Abstract

Four numerical methods useful for determining the band parameters of infrared transmittance models are presented. A double exponential function described by three absorber parameters and one spectral parameter is adopted as the transmittance model. The methods are compared for accuracy and convenience through an application to line-by-line and measured data for nitrous oxide (N2O) of 20-cm−1 resolution over a wide range of atmospheric conditions. Sufficient mathematical details are provided so that the methods may be easily adapted to other transmittance functions. One of these—the weighted average method—is shown to be clearly superior to the others.

© 1984 Optical Society of America

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References

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  1. A. J. La Rocca, Proc. IEEE 63, 75 (1975).
    [Crossref]
  2. J. H. Pierluissi, K. Tomiyama, Appl. Opt. 19, 2298 (1980).
    [Crossref] [PubMed]
  3. J. H. Pierluissi, K. Tomiyama, F. X. Kneizys, Appl. Opt. 20, 2517 (1981).
    [Crossref] [PubMed]
  4. J. H. Pierluissi, K. Tomiyama, Appl. Opt. 22, 1628 (1983).
    [Crossref] [PubMed]

1983 (1)

1981 (1)

1980 (1)

1975 (1)

A. J. La Rocca, Proc. IEEE 63, 75 (1975).
[Crossref]

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Figures (3)

Fig. 1
Fig. 1

Degraded transmittance comparisons between four numerical methods for the determination of model parameters and the developing data for the ν1 band of N2O at P = 0.8869 atm, T = 281.6 K, and U = 1.203 atm. cm.

Fig. 2
Fig. 2

Degraded transmittance compaisons between four numerical methods for the determination of model parameters and the developing data for the ν2 band of N2O at P = 0.3040 atm, T = 229.7 K, and U = 4.800 atm cm.

Fig. 3
Fig. 3

Degraded transmittance comparisons between four numerical methods for the determination of model parameters and the developing data for the ν3 band of N2O at P = 1.0 atm, T = 257.1 K, and U = 0.0564 atm cm.

Tables (2)

Tables Icon

Table I Nonspectral Parameters for the Model in Eq. (1) for the ν1, ν2 and ν3 Bands of N2O, as Obtained From Four Numerical Methods

Tables Icon

Table II Total Mean Square Errors (Top Half of Entry) and Individual Sample rms Errors (Bottom Half of Entry) for the ν1, ν2, and ν3 Bands of N2O, as Obtained From the Use of Four Numerical Methods

Equations (27)

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τ M = exp [ - 10 a ( C 1 + log 10 W ) ] i = 1 , 2 , , I ,
log 10 W = n P + m T + U , P = log 10 ( P P 0 ) , T = log 10 ( T 0 T ) , U = log 10 U ,
ɛ = k = 1 4 j = 1 J [ τ ( i k , j ) - τ M ( i k , j ) ] 2 ,
ɛ a = 2 ln ( 10 ) k = 1 4 j = 1 J D ( i k , j ) .             [ C i k + n P ( j ) + m T ( j ) + U ( j ) ] ,
ɛ n = 2 a ln ( 10 ) k = 1 4 j = 1 J D ( i k , j ) P ( j ) ,
ɛ m = 2 a ln ( 10 ) k = 1 4 j = 1 J D ( i k , j ) T ( j ) ,
ɛ C i k = 2 a ln ( 10 ) j = 1 J D ( i k , j ) ,
D ( i , j ) = [ τ ( i , j ) - τ M ( i , j ) ] [ τ M ( i , j ) 10 F ( i , j ) , F ( i , j ) = a [ C i + n P ( j ) + m T ( j ) + U ( j ) ] ,
C i = 1 J J = 1 J ( { log [ - ln τ ( i , j ) ] } / a - n P ( j ) - m T ( j ) - U ( j ) ) , i = 1 , 2 , , I .
ɛ = i = 1 I j = 1 J [ τ ( i , j ) - τ M ( i , j ) ] 2 .
ɛ a = 2 ln ( 10 ) i = 1 I j = 1 J D ( i , j ) × { n [ P ( j ) - P ¯ ] + m [ T ( j ) - T ¯ ] + U ( j ) - U ¯ } ,
ɛ n = 2 a ln ( 10 ) i = 1 I j = 1 J D ( i , j ) [ P ( j ) - P ¯ ] ,
ɛ m = 2 a ln ( 10 ) i = 1 I j = 1 J D ( i , j ) [ T ( j ) - T ¯ ] ,
P ¯ = 1 J j = 1 J P ( j ) , T ¯ = 1 J j = 1 J T ( j ) , J ¯ = 1 J j = 1 J U ( j ) .
ɛ ( i ) = j = 1 J [ τ ( i , j ) - τ M ( i , j ) ] 2 , i = 1 , , I ,
0 = ɛ ( i ) C i = 2 a ln ( 10 ) j = 1 J D ( i , j ) ,
0 = j = 1 J D ( i , j ) ,             i = 1 , 2 , , I .
ɛ a , ɛ n , and ɛ m
C i a , C i n , and C i m
ɛ a = 2 ln ( 10 ) i = 1 I j = 1 J D ( i , j ) × [ n P W ( i , j ) + m T W ( i , j ) + U W ( i , j ) ] ,
ɛ n = 2 a ln ( 10 ) i = 1 I j = 1 J D ( i , j ) P W ( i , j ) ,
ɛ m = 2 a ln ( 10 ) i = 1 I j = 1 J D ( i , j ) T W ( i , j ) ,
P W ( i , j ) = P ( j ) - j = 1 J W F ( i , j ) P ( j ) j = 1 J W F ( i , j ) ,
T W ( i , j ) = T ( j ) - j = 1 J W F ( i , j ) T ( j ) j = 1 J W F ( i , j ) ,
U W ( i , j ) = U ( j ) - j = 1 J W F ( i , j ) U ( j ) j = 1 J W F ( i , j ) ,
W F ( i , j ) = τ M ( i , j ) 2 10 2 F ( i , j ) + D ( i , j ) [ 1 - 10 F ( i , j ) ] .
ɛ rms = ɛ I J .

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