Abstract

We present a novel method for the quantitative analysis of spectra based on the properties of the cross correlation between a real spectrum and either a numerical synthesis or a laboratory simulation. We propose a new goodness-of-fit criterion called the heteromorphic coefficient H that has the property of being zero when a fit is achieved and varying smoothly through zero as the iteration proceeds, providing a powerful tool for automatic or near-automatic analysis. We also show that H can be rendered substantially noise-immune, permitting the analysis of very weak spectra well below the apparent noise level and, as a by-product, providing Doppler shift and radial velocity information with excellent precision. The technique is in regular use in the Atmospheric Trace Molecule Spectroscopy (ATMOS) project and operates in an interactive, real-time computing environment with turn-around times of a few seconds or less.

© 1988 Optical Society of America

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  1. H. P. Larson, R. R. Treffers, U. Fink, “Phosphine in Jupiter’s Atmosphere: the Evidence from High-Altitude Observations at 5 Micrometers,” Astrophys. J. 211, 972 (1977).
    [CrossRef]
  2. U. Fink, H. P. Larson, R. R. Treffers, “Germane in the Atmosphere of Jupiter,” Icarus 34, 344 (1978).
    [CrossRef]
  3. Y. W. Lee, Statistical Theory of Communication (Wiley, New York, 1960).
  4. Y. S. Chang, J. H. Shaw, “A Non-Linear Least Squares Method of Determining Line Intensities and Half-Widths,” Appl. Spectrosc. 31, 213 (1977).
    [CrossRef]
  5. C. L. Lin, J. H. Shaw, J. G. Calvert, “Least Squares Analysis of Voigt-Shaped Lines,” J. Quant. Spectrosc. Radiat. Transfer 22, 253 (1979).
    [CrossRef]
  6. E. R. Niple, “Nonlinear Least Squares Analysis of Atmospheric Absorption Spectra,” Appl. Opt. 19, 3481 (1980).
    [CrossRef] [PubMed]
  7. L. R. Brown, J. S. Margolis, R. H. Norton, B. D. Stedry, “Computer Measurements of Line Strengths with Application to the Methane Spectrum,” Appl. Spectrosc. 37, 287 (1983).
    [CrossRef]
  8. J. H. Park, B. Carli, “Analysis of Far-Infrared Emission Fourier Transform Spectra,” Appl. Opt. 25, 3490 (1986).
    [CrossRef] [PubMed]
  9. G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968).
  10. C. B. Farmer, O. F. Raper, “High Resolution Infrared Spectroscopy from Space: a Preliminary Report on the Results of the Atmospheric Trace Molecule Spectroscopy (ATMOS) Experiment on SPACELAB 3,” NASA Conference Proceedings CP-2429 (May1986).
  11. G. C. Toon, C. B. Farmer, R. H. Norton, “Detection of Stratospheric N2O5 by Infrared Remote Sounding,” Nature London 319, 570 (1986).
    [CrossRef]
  12. See a group of papers in Geophys. Res. Lett. 13, 757ff(1986).

1986

G. C. Toon, C. B. Farmer, R. H. Norton, “Detection of Stratospheric N2O5 by Infrared Remote Sounding,” Nature London 319, 570 (1986).
[CrossRef]

See a group of papers in Geophys. Res. Lett. 13, 757ff(1986).

J. H. Park, B. Carli, “Analysis of Far-Infrared Emission Fourier Transform Spectra,” Appl. Opt. 25, 3490 (1986).
[CrossRef] [PubMed]

1983

1980

1979

C. L. Lin, J. H. Shaw, J. G. Calvert, “Least Squares Analysis of Voigt-Shaped Lines,” J. Quant. Spectrosc. Radiat. Transfer 22, 253 (1979).
[CrossRef]

1978

U. Fink, H. P. Larson, R. R. Treffers, “Germane in the Atmosphere of Jupiter,” Icarus 34, 344 (1978).
[CrossRef]

1977

H. P. Larson, R. R. Treffers, U. Fink, “Phosphine in Jupiter’s Atmosphere: the Evidence from High-Altitude Observations at 5 Micrometers,” Astrophys. J. 211, 972 (1977).
[CrossRef]

Y. S. Chang, J. H. Shaw, “A Non-Linear Least Squares Method of Determining Line Intensities and Half-Widths,” Appl. Spectrosc. 31, 213 (1977).
[CrossRef]

Brown, L. R.

Calvert, J. G.

C. L. Lin, J. H. Shaw, J. G. Calvert, “Least Squares Analysis of Voigt-Shaped Lines,” J. Quant. Spectrosc. Radiat. Transfer 22, 253 (1979).
[CrossRef]

Carli, B.

Chang, Y. S.

Farmer, C. B.

G. C. Toon, C. B. Farmer, R. H. Norton, “Detection of Stratospheric N2O5 by Infrared Remote Sounding,” Nature London 319, 570 (1986).
[CrossRef]

C. B. Farmer, O. F. Raper, “High Resolution Infrared Spectroscopy from Space: a Preliminary Report on the Results of the Atmospheric Trace Molecule Spectroscopy (ATMOS) Experiment on SPACELAB 3,” NASA Conference Proceedings CP-2429 (May1986).

Fink, U.

U. Fink, H. P. Larson, R. R. Treffers, “Germane in the Atmosphere of Jupiter,” Icarus 34, 344 (1978).
[CrossRef]

H. P. Larson, R. R. Treffers, U. Fink, “Phosphine in Jupiter’s Atmosphere: the Evidence from High-Altitude Observations at 5 Micrometers,” Astrophys. J. 211, 972 (1977).
[CrossRef]

Korn, G. A.

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968).

Korn, T. M.

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968).

Larson, H. P.

U. Fink, H. P. Larson, R. R. Treffers, “Germane in the Atmosphere of Jupiter,” Icarus 34, 344 (1978).
[CrossRef]

H. P. Larson, R. R. Treffers, U. Fink, “Phosphine in Jupiter’s Atmosphere: the Evidence from High-Altitude Observations at 5 Micrometers,” Astrophys. J. 211, 972 (1977).
[CrossRef]

Lee, Y. W.

Y. W. Lee, Statistical Theory of Communication (Wiley, New York, 1960).

Lin, C. L.

C. L. Lin, J. H. Shaw, J. G. Calvert, “Least Squares Analysis of Voigt-Shaped Lines,” J. Quant. Spectrosc. Radiat. Transfer 22, 253 (1979).
[CrossRef]

Margolis, J. S.

Niple, E. R.

Norton, R. H.

G. C. Toon, C. B. Farmer, R. H. Norton, “Detection of Stratospheric N2O5 by Infrared Remote Sounding,” Nature London 319, 570 (1986).
[CrossRef]

L. R. Brown, J. S. Margolis, R. H. Norton, B. D. Stedry, “Computer Measurements of Line Strengths with Application to the Methane Spectrum,” Appl. Spectrosc. 37, 287 (1983).
[CrossRef]

Park, J. H.

Raper, O. F.

C. B. Farmer, O. F. Raper, “High Resolution Infrared Spectroscopy from Space: a Preliminary Report on the Results of the Atmospheric Trace Molecule Spectroscopy (ATMOS) Experiment on SPACELAB 3,” NASA Conference Proceedings CP-2429 (May1986).

Shaw, J. H.

C. L. Lin, J. H. Shaw, J. G. Calvert, “Least Squares Analysis of Voigt-Shaped Lines,” J. Quant. Spectrosc. Radiat. Transfer 22, 253 (1979).
[CrossRef]

Y. S. Chang, J. H. Shaw, “A Non-Linear Least Squares Method of Determining Line Intensities and Half-Widths,” Appl. Spectrosc. 31, 213 (1977).
[CrossRef]

Stedry, B. D.

Toon, G. C.

G. C. Toon, C. B. Farmer, R. H. Norton, “Detection of Stratospheric N2O5 by Infrared Remote Sounding,” Nature London 319, 570 (1986).
[CrossRef]

Treffers, R. R.

U. Fink, H. P. Larson, R. R. Treffers, “Germane in the Atmosphere of Jupiter,” Icarus 34, 344 (1978).
[CrossRef]

H. P. Larson, R. R. Treffers, U. Fink, “Phosphine in Jupiter’s Atmosphere: the Evidence from High-Altitude Observations at 5 Micrometers,” Astrophys. J. 211, 972 (1977).
[CrossRef]

Appl. Opt.

Appl. Spectrosc.

Astrophys. J.

H. P. Larson, R. R. Treffers, U. Fink, “Phosphine in Jupiter’s Atmosphere: the Evidence from High-Altitude Observations at 5 Micrometers,” Astrophys. J. 211, 972 (1977).
[CrossRef]

Geophys. Res. Lett.

See a group of papers in Geophys. Res. Lett. 13, 757ff(1986).

Icarus

U. Fink, H. P. Larson, R. R. Treffers, “Germane in the Atmosphere of Jupiter,” Icarus 34, 344 (1978).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer

C. L. Lin, J. H. Shaw, J. G. Calvert, “Least Squares Analysis of Voigt-Shaped Lines,” J. Quant. Spectrosc. Radiat. Transfer 22, 253 (1979).
[CrossRef]

Nature London

G. C. Toon, C. B. Farmer, R. H. Norton, “Detection of Stratospheric N2O5 by Infrared Remote Sounding,” Nature London 319, 570 (1986).
[CrossRef]

Other

Y. W. Lee, Statistical Theory of Communication (Wiley, New York, 1960).

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968).

C. B. Farmer, O. F. Raper, “High Resolution Infrared Spectroscopy from Space: a Preliminary Report on the Results of the Atmospheric Trace Molecule Spectroscopy (ATMOS) Experiment on SPACELAB 3,” NASA Conference Proceedings CP-2429 (May1986).

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Figures (9)

Fig. 1
Fig. 1

Failure of the conventional correlation coefficient as the matching criterion between two spectra (both synthetic for this illustration): (a) synthetic real spectrum; (b) synthetic test spectrum (offset); (c) difference (f1f2), 5× scale expanded. The correlation coefficient between the two spectra is 0.995, but they are visibly different.

Fig. 2
Fig. 2

Further evidence for the failure of the correlation coefficient as a matching criterion: (a) a real ATMOS spectrum acquired at 122-km tangent height; (b) a synthetic spectrum (offset) that is believed to represent a good fit to the data. The correlation coefficient between the two is, however, only 0.615.

Fig. 3
Fig. 3

Examples of the autocorrelation of the noise on a real ATMOS spectrum computed over two widely differing ranges: (a) 4 cm−1; peak autocorrelation = 9.38 ± 0.48 × 10−6; (b) 80 cm−1; peak autocorrelation = 9.76 ± 0.13 × 10−6. Note how the noise as seen at leads and lags removed from zero declines as the range increases whereas the peak autocorrelation (noise variance) is quite stable.

Fig. 4
Fig. 4

Plot of the rms noise on the autocorrelations of Fig. 3 vs spectral range. The points are the measured (relative) uncertainties and the line is a best-fit straight line of slope 0.5, demonstrating that the noise (and thus the autocorrelations themselves) has good statistical properties.

Fig. 5
Fig. 5

Real and synthetic spectra in the region of the CO2 band near 2350 cm−1 for a tangent height of 138 km: (a) ATMOS data; (b) synthetic spectrum based on existing atmospheric models; (c) best-fit spectrum using the technique described in this paper (spectra offset for clarity). Note that no spectral features are evident in the real data (certainly not at the level required by the existing models) and that lines of the strength required by the present technique could never be seen using point-by-point comparison methods.

Fig. 6
Fig. 6

Cross correlation of spectra (a) and (c) in Fig. 5. The major peak at a shift of 0.0491 cm−1 (≡ radial velocity of 6283 ± 60 m · s−1) indicates that the real data do, indeed, contain features of CO2 buried in noise. Note that the expected radial velocity (based on the orbital ephemeris of SPACELAB 3) is 6250 m · s−1, well within the measurement error of the peak-finding algorithm.

Fig. 7
Fig. 7

As Fig. 5, but acquired some 9 s later at a tangent height of 123 km. Although CO2 lines are now evident in the data [curve (a)], they are still much weaker than predicted [curve (b)]. Curve (c) is the best fit derived by the H coefficient method.

Fig. 8
Fig. 8

Cross correlation of curves (a) and (c) in Fig. 7. Note that as the real data become more structured, the cross correlation becomes more pronounced (as expected), and the radial velocity precision (6257 ± 8 m · s−1) improves.

Fig. 9
Fig. 9

Trend of the ac heteromorphic coefficient vs tangent layer volume mixing ratio of CO2 (note that higher layers were previously measured in an onion-peeling approach). The smoothness of the trend through zero (the best-fit condition) is evident and leads to a value for the CO2 volume mixing ratio at this level of 8.2 ppm.

Tables (2)

Tables Icon

Table I Noise Variance and Uncertainty as a Function of Spectral Range for Real ATMOS Data

Tables Icon

Table II General Characteristics of the ATMOS Fourier Transform Spectrometer

Equations (13)

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C ( x ) = + f 1 ( x ) · f 2 ( x + x ) · d x .
C ( x ) = f 1 ( x ) * f 2 ( x ) ,
A 1 ( x ) + A 2 ( x ) 2 · | C ( x ) | for all x ,
C ( x ) = [ f 1 a ( x ) + f 1 b ( x ) ] * [ f 2 a ( x ) + f 2 b ( x ) ] = f 1 a * f 2 a + f 1 a * f 2 b + f 1 b * f 2 a + f 1 b * f 2 b ,
ρ = C max [ A 1 ( 0 ) · A 2 ( 0 ) ] 1 / 2 ,
H = A 1 ( 0 ) A 2 ( 0 ) C max ,
A ( 0 ) = A ( 0 ) A n ( 0 ) ,
H = [ A 1 ( 0 ) A 1 n ( 0 ) ] [ A 2 ( 0 ) A 2 n ( 0 ) ] C max ,
H lim = [ ( δ A 1 n ) 2 + ( δ A 2 n ) 2 ] 1 / 2 C max .
F ( x ) = F 0 ( x ) · exp ( 2 π i f ) ,
ϕ ( k ) = 2 π Δ f k ,
= 1 2 π Δ f d ϕ d k ,
δ = | Σ w ( k ) · r 2 ( k ) [ Σ w ( k ) 2 ] · Σ [ w ( k ) · r ( k ) 2 ] | 1 / 2 ,

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