 1 ( Q ) ,
S r f = W r f W s d = a 0 f × { exp ( Q / cos θ 0 ) + [ 1 exp ( Q / cos θ 0 ) ] f } 2 C 1 ( Q ) b [ 1 2 a 0 b C 1 ( Q ) ] [ 1 exp ( Q / cos θ 0 ) ] = a 0 f 2 b { 1 + exp ( Q / cos θ 0 ) f [ 1 exp ( Q / cos θ 0 ) ] } 2 C 1 ( Q ) 1 2 a 0 b C 1 ( Q ) .
S r f 2 a 0 f cos θ 0 / b .
C m ( Q ) = 0 π / 2 sin ψ cos m ψ [ 1 exp ( Q / cos ψ ) ] d ψ = 0 π / 2 sin ψ cos m ψ d ψ 0 π / 2 sin ψ cos m ψ exp ( Q / cos ψ ) d ψ = 1 m + 1 1 exp ( Q x ) x m + 2 d x ,
1 exp ( Q x ) x m + 2 d x = E m + 2 ( Q ) ,
E m + 1 ( Q ) = 1 m [ exp ( Q ) Q E m ( Q ) ] .
E 3 ( Q ) = ( 1 Q ) 2 exp ( Q ) + Q 2 2 E 1 ( Q ) .
E 1 ( Q ) = 0.5772 ln Q 1 ( 1 ) n Q n n n !
C 1 ( Q ) = 1 2 [ 1 ( 1 Q ) exp ( Q ) Q 2 E 1 ( Q ) ] = 1 2 { 1 ( 1 Q ) exp ( Q ) Q 2 [ 0.5772 ln Q 1 ( 1 ) n Q n n n ! ] } .

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