Abstract

A numerical approach is taken in the study of two methods commonly used in the development of band models for the calculation of gaseous molecular transmittance in the IR region. The first method considered is for the determination of a discrete transmittance function without the use of an analytical band model. This method is then modified assuming a piecewise continuous function to provide for interpolation between the discrete points. The second method relaxes restrictions inherent to the first and assumes an analytical function over the entire range of transmittance values. Although the theory is generally applicable to other gaseous absorbers, it is specifically applied to 20-cm−1 resolution data for the major bands of the atmospheric trace gases SO2, NH3, NO, and NO2. The spectral parameters are listed for the convenience of model users at 5-cm−1 intervals throughout the bands.

© 1980 Optical Society of America

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References

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  1. W. M. Elsasser, Phys. Rev. 34, 126 (1938).
    [CrossRef]
  2. R. A. McClatchey, R. W. Fenn, J. E. A. Selby, F. E. Volz, J. S. Garing, “Optical Properties of the Atmosphere,” AFCRL Environmental Research Paper 331 (AFCRL, Hanscom AFB, Mass., 1970).
  3. J. E. A. Selby, F. X. Kneizys, J. H. Chetwynd, R. A. McClatchey, “Atmospheric Transmittance/Radiance: Computer Code Lowtran 4,” AFGL Environmental Research Paper 626 (AFGL, Hanscom AFB, Mass., 1978).
  4. R. M. Goody, Q. J. R. Meteorol. Soc. 78, 165 (1952).
    [CrossRef]
  5. C. D. Rodgers, “Approximate Methods of Calculating Transmission by Bands of Spectral Lines,” National Center for Atmospheric Research Technical Note 116+1A (NCAR, Boulder, Colo., 1976).
  6. J. I. F. King, “Statistical Transmission Models of Arbitrary Variance,” Proc. IRIS 4 (1959).
  7. R. R. Gruenzel, Appl. Opt. 17, 2591 (1978).
    [PubMed]
  8. M. Aoki, Introduction to Optimization Theory (Macmillan, New York, 1971).
  9. IBM Manual System/360, Scientific Subroutine Package H20-0205-3 (IBM, New York, 1968).
  10. R. M. Goody, Atmospheric Radiation (Oxford U.P., London, 1961).
  11. J. H. Pierluissi, K. Tomiyama, R. B. Gomez, Appl. Opt. 18, 1607 (1979).
    [CrossRef] [PubMed]
  12. W. L. Smith, “A Polynomial Representation of Carbon Dioxide and Water Vapor Transmission,” ESSA Technical Report NESC47 (National Environmental Satellite Center, Washington, D.C., 1969).
  13. L. S. Rothman, S. A. Clough, R. A. McClatchey, L. G. Young, D. E. Snider, A. Goldman, Appl. Opt. 17, 507 (1978).
    [CrossRef] [PubMed]
  14. J. H. Pierluissi, G. A. Gibson, R. B. Gomez, Appl. Opt. 17, 1425 (1978).
    [CrossRef] [PubMed]
  15. R. A. McClatchey, A. P. D'Agati, “Atmospheric Transmission of Laser Radiation: Computer Code LASER,” AFGL Environmental Research Paper 622 (AFGL, Hanscom AFB, Mass., 1978).
  16. S. L. Valley, Ed., Handbook of Geophysics and Space Environments (McGraw-Hill, New York, 1965).

1979 (1)

1978 (3)

1959 (1)

J. I. F. King, “Statistical Transmission Models of Arbitrary Variance,” Proc. IRIS 4 (1959).

1952 (1)

R. M. Goody, Q. J. R. Meteorol. Soc. 78, 165 (1952).
[CrossRef]

1938 (1)

W. M. Elsasser, Phys. Rev. 34, 126 (1938).
[CrossRef]

Aoki, M.

M. Aoki, Introduction to Optimization Theory (Macmillan, New York, 1971).

Chetwynd, J. H.

J. E. A. Selby, F. X. Kneizys, J. H. Chetwynd, R. A. McClatchey, “Atmospheric Transmittance/Radiance: Computer Code Lowtran 4,” AFGL Environmental Research Paper 626 (AFGL, Hanscom AFB, Mass., 1978).

Clough, S. A.

D'Agati, A. P.

R. A. McClatchey, A. P. D'Agati, “Atmospheric Transmission of Laser Radiation: Computer Code LASER,” AFGL Environmental Research Paper 622 (AFGL, Hanscom AFB, Mass., 1978).

Elsasser, W. M.

W. M. Elsasser, Phys. Rev. 34, 126 (1938).
[CrossRef]

Fenn, R. W.

R. A. McClatchey, R. W. Fenn, J. E. A. Selby, F. E. Volz, J. S. Garing, “Optical Properties of the Atmosphere,” AFCRL Environmental Research Paper 331 (AFCRL, Hanscom AFB, Mass., 1970).

Garing, J. S.

R. A. McClatchey, R. W. Fenn, J. E. A. Selby, F. E. Volz, J. S. Garing, “Optical Properties of the Atmosphere,” AFCRL Environmental Research Paper 331 (AFCRL, Hanscom AFB, Mass., 1970).

Gibson, G. A.

Goldman, A.

Gomez, R. B.

Goody, R. M.

R. M. Goody, Q. J. R. Meteorol. Soc. 78, 165 (1952).
[CrossRef]

R. M. Goody, Atmospheric Radiation (Oxford U.P., London, 1961).

Gruenzel, R. R.

King, J. I. F.

J. I. F. King, “Statistical Transmission Models of Arbitrary Variance,” Proc. IRIS 4 (1959).

Kneizys, F. X.

J. E. A. Selby, F. X. Kneizys, J. H. Chetwynd, R. A. McClatchey, “Atmospheric Transmittance/Radiance: Computer Code Lowtran 4,” AFGL Environmental Research Paper 626 (AFGL, Hanscom AFB, Mass., 1978).

McClatchey, R. A.

L. S. Rothman, S. A. Clough, R. A. McClatchey, L. G. Young, D. E. Snider, A. Goldman, Appl. Opt. 17, 507 (1978).
[CrossRef] [PubMed]

R. A. McClatchey, A. P. D'Agati, “Atmospheric Transmission of Laser Radiation: Computer Code LASER,” AFGL Environmental Research Paper 622 (AFGL, Hanscom AFB, Mass., 1978).

J. E. A. Selby, F. X. Kneizys, J. H. Chetwynd, R. A. McClatchey, “Atmospheric Transmittance/Radiance: Computer Code Lowtran 4,” AFGL Environmental Research Paper 626 (AFGL, Hanscom AFB, Mass., 1978).

R. A. McClatchey, R. W. Fenn, J. E. A. Selby, F. E. Volz, J. S. Garing, “Optical Properties of the Atmosphere,” AFCRL Environmental Research Paper 331 (AFCRL, Hanscom AFB, Mass., 1970).

Pierluissi, J. H.

Rodgers, C. D.

C. D. Rodgers, “Approximate Methods of Calculating Transmission by Bands of Spectral Lines,” National Center for Atmospheric Research Technical Note 116+1A (NCAR, Boulder, Colo., 1976).

Rothman, L. S.

Selby, J. E. A.

R. A. McClatchey, R. W. Fenn, J. E. A. Selby, F. E. Volz, J. S. Garing, “Optical Properties of the Atmosphere,” AFCRL Environmental Research Paper 331 (AFCRL, Hanscom AFB, Mass., 1970).

J. E. A. Selby, F. X. Kneizys, J. H. Chetwynd, R. A. McClatchey, “Atmospheric Transmittance/Radiance: Computer Code Lowtran 4,” AFGL Environmental Research Paper 626 (AFGL, Hanscom AFB, Mass., 1978).

Smith, W. L.

W. L. Smith, “A Polynomial Representation of Carbon Dioxide and Water Vapor Transmission,” ESSA Technical Report NESC47 (National Environmental Satellite Center, Washington, D.C., 1969).

Snider, D. E.

Tomiyama, K.

Volz, F. E.

R. A. McClatchey, R. W. Fenn, J. E. A. Selby, F. E. Volz, J. S. Garing, “Optical Properties of the Atmosphere,” AFCRL Environmental Research Paper 331 (AFCRL, Hanscom AFB, Mass., 1970).

Young, L. G.

Appl. Opt. (4)

Phys. Rev. (1)

W. M. Elsasser, Phys. Rev. 34, 126 (1938).
[CrossRef]

Proc. IRIS (1)

J. I. F. King, “Statistical Transmission Models of Arbitrary Variance,” Proc. IRIS 4 (1959).

Q. J. R. Meteorol. Soc. (1)

R. M. Goody, Q. J. R. Meteorol. Soc. 78, 165 (1952).
[CrossRef]

Other (9)

C. D. Rodgers, “Approximate Methods of Calculating Transmission by Bands of Spectral Lines,” National Center for Atmospheric Research Technical Note 116+1A (NCAR, Boulder, Colo., 1976).

R. A. McClatchey, R. W. Fenn, J. E. A. Selby, F. E. Volz, J. S. Garing, “Optical Properties of the Atmosphere,” AFCRL Environmental Research Paper 331 (AFCRL, Hanscom AFB, Mass., 1970).

J. E. A. Selby, F. X. Kneizys, J. H. Chetwynd, R. A. McClatchey, “Atmospheric Transmittance/Radiance: Computer Code Lowtran 4,” AFGL Environmental Research Paper 626 (AFGL, Hanscom AFB, Mass., 1978).

M. Aoki, Introduction to Optimization Theory (Macmillan, New York, 1971).

IBM Manual System/360, Scientific Subroutine Package H20-0205-3 (IBM, New York, 1968).

R. M. Goody, Atmospheric Radiation (Oxford U.P., London, 1961).

W. L. Smith, “A Polynomial Representation of Carbon Dioxide and Water Vapor Transmission,” ESSA Technical Report NESC47 (National Environmental Satellite Center, Washington, D.C., 1969).

R. A. McClatchey, A. P. D'Agati, “Atmospheric Transmission of Laser Radiation: Computer Code LASER,” AFGL Environmental Research Paper 622 (AFGL, Hanscom AFB, Mass., 1978).

S. L. Valley, Ed., Handbook of Geophysics and Space Environments (McGraw-Hill, New York, 1965).

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Figures (3)

Fig. 1
Fig. 1

Qualitative representation of the cut and layer structure of input transmittance data.

Fig. 2
Fig. 2

Generalized form of the coefficient matrix in Eq. (17).

Fig. 3
Fig. 3

Composite plot of empirical and analytical transmittance functions together with 65-point data for SO2.

Tables (8)

Tables Icon

Table I Empirical Transmlttance Function τ(x) for Trace Gases

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Table II Absorber and Spectral Parameters for Various Bands of Trace Gases

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Table III Coefficients of the Plecewise Continuous Function in Eq. (21) as Used for Interpolation Between the Tabulated Values for τ(x) in Table I

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Table IV Spectral Parameters for SO2

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Table V Spectral Parameters for NH3

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Table VI Spectral Parameters for NO

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Table VII Spectral Parameters for NO2

Tables Icon

Table VIII Absorber and Spectral Parameters for the Analytical Band Model in Eq. (23)

Equations (42)

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τ ν = exp [ K ν ( P , T ) d U ( Z ) ] ,
τ = τ ν ϕ ν d ν / ϕ ν d ν .
τ = g ( S α n U ) ,
S α n U = S ( Z ) α n ( Z ) d U ( Z ) .
S = S 0 ( T 0 T ) a ,
α = α 0 P P 0 ( T 0 T ) 1 / 2 ,
τ = g [ C ( P P 0 ) n ( T 0 T ) m U ] ,
τ = f { x } ,
x = C + log 10 W ,
C = log 10 C ,
W = ( P P 0 ) n ( T 0 T ) m U .
x j = f 1 ( τ j ) ,
n log 10 ( P ijk P 0 ) + m log 10 ( T 0 T ijk ) + C i x j = log 10 U ijk ,
n log 10 ( P ijk P 0 ) + m log 10 ( T 0 T ijk ) + u 1 , ijk C 1 + . . . + u I , ijk C I υ 1 , ijk x 1 . . . υ J , ijk x J = log 10 U ijk ,
E ijk = [ n log 10 ( P ijk P 0 ) + m log 10 ( T 0 T ijk ) + u 1 , ijk C 1 + . . . + u I , ijk C I + υ 1 , ijk y 1 . . . + υ J , ijk y J ( log 10 U ijk ) ] 2 ,
E = i = 1 I j = 1 J i k = 1 K i j E ijk ,
A X = B ,
B = [ ( υ j log U ) , . . . , ( υ 1 log U ) , ( u I log U ) , . . . , ( u 2 log U ) , ( log ( T 0 T ) log U ) , ( log ( P P 0 ) log U ) ] t ,
X = [ y J , . . . , y 1 , C I , . . . , C 2 , m , n ] t .
τ ( x ) = exp ( 10 a 1 + a 2 x ) ,
C = 1 L l = 1 L ( x l log 10 W l ) ,
τ ( x ) = exp ( 10 a 1 + a 2 x + a 3 x 2 ) ,
E i k = [ τ i k exp ( 10 a 1 + a 2 x i k + a 3 x i k 2 ) ] 2 ,
x i k = n log 10 ( P i k P 0 ) + m log 10 ( T 0 T i k ) + log 10 U i k + u 1 , i k C 1 + . . . + u I , i k C I .
E = i = 1 I k = 1 K j E i k ,
E a 1 = 2 D i k δ τ i k E a 2 = 2 D i k δ τ i k x i k E a 3 = 2 D i k δ τ i k x i k 2 E n = 2 D i k δ τ i k ( a 2 + 2 a 3 x i k ) log ( P i k P 0 ) , E m = 2 D i k δ τ i k ( a 2 + 2 a 3 x i k ) log ( T 0 T i k ) E C i = 2 D i k δ τ i k ( a 2 + 2 a 3 x i k ) u i
D i k = { E i k } 1 / 2 ,
δ τ i k = ( ln 10 ) 10 a 1 + a 2 x i k + a 3 x i k 2 τ i k .
a 2 E a 1 + 2 a 3 E a 2 = i E C i ,
U ( cm atm ) = ppmv × 10 12 ρ a ( g / m 3 ) Z ( km ) 2.24 × 10 9 m a ,
U ( cm atm ) = 0.2 × 10 12 × 1.167 × 10 3 × 5 × 2.24 × 10 9 28.97 = 9.02 × 10 2 .
x = C + log 10 [ ( P P 0 ) n ( T 0 T ) m U ] = 0.014 + log 10 [ ( 1013.00 1013.25 ) n ( 273.16 300 ) m 9.02 × 10 2 ] .
τ = exp ( 10 a 1 + a 2 x ) = exp [ 10 0.0594 + 0.9811 ( 1.0333 ) ] = 0.8948 .
τ = exp ( 10 a 1 + a 2 x + a 3 x 2 ) = exp [ 10 0.02292 + 0.86759 ( 1.0283 ) 0.08578 ( 1.0283 ) 2 ] = 0.8961 .
C 1
C 2
C 3
C 4
C 1
C 2
C 3
C 4

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