Abstract

Atmospheric transmittance models for absorbing gases with constant mixing ratios were described in the two preceding papers of this series. In this paper a method for calculating atmospheric transmittances for absorbing gases with variable mixing ratios is described. Because the model uses only arithmetic operations, it is computationally fast as well as accurate. Details of the computational algorithm are given, including the calculation of the expansion coefficients. In a test of eleven independent profiles, the resulting transmittances agreed with line-by-line calculations in an rms sense to within 0.0090 in the worst case and to within 0.0018 in all other cases. This paper also includes a discussion for computing transmittances when several gases absorb in the same spectral interval. These three papers provide a complete treatment for modeling transmittances in inhomogeneous atmospheres.

© 1979 Optical Society of America

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References

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  1. L. M. McMillin, H. E. Fleming, Appl. Opt. 15, 358 (1976).
    [CrossRef] [PubMed]
  2. H. E. Fleming, L. M. McMillin, Appl. Opt. 16, 1366 (1977).
    [CrossRef] [PubMed]
  3. R. M. Goody, Atmospheric Radiation (Clarendon, Oxford, 1964), Chap. 6.
  4. K. Ya. Kondratyev, Yu. M. Timofeev, Atmos. Oceanic Phys. 5, 208 (1969) (English translation).
  5. J. Y. Wang, Opt. Lett. 2, 169 (1978).
    [CrossRef] [PubMed]
  6. M.-D. Chow, A. Arking, Preprint Volume of the Third Conference on Atmospheric Radiation (American Meteorological Society, Boston, Mass., 1978).
  7. M. P. Weinreb, A. C. Neuendorffer, J. Atmos. Sci. 30, 662 (1973).
    [CrossRef]
  8. J. H. Pierluissi, R. B. Gomez, R. E. Bruce, Appl. Opt. 16, 18 (1977).
    [CrossRef] [PubMed]
  9. S. R. Drayson, Appl. Opt. 5, 385 (1966).
    [CrossRef] [PubMed]
  10. W. L. Smith, H. M. Woolf, J. Atmos. Sci. 33, 1127 (1976).
    [CrossRef]

1978 (1)

1977 (2)

1976 (2)

L. M. McMillin, H. E. Fleming, Appl. Opt. 15, 358 (1976).
[CrossRef] [PubMed]

W. L. Smith, H. M. Woolf, J. Atmos. Sci. 33, 1127 (1976).
[CrossRef]

1973 (1)

M. P. Weinreb, A. C. Neuendorffer, J. Atmos. Sci. 30, 662 (1973).
[CrossRef]

1969 (1)

K. Ya. Kondratyev, Yu. M. Timofeev, Atmos. Oceanic Phys. 5, 208 (1969) (English translation).

1966 (1)

Arking, A.

M.-D. Chow, A. Arking, Preprint Volume of the Third Conference on Atmospheric Radiation (American Meteorological Society, Boston, Mass., 1978).

Bruce, R. E.

Chow, M.-D.

M.-D. Chow, A. Arking, Preprint Volume of the Third Conference on Atmospheric Radiation (American Meteorological Society, Boston, Mass., 1978).

Drayson, S. R.

Fleming, H. E.

Gomez, R. B.

Goody, R. M.

R. M. Goody, Atmospheric Radiation (Clarendon, Oxford, 1964), Chap. 6.

Kondratyev, K. Ya.

K. Ya. Kondratyev, Yu. M. Timofeev, Atmos. Oceanic Phys. 5, 208 (1969) (English translation).

McMillin, L. M.

Neuendorffer, A. C.

M. P. Weinreb, A. C. Neuendorffer, J. Atmos. Sci. 30, 662 (1973).
[CrossRef]

Pierluissi, J. H.

Smith, W. L.

W. L. Smith, H. M. Woolf, J. Atmos. Sci. 33, 1127 (1976).
[CrossRef]

Timofeev, Yu. M.

K. Ya. Kondratyev, Yu. M. Timofeev, Atmos. Oceanic Phys. 5, 208 (1969) (English translation).

Wang, J. Y.

Weinreb, M. P.

M. P. Weinreb, A. C. Neuendorffer, J. Atmos. Sci. 30, 662 (1973).
[CrossRef]

Woolf, H. M.

W. L. Smith, H. M. Woolf, J. Atmos. Sci. 33, 1127 (1976).
[CrossRef]

Appl. Opt. (4)

Atmos. Oceanic Phys. (1)

K. Ya. Kondratyev, Yu. M. Timofeev, Atmos. Oceanic Phys. 5, 208 (1969) (English translation).

J. Atmos. Sci. (2)

W. L. Smith, H. M. Woolf, J. Atmos. Sci. 33, 1127 (1976).
[CrossRef]

M. P. Weinreb, A. C. Neuendorffer, J. Atmos. Sci. 30, 662 (1973).
[CrossRef]

Opt. Lett. (1)

Other (2)

M.-D. Chow, A. Arking, Preprint Volume of the Third Conference on Atmospheric Radiation (American Meteorological Society, Boston, Mass., 1978).

R. M. Goody, Atmospheric Radiation (Clarendon, Oxford, 1964), Chap. 6.

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Tables (1)

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Table I Errors in Derived Transmittance as a Function of Channel

Equations (40)

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τ i = τ i - 1 f i ( T , T ^ , p , p ^ ) ,
τ ( ν , T , θ , p ) = exp [ - ( sec θ / g ) 0 p k ( ν , T , p ) q ( p ) d p ] ,
d u = ( sec θ / g ) q ( p ) d p .
τ ( ν , T , p , u ) = exp { - 0 u k [ ν , T ( u ) , p ( u ) ] d u } .
τ ( ν , T , p , u + Δ u ) = τ ( ν , T , p , u ) exp [ - u u + Δ u k ( ν , T , p ) d u ] .
τ ( ν , T , p , u + Δ u ) τ ( ν , T , p , u ) · τ ( ν , T ^ , p ^ , u ) τ ( ν , T ^ , p ^ , u + Δ u ) = exp { - u u + Δ u [ k ( ν , T , p ) - k ( ν , T ^ , p ^ , ) ] d u } ,
τ ( ν , T , p , u + Δ u ) τ ( ν , T , p , u ) · τ ( ν , T ^ , p ^ , u ) τ ( ν , T ^ , p ^ , u + Δ u ) = exp [ - Δ u m = 1 M 1 m ! ( Δ T T + Δ p p ) m k ( ν , T ^ , p ^ ) ] ,
τ ( ν , T , p , u + Δ u ) τ ( ν , T , p , u ) · τ ( ν , T ^ , p ^ , u ) τ ( ν , T ^ , p ^ , u + Δ u ) = 1 + a Δ T + b Δ p + c Δ T 2 + d Δ T Δ p + e Δ p 2 + ,
τ i = τ ( ν , T , p , u i ) ; τ ^ i = τ ( ν , T ^ , p ^ , u i ) ; Δ p i = p ( u i ) - p ^ ( u i ) ; Δ T i = T ( u i ) - T ^ ( u i ) ; α i = τ ^ i / τ ^ i - 1 ; A i = α i a i ;             E i = α i e i .
τ i = τ i - 1 ( α i + A i Δ T i + B i Δ p i + C i Δ T i 2 + D i Δ T i Δ p i + E i Δ p i 2 + ) , i = 1 , , n ,
τ i ( Δ ν ) = Δ ν τ i ( ν ) ϕ ( ν ) d ν ,
τ i ( Δ ν ) = τ i ( ν 0 ) [ τ i ( Δ ν ) / τ i ( ν 0 ) ] .
τ i ( Δ ν ) / τ i ( ν 0 ) = Δ ν τ i ( ν ) ϕ ( ν ) d ν / τ i ( ν 0 ) = Δ ν exp { - 0 u i [ k ( ν , T , p ) - k ( ν 0 , T , p ) ] d u } ϕ ( ν ) d ν .
τ i ( Δ ν ) / τ i ( ν 0 ) = Δ ν [ 1 - 0 u i Δ k ( ν , T , p ) d u ] ϕ ( ν ) d ν = 1 - 0 u i [ Δ ν Δ k ( ν , T , p ) ϕ ( ν ) d ν ] d u ,
f ( T , p ) = Δ ν Δ k ( ν , T , p ) ϕ ( ν ) d ν ,
f ( T , p ) = f ( T ^ , p ^ ) + f ( T ^ , p ^ ) T Δ T + f ( T ^ , p ^ ) p Δ p ,
τ i ( Δ ν ) / τ i ( ν 0 ) = β i + 0 u i [ f ( T ^ , p ^ ) T Δ T + f ( T ^ , p ^ ) p Δ p ] d u ,
τ i ( Δ ν ) / τ i ( ν 0 ) = β i + 0 u i ( λ 1 Δ T + λ 2 u Δ T + λ 3 u 2 Δ T + λ 4 Δ p + λ 5 u Δ p ) d u = β i + γ i Δ T i * + δ i Δ T i * * + i Δ T i * * * + ζ i Δ p i * + η i Δ p i * * ,
γ i = λ 1 u i , Δ T i * = 0 u i Δ T ( u ) d u / 0 u i d u , δ i = ( λ 2 u i 2 ) / 2 , Δ T i * * = 0 u i u Δ T ( u ) d u / 0 u i u i d u , i = ( λ 3 u i 3 ) / 3 , Δ T i * * * = 0 u i u 2 Δ T ( u ) d u / 0 u i u 2 d u , ζ i = λ 4 u i , Δ p i * = 0 u i Δ p ( u ) d u / 0 u i d u , η i = ( λ 5 u i 2 ) / 2 , Δ p i * * = 0 u i u Δ p ( u ) d u / 0 u i u d u , i = 1 , , n . }
τ i ( Δ ν ) = τ i ( ν 0 ) ( β i + γ i Δ T i * + δ i Δ T * * + i Δ T i * * * + ζ i Δ p i * + η i Δ p i * * ) .
τ i ( Δ ν ) = τ i - 1 ( ν 0 ) ( a i 1 + a i 2 Δ T i + a i 3 Δ p i + a i 4 Δ T i 2 + a i 5 Δ T i Δ p i + a i 6 Δ p i 2 + a i 7 Δ T i Δ p i 2 + a i 8 Δ T i * + a i 9 Δ p i * + a i , 10 Δ T i * * + a i , 11 Δ p i * * + a i , 12 Δ T i * * * ) , i = 1 , , n .
u ( p ) = ( sec θ / g ) 0 p q ( p ) d p .
Δ T i = T i - T ^ i ,             Δ p i = p i - p ^ i ,
Δ T i ( * k ) = k u i k j = 1 i u j k - 1 Δ T j Δ u j ,             k = 1 , 2 , 3 ,
Δ p i ( * k ) = k u i k j = 1 i u j k - 1 Δ p j Δ u j ,             k = 1 , 2 ,
x i 1 = 1 , x i 2 = Δ T i , x i 3 = Δ p i , x i 4 = Δ T i 2 , x i 5 = Δ T i Δ p i , x i 6 = Δ p i 2 , x i 7 = Δ T i Δ p i 2 , x i 8 = Δ T i * , x i 9 = Δ p i * , x i , 10 = Δ T i * * , x i , 11 = Δ p i * * , x i , 12 = Δ T i * * * , τ i ( Δ ν ) = τ i . }
τ i = τ i - 1 ( j = 1 12 c i j x i j ) .
τ 1 = k = 1 7 c i k x 1 k .
τ 2 = τ 1 ( k = 1 7 c i j x i j ) .
w i = u i - 1 / u i , w ¯ i = 1 - w i , Δ T i k = ( T i - T ^ i ) k , k = 1 , 2 , Δ p i k = ( p i - p ^ i ) k , k = 1 , 2 , Δ T i ( * k ) = w i k Δ T i - 1 ( * k ) + k w ¯ i Δ T i , k = 1 , 2 , 3 , Δ p i ( * k ) = w i k Δ p i - 1 ( * k ) + k w ¯ i Δ p i , k = 1 , 2 ,
τ i = τ i - 1 ( j = 1 12 c i j x i j ) ,             i = 1 , , n .
τ i τ i - 1 - τ ^ i τ ^ i - 1 = j = 2 12 c i j x i j ,             i = 1 , , n .
c i 1 = τ ^ i / τ ^ i - 1 .
c i = ( c i 2 , , c i , 12 ) T ,
c i = ( A i T A i ) - 1 A i T y i ,
A i = x i 2 1 · · · · x i 12 1 x i 2 m x i 12 m
y i = [ ( τ i / τ i - 1 - τ ^ i / τ ^ i - 1 ) 1 , , ( τ i / τ i - 1 - τ ^ i / τ ^ i - 1 ) m ] T
I = 0 1 B d τ
τ i = τ i ( α ) · α i ( β ) · τ i ( γ ) ,
τ i ( γ ) = τ i / [ τ i ( α ) · τ i ( β ) ] .

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