Abstract

An analytical solution is discussed for the nadir radiance as measured from a satellite, based on a simplified single-scattering approximation in which the scattered radiation is not subject to extinction. In the solution, terms can be identified as due to a reflection from the vicinity of the object pixel, and, respectively, (1) upward scattering to zenith above the object pixel, and (2) downward scattering from the entire atmosphere to the object pixel. The first term is referred to as the cross radiance, the second as the cross irradiance. The cross radiance is proportional to the forward scattering optical thickness, as defined, and the cross irradiance to the backscattering optical thickness. The cross radiance usually constitutes the predominant effect. The effect, even at low atmospheric turbidity, can be large enough to constitute a significant fraction of the radiance registered at the satellite, thus hampering determination of spectral signature of the object pixel or identification of pixels with inherently the same spectral signature. Explicit expressions and computer solutions for the cross radiance from annular or from rectangular reflecting areas are presented. The effect depends on the height distribution and on the sharpness of the forward peak of the scattering particles.

© 1979 Optical Society of America

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References

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  1. J. Otterman, Appl. Opt. 17, 3431 (1978).
    [CrossRef] [PubMed]
  2. The surface irradiance Gt depends on solar zenith angle, atmospheric parameters, and also on reflection from terrain surrounding the object pixel and subsequent scattering.
  3. For low surface reflectivities, the slope of space measured reflectivity against the surface reflectivity is significantly smaller than unity, and there is a loss in discrimination capability.
  4. The phenomenon has some similarities to an effect known as the cross talk, i.e., an unwanted coupling between separate telephone channels. Here the channels are the vertical propagation paths through the atmosphere, and pixels on the surface correspond to the individual speakers. The object pixel corresponds to a speaker for whom we analyze the quality of transmission.
  5. See Sec. VII where the validity of this statement is qualified.
  6. If the size distribution varies with height, β is a function of h. The treatment can be generalized by introducing βjΔM(h)=Fj(h/Hj)MΔh/Hj.
  7. Gt is assumed constant, i.e., variability of the cross irradiance between sources, which occurs when a is variable, is neglected.
  8. E. P. Mercanti, Ed., Astronaut. Aeronaut.36 (1973).
  9. J. Otterman, P. D. Lowman, V. V. Salomonson, Geophys. Surv. 2, 431 (1976).
    [CrossRef]
  10. A. Park, in Contribution of Space Observations to Global Food Information Systems, E. A. Godby, J. Otterman, Eds. (Pergamon, New York, 1978), pp. 105–113.
  11. R. S. Fraser, O. P. Bahethi, A. H. Al-Abbas, Remote Sensing Environ. 6, 229 (1977).
    [CrossRef]
  12. J. Otterman, J. Br. Interplanet. Soc. 23, 349 (1970).
  13. J. Otterman, Y. Kaufman, M. Podolak, S. Rehavi, “Adjacency Effects on Imaging by Surface Reflection and Atmospheric Scattering: Sky Radiance and Cross Irradiance,” in preparation (1979).

1978 (1)

1977 (1)

R. S. Fraser, O. P. Bahethi, A. H. Al-Abbas, Remote Sensing Environ. 6, 229 (1977).
[CrossRef]

1976 (1)

J. Otterman, P. D. Lowman, V. V. Salomonson, Geophys. Surv. 2, 431 (1976).
[CrossRef]

1970 (1)

J. Otterman, J. Br. Interplanet. Soc. 23, 349 (1970).

Al-Abbas, A. H.

R. S. Fraser, O. P. Bahethi, A. H. Al-Abbas, Remote Sensing Environ. 6, 229 (1977).
[CrossRef]

Bahethi, O. P.

R. S. Fraser, O. P. Bahethi, A. H. Al-Abbas, Remote Sensing Environ. 6, 229 (1977).
[CrossRef]

Fraser, R. S.

R. S. Fraser, O. P. Bahethi, A. H. Al-Abbas, Remote Sensing Environ. 6, 229 (1977).
[CrossRef]

Kaufman, Y.

J. Otterman, Y. Kaufman, M. Podolak, S. Rehavi, “Adjacency Effects on Imaging by Surface Reflection and Atmospheric Scattering: Sky Radiance and Cross Irradiance,” in preparation (1979).

Lowman, P. D.

J. Otterman, P. D. Lowman, V. V. Salomonson, Geophys. Surv. 2, 431 (1976).
[CrossRef]

Otterman, J.

J. Otterman, Appl. Opt. 17, 3431 (1978).
[CrossRef] [PubMed]

J. Otterman, P. D. Lowman, V. V. Salomonson, Geophys. Surv. 2, 431 (1976).
[CrossRef]

J. Otterman, J. Br. Interplanet. Soc. 23, 349 (1970).

J. Otterman, Y. Kaufman, M. Podolak, S. Rehavi, “Adjacency Effects on Imaging by Surface Reflection and Atmospheric Scattering: Sky Radiance and Cross Irradiance,” in preparation (1979).

Park, A.

A. Park, in Contribution of Space Observations to Global Food Information Systems, E. A. Godby, J. Otterman, Eds. (Pergamon, New York, 1978), pp. 105–113.

Podolak, M.

J. Otterman, Y. Kaufman, M. Podolak, S. Rehavi, “Adjacency Effects on Imaging by Surface Reflection and Atmospheric Scattering: Sky Radiance and Cross Irradiance,” in preparation (1979).

Rehavi, S.

J. Otterman, Y. Kaufman, M. Podolak, S. Rehavi, “Adjacency Effects on Imaging by Surface Reflection and Atmospheric Scattering: Sky Radiance and Cross Irradiance,” in preparation (1979).

Salomonson, V. V.

J. Otterman, P. D. Lowman, V. V. Salomonson, Geophys. Surv. 2, 431 (1976).
[CrossRef]

Appl. Opt. (1)

Geophys. Surv. (1)

J. Otterman, P. D. Lowman, V. V. Salomonson, Geophys. Surv. 2, 431 (1976).
[CrossRef]

J. Br. Interplanet. Soc. (1)

J. Otterman, J. Br. Interplanet. Soc. 23, 349 (1970).

Remote Sensing Environ. (1)

R. S. Fraser, O. P. Bahethi, A. H. Al-Abbas, Remote Sensing Environ. 6, 229 (1977).
[CrossRef]

Other (9)

J. Otterman, Y. Kaufman, M. Podolak, S. Rehavi, “Adjacency Effects on Imaging by Surface Reflection and Atmospheric Scattering: Sky Radiance and Cross Irradiance,” in preparation (1979).

A. Park, in Contribution of Space Observations to Global Food Information Systems, E. A. Godby, J. Otterman, Eds. (Pergamon, New York, 1978), pp. 105–113.

The surface irradiance Gt depends on solar zenith angle, atmospheric parameters, and also on reflection from terrain surrounding the object pixel and subsequent scattering.

For low surface reflectivities, the slope of space measured reflectivity against the surface reflectivity is significantly smaller than unity, and there is a loss in discrimination capability.

The phenomenon has some similarities to an effect known as the cross talk, i.e., an unwanted coupling between separate telephone channels. Here the channels are the vertical propagation paths through the atmosphere, and pixels on the surface correspond to the individual speakers. The object pixel corresponds to a speaker for whom we analyze the quality of transmission.

See Sec. VII where the validity of this statement is qualified.

If the size distribution varies with height, β is a function of h. The treatment can be generalized by introducing βjΔM(h)=Fj(h/Hj)MΔh/Hj.

Gt is assumed constant, i.e., variability of the cross irradiance between sources, which occurs when a is variable, is neglected.

E. P. Mercanti, Ed., Astronaut. Aeronaut.36 (1973).

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Figures (5)

Fig. 1
Fig. 1

The geometry of surface reflection and subsequent scattering (a) to the zenith and (b) to the surface. Three components of radiance at the zenith are schematically indicated: (i) Lnr, reflection from the object pixel (ii) Lnd, radiance scattered from the direct beam and (iii) Lna, radiance due to scattering from reflection in the vicinity of the object pixel.

Fig. 2
Fig. 2

Relative contribution (in percent) of two rings of Landsat pixels (24 pixels) to the cross radiance over the object pixel, for m = 400 (upper numbers), and m = 8 (lower numbers), H = 2000 m.

Fig. 3
Fig. 3

The plot of the m!!/(m − 1)!! vs m, the slope of Lna (normalized to 1 at infinity) vs the dimensionless distance at a boundary between reflecting and black half-planes, in the direction perpendicular to the boundary, a case of thin layer of scatterers.

Fig. 4
Fig. 4

Contribution to the cross radiance of the reflection from an infinite annular region further than κ1, exponentially distributed scatterers, numerical integration of 0 ( σ 2 σ 2 + κ 1 2 ) ( m + 1 ) / 2 e - σ d σ

Fig. 5
Fig. 5

The radiance Lna, normalized to 1 at + infinity vs the dimensionless distance ν1 perpendicular to a boundary between a reflecting half-plane and a black half-plane; the reflecting half-plane extends from ν = 0 to +∞; exponentially distributed scatterers; numerical integration of m ! ! π ( m - 1 ) ! ! 0 e - σ σ m + 1 d σ ν 1 d ν ( σ 2 + ν 2 ) ( m + 2 ) / 2

Tables (2)

Tables Icon

Table I Two α and Two β Term Representations of an Aerosol Scattering Phase Function for Two Wavelengths

Tables Icon

Table II Effects of a Ground-Hugging Absorbing Layer with an Optical Thickness W0 on Cross Radiance for a Term in the Phase Function with an Exponent m

Equations (43)

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L n r = r G t exp ( - Q ) / π ,
L n d = μ 0 [ 1 - exp ( - Q / μ 0 ) ] [ R P R ( 180 ° - θ 0 ) + M P M ( 180 ° - θ 0 ) ] / Q .
L n d = μ 0 { 1 - exp [ - Q ( 1 + sec θ 0 ) ] } [ R P R ( 180 ° - θ 0 ) + M P M ( 180 ° - θ 0 ) ] Q ( 1 + μ 0 ) .
L n a = ( a G t / π Q ) 0 π / 2 [ 1 - exp ( - Q / cos ϕ ) ] cos ϕ [ R P R ( ϕ ) + M P M ( ϕ ) ] 2 π sin ϕ d ϕ ,
P R ( ϕ ) = 3 ( 1 + cos 2 ϕ ) / 16 π .
P M ( ϕ ) = [ 2 i α i ( n i + 1 ) cos n i ϕ + j β j ( m j + 1 ) cos m j ϕ ] / 4 π             0 ϕ ( π / 2 ) = j β j ( m j + 1 ) cos m j ϕ / 4 π             ( π / 2 ) ϕ π ,
i α i + j β j = 1.
L n a = a G t 2 π Q 0 π / 2 cos ϕ [ 1 - exp ( - Q / cos ϕ ) ] { [ 2 i α i ( n i + 1 ) cos n i ϕ + j β j ( m j + 1 ) cos m j ϕ ] + 3 4 ( 1 + cos 2 ϕ ) R } sin ϕ d ϕ = a G t 2 π G { [ 2 i α i ( n i + 1 ) C n i + 1 ( Q ) + j β j ( m j + 1 ) C m j + 1 ( Q ) ] M + 3 4 [ C 1 ( Q ) + C 3 ( Q ) ] R } ,
C n ( Q ) = 0 π / 2 [ 1 - exp ( - Q / cos ϕ ) ] cos n ϕ sin ϕ d ϕ .
L n = L n r + L n d + L n a .
G t = μ 0 exp ( - Q / μ 0 ) + [ 1 - exp ( - Q / μ 0 ) ] f 1 - 2 a b C 1 ( Q ) ,
f = [ ( 1 + α ) M + R ] / 2 Q ,
b = [ ( 1 - α ) M + R ] / 2 Q ,
α = i α i
L n = ( r exp ( - Q ) π + a π { [ i 2 α i ( n i + 1 ) C n i + 1 + j β i ( m j + 1 ) C m j + 1 ] M 2 Q + 3 4 ( C 1 + C 3 ) R 2 Q } ) × μ 0 exp ( - Q / μ 0 ) + [ 1 - exp ( - Q / μ 0 ) ] f 1 - 2 a b C 1 + μ 0 [ 1 - exp ( - Q / μ 0 ) ] × j β j ( m j + 1 ) μ 0 m M + 3 4 ( 1 + μ 0 2 ) R 4 π Q ,
( n + 1 ) C n + 1 ( Q ) Q ,
L n = r exp ( - Q ) + a [ ( 1 + α ) M 2 + R 2 ] π ( 1 - 2 a b Q ) × μ 0 { exp ( - Q / μ 0 ) + [ 1 - exp ( - Q / μ 0 ) ] f } + L n d r ( 1 - Q ) + a f Q π ( 1 - 2 a b Q ) μ 0 { exp ( - Q / μ 0 ) + [ 1 - exp ( - Q / μ 0 ) ] f } + L n d .
L n = r π ( 1 - Q + a r f Q + 2 a b Q ) μ 0 × { exp ( - Q / μ 0 ) + [ 1 - exp ( - Q / μ 0 ) f ] } + L n d .
Δ L n a u ( ϕ ) = a G t cos ϕ ( Δ M ) P M ( ϕ ) / π ( h 2 + y 2 ) = a G t cos 3 ϕ ( Δ M ) P M ( ϕ ) / π h 2 ,
cos 2 ϕ = h 2 / ( h 2 + y 2 ) .
Δ L n a u ( y ) = ( a G t π ) ( m + 1 ) β Δ M 4 π h 2 ( h 2 h 2 + y 2 ) ( m + 3 ) / 2
Δ M ( h ) = F ( h / H ) M Δ h / H ,
L n a u = a G t π 0 ( m + 1 ) β M F ( h / H ) 4 π h 2 H ( h 2 h 2 + y 2 ) ( m + 3 ) / 2 d h .
L n a = L n a u ( y ) d A .
σ = h / H
κ = y / H
L n a = a G t π 0 F ( σ ) d σ σ 2 β M m + 1 4 π A ( σ 2 σ 2 + κ 2 ) ( m + 3 ) / 2 d A H 2 .
0 F ( σ ) f ( σ 2 , κ 2 ) d σ
l n a = β M [ ( m + 1 ) / 4 π ] ( 1 + κ 2 ) - ( m + 3 ) / 2 A / H 2 .
( 1 + κ 2 ) - ( m + 3 ) / 2
l n a = β M [ ( m + 1 ) / 4 π ] κ 1 2 π κ ( 1 + κ 2 ) - ( m + 3 ) / 2 d κ = ( β M / 2 ) ( 1 + κ 1 2 ) - ( m + 1 ) / 2 .
l n a = β M [ ( m + 1 ) / 4 π ] ν 1 ν 2 d ν χ 1 χ 2 ( 1 + ν 2 + χ 2 ) - ( m + 3 ) / 2 d χ .
l n a = β M [ ( m + 1 ) / 4 π ] ν 1 d ν - ( 1 + ν 2 + χ 2 ) - ( m + 3 ) / 2 d χ = β M m + 1 4 π ν 1 ( 1 + ν 2 ) - ( m + 2 ) / 2 2 ( m + 2 ) / 2 ( m 2 ) ! ( m + 1 ) ! ! d ν = β M m ! ! 2 π ( m - 1 ) ! ! ν 1 d ν ( 1 + ν 2 ) ( m + 2 ) / 2 .
m ! ! π ( m - 1 ) ! ! 0 d ν ( 1 + ν 2 ) ( m + 2 ) / 2 = 0.5 ,
0 M exp ( - q / cos ϕ ) exp ( - M + q ) d q / cos ϕ = exp ( - M ) [ 1 - exp ( - M sec ϕ + M ) ] cos ϕ ( sec ϕ - 1 ) ,
L n a = β M [ ( m + 1 ) / 4 ] κ 1 2 × exp [ - W 0 ( 1 + κ 2 ) 1 / 2 ] ( 1 + κ 2 ) - ( m + 3 ) / 2 d κ 2 .
L n a = β M 2 ξ 1 exp ( - W 0 ξ ) ξ - ( m + 2 ) d ξ = β M 2 [ ( - 1 ) m + 2 W 0 m + 1 E 1 ( - W 0 ξ 1 ) ( m + 1 ) ! + exp ( - W 0 ξ 1 ) ξ 1 m + 1 × k = 0 m ( - 1 ) k W 0 k ξ 1 k ( m + 1 ) m ( m + 1 - k ) ] ,
l n a = β M [ ( m + 1 ) / 4 π ] 0 σ m + 1 exp ( - σ ) d σ × A ( σ 2 + κ 2 ) - ( m + 3 ) / 2 d A / H 2 ,
l n a ( κ 1 , ) = β M [ ( m + 1 ) / 4 π ] 0 σ m + 1 exp ( - σ ) d σ × κ 1 2 π ( σ 2 + κ 2 ) - ( m + 3 ) / 2 κ d κ = β M 2 0 exp ( - σ ) ( σ 2 σ 2 + κ 1 2 ) ( m + 1 ) / 2 d σ .
l n a = β M [ ( m + 1 ) / 4 π ] 0 σ m + 1 exp ( - σ ) d σ × ν 1 d ν - d χ ( σ 2 + ν 2 + χ 2 ) ( m + 3 ) / 2 = β M [ ( m + 1 ) / 4 π ] 0 σ m + 1 exp ( - σ ) d σ × ν 1 2 ( m + 2 ) / 2 ( m 2 ) ! d ν ( m + 1 ) ! ! ( σ 2 + ν 2 ) ( m + 2 ) / 2 = β M m ! ! 2 π ( m - 1 ) ! ! 0 σ m + 1 exp ( - σ ) d σ × ν 1 d ν ( σ 2 + ν 2 ) ( m + 2 ) / 2 .
0 ( σ 2 σ 2 + κ 1 2 ) ( m + 1 ) / 2 e - σ d σ
m ! ! π ( m - 1 ) ! ! 0 e - σ σ m + 1 d σ ν 1 d ν ( σ 2 + ν 2 ) ( m + 2 ) / 2
βjΔM(h)=Fj(h/Hj)MΔh/Hj.

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