Abstract

The essential elements of practical nonlinear least squares analysis of atmospheric absorption spectra are discussed. These include speed of analysis, flexibility of spectrum type, and stability in the presence of noise. A computer program which embodies these elements is described, and two types of results are presented. The first is a series of least squares analyses of synthetic spectra, both noise free and noisy. The second is a summary of results from analyses of high resolution solar spectra. Among the parameters which have been fit are trace gas mixing ratios, temperatures, spectral background, instrument resolution, phase error and channel spectrum parameters, line shifts, intensities, and broadening coefficients.

© 1980 Optical Society of America

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References

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  1. R. Mireles, J. Opt. Soc. Am. 56, 644 (1966).
    [CrossRef]
  2. J. Pourcin, R. Romanetti, Infrared Phys. 13, 161 (1973).
    [CrossRef]
  3. Y. S. Chang, J. H. Shaw, Appl. Spectrosc. 31, 213 (1977).
    [CrossRef]
  4. R. J. Anderson, P. R. Griffiths, J. Quant. Spectrosc. Radiat. Transfer 17, 393 (1977).
    [CrossRef]
  5. Y. S. Chang, J. H. Shaw, J. Quant. Spectrosc. Radiat. Transfer 18, 491 (1977).
    [CrossRef]
  6. Y. S. Chang, J. H. Shaw, J. G. Calvert, W. M. Uselman, J. Quant. Spectrosc. Radiat. Transfer 18, 589 (1977).
    [CrossRef]
  7. Y. S. Chang, J. H. Shaw, J. G. Calvert, W. M. Uselman, J. Quant. Spectrosc. Radiat. Transfer 19, 599 (1978).
    [CrossRef]
  8. P. G. Eitner, F. G. Smith, D. R. Woods, Proc. Soc. Photo-Opt. Instrum. Eng. 148, 112 (1978).
  9. A. S. Pine, J. W. C. Jones, A. G. Robiette, J. Mol. Spectrosc. 74, 52 (1979).
    [CrossRef]
  10. G. Eichen, J. Laurent, in Remote Sensing of the Atmosphere: Inversion Methods and Applications, A. L. Fymat, V. E. Zuev, Eds. (Elsevier, Amsterdam, 1978).
  11. R. J. Noll, A. Pires, Appl. Spectrosc. (1980), in press.
  12. J. H. Shaw, Ohio State U.; private communication.
  13. W. G. Mankin, Appl. Opt. 18, 3426 (1979).
    [CrossRef] [PubMed]
  14. L. S. Rothman, S. A. Clough, R. A. McClatchey, L. G. Young, D. E. Snider, A. Goldman, Appl. Opt. 17, 507 (1978).
    [CrossRef] [PubMed]
  15. C. R. Rao, Linear Statistical Inference and Its Applications (Wiley, New York, 1973), Chap. 4.
    [CrossRef]
  16. L. Nazareth, SIAM (Soc. Ind. Appl. Math.) Rev. 22, 1 (1980).
  17. M. L. Ralston, R. I. Jennrich, Technometrics 20, 7 (1978).
    [CrossRef]
  18. J. J. Dongarra, C. B. Moler, J. R. Bunch, G. W. Stewart, Linpack Users’ Guide (Society for Industrial and Applied Mathematics, Philadelphia, 1979), Chap. 11.
    [CrossRef]
  19. H. O. Hartley, Technometrics 3, 269 (1961).
    [CrossRef]
  20. G. E. P. Box, G. C. Tiao, Bayesian Inference in Statistical Analysis (Addison-Wesley, Reading, Mass., 1973).
  21. C. L. Lin, E. Niple, J. H. Shaw, J. G. Calvert, Appl. Spectrosc. 33, 481 (1979).
    [CrossRef]
  22. E. Niple, W. G. Mankin, A. Goldman, D. G. Murcray, F. J. Murcray, Geophys. Rev. Lett. 7, 489 (1980).
    [CrossRef]
  23. C. B. Farmer, Jet Propulsion Lab.; private communication.
  24. R. J. Bell, Introductory Fourier Transform Spectroscopy (Academic, New York, 1972), Chaps. 8 and 12.
  25. The H2O parameters used in the spectra in Figs. 7–9 were taken from C. Camy-Peyret, J. M. Flaud, Mol. Phys. 32, 523 (1976); J. M. Flaud, C. Camy-Peyret, J. Y. Mandin, G. Guelachvili, Mol. Phys. 34, 413 (1977).
    [CrossRef]
  26. A. Goldman, U. Denver; private communication.

1980

L. Nazareth, SIAM (Soc. Ind. Appl. Math.) Rev. 22, 1 (1980).

E. Niple, W. G. Mankin, A. Goldman, D. G. Murcray, F. J. Murcray, Geophys. Rev. Lett. 7, 489 (1980).
[CrossRef]

1979

1978

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

M. L. Ralston, R. I. Jennrich, Technometrics 20, 7 (1978).
[CrossRef]

Y. S. Chang, J. H. Shaw, J. G. Calvert, W. M. Uselman, J. Quant. Spectrosc. Radiat. Transfer 19, 599 (1978).
[CrossRef]

P. G. Eitner, F. G. Smith, D. R. Woods, Proc. Soc. Photo-Opt. Instrum. Eng. 148, 112 (1978).

1977

Y. S. Chang, J. H. Shaw, Appl. Spectrosc. 31, 213 (1977).
[CrossRef]

R. J. Anderson, P. R. Griffiths, J. Quant. Spectrosc. Radiat. Transfer 17, 393 (1977).
[CrossRef]

Y. S. Chang, J. H. Shaw, J. Quant. Spectrosc. Radiat. Transfer 18, 491 (1977).
[CrossRef]

Y. S. Chang, J. H. Shaw, J. G. Calvert, W. M. Uselman, J. Quant. Spectrosc. Radiat. Transfer 18, 589 (1977).
[CrossRef]

1976

The H2O parameters used in the spectra in Figs. 7–9 were taken from C. Camy-Peyret, J. M. Flaud, Mol. Phys. 32, 523 (1976); J. M. Flaud, C. Camy-Peyret, J. Y. Mandin, G. Guelachvili, Mol. Phys. 34, 413 (1977).
[CrossRef]

1973

J. Pourcin, R. Romanetti, Infrared Phys. 13, 161 (1973).
[CrossRef]

1966

1961

H. O. Hartley, Technometrics 3, 269 (1961).
[CrossRef]

Anderson, R. J.

R. J. Anderson, P. R. Griffiths, J. Quant. Spectrosc. Radiat. Transfer 17, 393 (1977).
[CrossRef]

Bell, R. J.

R. J. Bell, Introductory Fourier Transform Spectroscopy (Academic, New York, 1972), Chaps. 8 and 12.

Box, G. E. P.

G. E. P. Box, G. C. Tiao, Bayesian Inference in Statistical Analysis (Addison-Wesley, Reading, Mass., 1973).

Bunch, J. R.

J. J. Dongarra, C. B. Moler, J. R. Bunch, G. W. Stewart, Linpack Users’ Guide (Society for Industrial and Applied Mathematics, Philadelphia, 1979), Chap. 11.
[CrossRef]

Calvert, J. G.

C. L. Lin, E. Niple, J. H. Shaw, J. G. Calvert, Appl. Spectrosc. 33, 481 (1979).
[CrossRef]

Y. S. Chang, J. H. Shaw, J. G. Calvert, W. M. Uselman, J. Quant. Spectrosc. Radiat. Transfer 19, 599 (1978).
[CrossRef]

Y. S. Chang, J. H. Shaw, J. G. Calvert, W. M. Uselman, J. Quant. Spectrosc. Radiat. Transfer 18, 589 (1977).
[CrossRef]

Camy-Peyret, C.

The H2O parameters used in the spectra in Figs. 7–9 were taken from C. Camy-Peyret, J. M. Flaud, Mol. Phys. 32, 523 (1976); J. M. Flaud, C. Camy-Peyret, J. Y. Mandin, G. Guelachvili, Mol. Phys. 34, 413 (1977).
[CrossRef]

Chang, Y. S.

Y. S. Chang, J. H. Shaw, J. G. Calvert, W. M. Uselman, J. Quant. Spectrosc. Radiat. Transfer 19, 599 (1978).
[CrossRef]

Y. S. Chang, J. H. Shaw, J. G. Calvert, W. M. Uselman, J. Quant. Spectrosc. Radiat. Transfer 18, 589 (1977).
[CrossRef]

Y. S. Chang, J. H. Shaw, J. Quant. Spectrosc. Radiat. Transfer 18, 491 (1977).
[CrossRef]

Y. S. Chang, J. H. Shaw, Appl. Spectrosc. 31, 213 (1977).
[CrossRef]

Clough, S. A.

Dongarra, J. J.

J. J. Dongarra, C. B. Moler, J. R. Bunch, G. W. Stewart, Linpack Users’ Guide (Society for Industrial and Applied Mathematics, Philadelphia, 1979), Chap. 11.
[CrossRef]

Eichen, G.

G. Eichen, J. Laurent, in Remote Sensing of the Atmosphere: Inversion Methods and Applications, A. L. Fymat, V. E. Zuev, Eds. (Elsevier, Amsterdam, 1978).

Eitner, P. G.

P. G. Eitner, F. G. Smith, D. R. Woods, Proc. Soc. Photo-Opt. Instrum. Eng. 148, 112 (1978).

Farmer, C. B.

C. B. Farmer, Jet Propulsion Lab.; private communication.

Flaud, J. M.

The H2O parameters used in the spectra in Figs. 7–9 were taken from C. Camy-Peyret, J. M. Flaud, Mol. Phys. 32, 523 (1976); J. M. Flaud, C. Camy-Peyret, J. Y. Mandin, G. Guelachvili, Mol. Phys. 34, 413 (1977).
[CrossRef]

Goldman, A.

E. Niple, W. G. Mankin, A. Goldman, D. G. Murcray, F. J. Murcray, Geophys. Rev. Lett. 7, 489 (1980).
[CrossRef]

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

A. Goldman, U. Denver; private communication.

Griffiths, P. R.

R. J. Anderson, P. R. Griffiths, J. Quant. Spectrosc. Radiat. Transfer 17, 393 (1977).
[CrossRef]

Hartley, H. O.

H. O. Hartley, Technometrics 3, 269 (1961).
[CrossRef]

Jennrich, R. I.

M. L. Ralston, R. I. Jennrich, Technometrics 20, 7 (1978).
[CrossRef]

Jones, J. W. C.

A. S. Pine, J. W. C. Jones, A. G. Robiette, J. Mol. Spectrosc. 74, 52 (1979).
[CrossRef]

Laurent, J.

G. Eichen, J. Laurent, in Remote Sensing of the Atmosphere: Inversion Methods and Applications, A. L. Fymat, V. E. Zuev, Eds. (Elsevier, Amsterdam, 1978).

Lin, C. L.

Mankin, W. G.

E. Niple, W. G. Mankin, A. Goldman, D. G. Murcray, F. J. Murcray, Geophys. Rev. Lett. 7, 489 (1980).
[CrossRef]

W. G. Mankin, Appl. Opt. 18, 3426 (1979).
[CrossRef] [PubMed]

McClatchey, R. A.

Mireles, R.

Moler, C. B.

J. J. Dongarra, C. B. Moler, J. R. Bunch, G. W. Stewart, Linpack Users’ Guide (Society for Industrial and Applied Mathematics, Philadelphia, 1979), Chap. 11.
[CrossRef]

Murcray, D. G.

E. Niple, W. G. Mankin, A. Goldman, D. G. Murcray, F. J. Murcray, Geophys. Rev. Lett. 7, 489 (1980).
[CrossRef]

Murcray, F. J.

E. Niple, W. G. Mankin, A. Goldman, D. G. Murcray, F. J. Murcray, Geophys. Rev. Lett. 7, 489 (1980).
[CrossRef]

Nazareth, L.

L. Nazareth, SIAM (Soc. Ind. Appl. Math.) Rev. 22, 1 (1980).

Niple, E.

E. Niple, W. G. Mankin, A. Goldman, D. G. Murcray, F. J. Murcray, Geophys. Rev. Lett. 7, 489 (1980).
[CrossRef]

C. L. Lin, E. Niple, J. H. Shaw, J. G. Calvert, Appl. Spectrosc. 33, 481 (1979).
[CrossRef]

Noll, R. J.

R. J. Noll, A. Pires, Appl. Spectrosc. (1980), in press.

Pine, A. S.

A. S. Pine, J. W. C. Jones, A. G. Robiette, J. Mol. Spectrosc. 74, 52 (1979).
[CrossRef]

Pires, A.

R. J. Noll, A. Pires, Appl. Spectrosc. (1980), in press.

Pourcin, J.

J. Pourcin, R. Romanetti, Infrared Phys. 13, 161 (1973).
[CrossRef]

Ralston, M. L.

M. L. Ralston, R. I. Jennrich, Technometrics 20, 7 (1978).
[CrossRef]

Rao, C. R.

C. R. Rao, Linear Statistical Inference and Its Applications (Wiley, New York, 1973), Chap. 4.
[CrossRef]

Robiette, A. G.

A. S. Pine, J. W. C. Jones, A. G. Robiette, J. Mol. Spectrosc. 74, 52 (1979).
[CrossRef]

Romanetti, R.

J. Pourcin, R. Romanetti, Infrared Phys. 13, 161 (1973).
[CrossRef]

Rothman, L. S.

Shaw, J. H.

C. L. Lin, E. Niple, J. H. Shaw, J. G. Calvert, Appl. Spectrosc. 33, 481 (1979).
[CrossRef]

Y. S. Chang, J. H. Shaw, J. G. Calvert, W. M. Uselman, J. Quant. Spectrosc. Radiat. Transfer 19, 599 (1978).
[CrossRef]

Y. S. Chang, J. H. Shaw, J. G. Calvert, W. M. Uselman, J. Quant. Spectrosc. Radiat. Transfer 18, 589 (1977).
[CrossRef]

Y. S. Chang, J. H. Shaw, Appl. Spectrosc. 31, 213 (1977).
[CrossRef]

Y. S. Chang, J. H. Shaw, J. Quant. Spectrosc. Radiat. Transfer 18, 491 (1977).
[CrossRef]

J. H. Shaw, Ohio State U.; private communication.

Smith, F. G.

P. G. Eitner, F. G. Smith, D. R. Woods, Proc. Soc. Photo-Opt. Instrum. Eng. 148, 112 (1978).

Snider, D. E.

Stewart, G. W.

J. J. Dongarra, C. B. Moler, J. R. Bunch, G. W. Stewart, Linpack Users’ Guide (Society for Industrial and Applied Mathematics, Philadelphia, 1979), Chap. 11.
[CrossRef]

Tiao, G. C.

G. E. P. Box, G. C. Tiao, Bayesian Inference in Statistical Analysis (Addison-Wesley, Reading, Mass., 1973).

Uselman, W. M.

Y. S. Chang, J. H. Shaw, J. G. Calvert, W. M. Uselman, J. Quant. Spectrosc. Radiat. Transfer 19, 599 (1978).
[CrossRef]

Y. S. Chang, J. H. Shaw, J. G. Calvert, W. M. Uselman, J. Quant. Spectrosc. Radiat. Transfer 18, 589 (1977).
[CrossRef]

Woods, D. R.

P. G. Eitner, F. G. Smith, D. R. Woods, Proc. Soc. Photo-Opt. Instrum. Eng. 148, 112 (1978).

Young, L. G.

Appl. Opt.

Appl. Spectrosc.

Geophys. Rev. Lett.

E. Niple, W. G. Mankin, A. Goldman, D. G. Murcray, F. J. Murcray, Geophys. Rev. Lett. 7, 489 (1980).
[CrossRef]

Infrared Phys.

J. Pourcin, R. Romanetti, Infrared Phys. 13, 161 (1973).
[CrossRef]

J. Mol. Spectrosc.

A. S. Pine, J. W. C. Jones, A. G. Robiette, J. Mol. Spectrosc. 74, 52 (1979).
[CrossRef]

J. Opt. Soc. Am.

J. Quant. Spectrosc. Radiat. Transfer

R. J. Anderson, P. R. Griffiths, J. Quant. Spectrosc. Radiat. Transfer 17, 393 (1977).
[CrossRef]

Y. S. Chang, J. H. Shaw, J. Quant. Spectrosc. Radiat. Transfer 18, 491 (1977).
[CrossRef]

Y. S. Chang, J. H. Shaw, J. G. Calvert, W. M. Uselman, J. Quant. Spectrosc. Radiat. Transfer 18, 589 (1977).
[CrossRef]

Y. S. Chang, J. H. Shaw, J. G. Calvert, W. M. Uselman, J. Quant. Spectrosc. Radiat. Transfer 19, 599 (1978).
[CrossRef]

Mol. Phys.

The H2O parameters used in the spectra in Figs. 7–9 were taken from C. Camy-Peyret, J. M. Flaud, Mol. Phys. 32, 523 (1976); J. M. Flaud, C. Camy-Peyret, J. Y. Mandin, G. Guelachvili, Mol. Phys. 34, 413 (1977).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng.

P. G. Eitner, F. G. Smith, D. R. Woods, Proc. Soc. Photo-Opt. Instrum. Eng. 148, 112 (1978).

SIAM (Soc. Ind. Appl. Math.)

L. Nazareth, SIAM (Soc. Ind. Appl. Math.) Rev. 22, 1 (1980).

Technometrics

M. L. Ralston, R. I. Jennrich, Technometrics 20, 7 (1978).
[CrossRef]

H. O. Hartley, Technometrics 3, 269 (1961).
[CrossRef]

Other

G. E. P. Box, G. C. Tiao, Bayesian Inference in Statistical Analysis (Addison-Wesley, Reading, Mass., 1973).

J. J. Dongarra, C. B. Moler, J. R. Bunch, G. W. Stewart, Linpack Users’ Guide (Society for Industrial and Applied Mathematics, Philadelphia, 1979), Chap. 11.
[CrossRef]

C. R. Rao, Linear Statistical Inference and Its Applications (Wiley, New York, 1973), Chap. 4.
[CrossRef]

G. Eichen, J. Laurent, in Remote Sensing of the Atmosphere: Inversion Methods and Applications, A. L. Fymat, V. E. Zuev, Eds. (Elsevier, Amsterdam, 1978).

R. J. Noll, A. Pires, Appl. Spectrosc. (1980), in press.

J. H. Shaw, Ohio State U.; private communication.

A. Goldman, U. Denver; private communication.

C. B. Farmer, Jet Propulsion Lab.; private communication.

R. J. Bell, Introductory Fourier Transform Spectroscopy (Academic, New York, 1972), Chaps. 8 and 12.

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Figures (9)

Fig. 1
Fig. 1

Synthetic spectrum calculated with the model in Eq. (4). See text for details of spectral conditions.

Fig. 2
Fig. 2

Comparison between the synthetic spectrum to be fit (same as in Fig. 1) and the initial guess spectrum.

Fig. 3
Fig. 3

First half of a sample fit to a noisy synthetic spectrum.

Fig. 4
Fig. 4

Second half of a sample fit.

Fig. 5
Fig. 5

First half of fit to a real spectrum. See text for details of experimental conditions.

Fig. 6
Fig. 6

Second half of fit to real spectrum.

Fig. 7
Fig. 7

Fit to solar spectrum using an unknown parameter set from synthetic spectrum fits.

Fig. 8
Fig. 8

Fit to same solar spectrum as Fig. 7 but including the effects of phase errors and channel spectra.

Fig. 9
Fig. 9

Fit to solar spectrum using the same model as Fig. 8 but after correcting line shifts and mystery lines.

Tables (6)

Tables Icon

Table I Physical Characteristics of the Atmosphere Used to Calculate the Spectrum in Fig. 1

Tables Icon

Table II Parameter Values for Fitting Synthetic Spectrum

Tables Icon

Table III Best Guess Parameter Values for the Best and Worst Fitting Runs on Noise-Free Synthetic Spectrum

Tables Icon

Table IV Summary of Parameter Estimates for Noisy Synthetic Spectrum Fitting, With and Without the Worst Run

Tables Icon

Table V Parameter Estimates for Spectrum in Figs. 5 and 6

Tables Icon

Table VI Comparison of Selected Spectrum Parameters Using Different Spectral Model

Equations (19)

Equations on this page are rendered with MathJax. Learn more.

I A ( σ ) = I 0 ( σ ) τ A ( σ ) ,
τ A ( σ ) = exp [ A i = 1 M n abs ( i ) ( s ) k ( i ) ( σ , s ) d s ] ,
I A ( σ ) = 0 I A ( σ ) R ( σ , σ ) d σ + I 0 ( σ ) ,
I A ( σ ) = [ I 0 ( σ ) R 1 ( σ , σ ) ] τ A ( σ ) R 2 ( σ σ ) d σ + I 0 ( σ ) .
Q ( θ ) i = 1 N [ y i I A ( σ i , ξ; θ ) ] 2 .
i = 1 N [ y i I A ( σ i , ξ; θ ˆ ) ] I A ( σ i , ξ; θ ) θ j | θ = θ ˆ = 0 , j = 1 , , P
W i j = k = 1 N I A ( σ k , ξ; θ ) θ j | θ = θ ˆ I A ( σ k , ξ; θ ) θ ˆ i | θ = Ө ˆ
Ө = θ 0 + Δ θ α ,
Δ θ k = θ k θ 0 , k = 1 , , P .
l ( α ) = I ( θ 0 ) + Δ I α ,
Δ I k = I ( θ k ) I ( θ 0 ) , k = 1 , , P .
α ˆ = ( Δ I T Δ I ) 1 Δ I T [ y I ( θ 0 ) ] .
Δ I = U S V T .
α ˆ = V S 1 U T [ y I ( θ 0 ) ] .
Q ( θ ) = Q ( θ ) + j = 1 P σ ˆ 2 [ ( θ j ( prior ) θ j ) / σ j ( prior ) ] 2 .
σ ˆ 2 = 1 / ( N P ) i = 1 N [ y i I ( σ i , ξ; θ 0 ) ] 2 .
I ( σ N + j , ξ; θ ) = σ ˆ [ θ j ( prior ) θ j ] / σ j ( prior ) .
( background ) σ i = B A 0 + i B A 1
σ i = σ 0 + i ( σ u σ 0 ) / N , i = 0 , N ,

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