Abstract

Differential optical absorption spectroscopy is a widely used technique for open-column atmospheric-gas pollution monitoring. The concentration retrieval is based on the fitting of the measured differential absorbance through the Lambert-Beer law. We present an alternative method for calculating the gas concentration on the basis of the proportionality between differential absorbance and differential absorption cross section of the gas under study. The method can be used on its own for single-component analysis or as a complement to the standard technique in multicomponent cases. The performance of the method for the case of cross interference between two gases is analyzed. The procedure can be used with differential absorption cross sections measured in the laboratory or taken from the literature. In addition, the method provides a criterion to discriminate against different species having absorption features in the same wavelength range.

© 2003 Optical Society of America

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References

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  1. D. Perner, U. Platt, “Detection of nitrous acid in the atmosphere by differential optical absorption,” Geophys. Res. Lett. 6, 917–920 (1979).
    [CrossRef]
  2. U. Platt, D. Perner, “Direct measurements of atmospheric CH2O, HNO2, O3, NO2 and SO2 by differential optical absorption in the near UV,” J. Geophys. Res. 85, 7453–7458 (1980).
    [CrossRef]
  3. U. Platt, D. Perner, H. Pätz, “Simultaneous measurements of atmospheric CH2O, O3, and NO2 by differential optical absorption,” J. Geophys. Res. 84, 6329–6335 (1979).
    [CrossRef]
  4. H. Axelsson, H. Edner, B. Galle, P. Ragnarson, M. Rudin, “Differential optical absorption spectroscopy (DOAS) measurements of ozone in the 280–290 nm wavelength region,” Appl. Spect. 44, 1654–1658 (1990).
    [CrossRef]
  5. H. Edner, P. Ragnarson, S. Spännare, S. Svanberg, “Differential optical absorption spectroscopy (DOAS) system for urban atmospheric pollution monitoring,” Appl. Opt. 32, 327–333 (1993).
    [CrossRef] [PubMed]
  6. H. Axelsson, A. Eilard, A. Emanuelsson, B. Galle, H. Edner, P. Ragnarson, H. Kloo, “Measurement of aromatic hydrocarbons with the DOAS technique,” Appl. Spect. 49, 1254–1260 (1995).
    [CrossRef]
  7. F. Evangelisti, A. Baroncelli, P. Bonasoni, G. Giovanelli, F. Ravegnani, “Differential optical absorption spectrometer for measurement of tropospheric pollutants,” Appl. Opt. 34, 2737–2744 (1995).
    [CrossRef] [PubMed]
  8. G. Giovanelli, P. Bonasoni, F. Evangelisti, F. Ravegnani, “Ozone ground-based measurements by the GASCOD near UV and visible DOAS system,” in Proceedings of the Quadrennial Ozone Symposium 1992, R. D. Hudson, ed. (Goddard Space Flight Center, Greenbelt, Md., 1992), pp. 707–710.
  9. P. Bonasoni, F. Evangelisti, G. Giovanelli, F. Ravegnani, “Remote sensing DOAS system for measurements of atmospheric trace gases,” Phys. Med. 9, 304–307 (1993).
  10. U. Platt, D. Perner, “Measurements of atmospheric trace gases by long path differential UV/visible absorption spectroscopy,” in Optical and Laser Remote Sensing Techniques, D. K. Killinger, A. Mooradian, eds., (Springer-Verlag, Berlin, 1983), pp. 97–105.
    [CrossRef]
  11. U. Platt, “Differential optical absorption spectroscopy,” in Monitoring by Spectroscopy Techniques, M. Sigrist, ed. (Wiley, New York, 1994), pp. 27–84.
  12. D. J. Brassington, “Sulfur dioxide absorption cross-section measurements from 290 nm to 317 nm,” Appl. Opt. 20, 3774–3779 (1981).
    [CrossRef] [PubMed]
  13. G. Herzberg, Electronic Spectra of Polyatomic Molecules (Van Nostrand Reinhold, New York, 1966).
  14. D. Schinca, J. Tocho, “Development of a DOAS system based on a cross dispersion spectrograph,” in Water, Ground, and Air Pollution Monitoring and Remediation, T. Vo-Dinh, R. Spellicy, eds., Proc. SPIE4199, 77–85 (2001).
    [CrossRef]
  15. D. Schinca, R. Di Paolo, F. Videla, E. Rodríguez, J. Tocho, Medidas en Planta de SO2 y NO2 en la Región de La Plata, (Asociación Física Argentina, Argentina, 1999).
  16. E. Lloyd, ed., Handbook of Applicable Mathematics, (Wiley, New York, 1984), Vol. VI, Part A, Chap. 8.
  17. L. T. Molina, M. J. Molina, “Absolute absorption cross sections of ozone in the 185- to 350-nm wavelength range,” J. Geophys. Res. 91, 14501–14508 (1986).
    [CrossRef]

1995 (2)

H. Axelsson, A. Eilard, A. Emanuelsson, B. Galle, H. Edner, P. Ragnarson, H. Kloo, “Measurement of aromatic hydrocarbons with the DOAS technique,” Appl. Spect. 49, 1254–1260 (1995).
[CrossRef]

F. Evangelisti, A. Baroncelli, P. Bonasoni, G. Giovanelli, F. Ravegnani, “Differential optical absorption spectrometer for measurement of tropospheric pollutants,” Appl. Opt. 34, 2737–2744 (1995).
[CrossRef] [PubMed]

1993 (2)

H. Edner, P. Ragnarson, S. Spännare, S. Svanberg, “Differential optical absorption spectroscopy (DOAS) system for urban atmospheric pollution monitoring,” Appl. Opt. 32, 327–333 (1993).
[CrossRef] [PubMed]

P. Bonasoni, F. Evangelisti, G. Giovanelli, F. Ravegnani, “Remote sensing DOAS system for measurements of atmospheric trace gases,” Phys. Med. 9, 304–307 (1993).

1990 (1)

H. Axelsson, H. Edner, B. Galle, P. Ragnarson, M. Rudin, “Differential optical absorption spectroscopy (DOAS) measurements of ozone in the 280–290 nm wavelength region,” Appl. Spect. 44, 1654–1658 (1990).
[CrossRef]

1986 (1)

L. T. Molina, M. J. Molina, “Absolute absorption cross sections of ozone in the 185- to 350-nm wavelength range,” J. Geophys. Res. 91, 14501–14508 (1986).
[CrossRef]

1981 (1)

1980 (1)

U. Platt, D. Perner, “Direct measurements of atmospheric CH2O, HNO2, O3, NO2 and SO2 by differential optical absorption in the near UV,” J. Geophys. Res. 85, 7453–7458 (1980).
[CrossRef]

1979 (2)

U. Platt, D. Perner, H. Pätz, “Simultaneous measurements of atmospheric CH2O, O3, and NO2 by differential optical absorption,” J. Geophys. Res. 84, 6329–6335 (1979).
[CrossRef]

D. Perner, U. Platt, “Detection of nitrous acid in the atmosphere by differential optical absorption,” Geophys. Res. Lett. 6, 917–920 (1979).
[CrossRef]

Axelsson, H.

H. Axelsson, A. Eilard, A. Emanuelsson, B. Galle, H. Edner, P. Ragnarson, H. Kloo, “Measurement of aromatic hydrocarbons with the DOAS technique,” Appl. Spect. 49, 1254–1260 (1995).
[CrossRef]

H. Axelsson, H. Edner, B. Galle, P. Ragnarson, M. Rudin, “Differential optical absorption spectroscopy (DOAS) measurements of ozone in the 280–290 nm wavelength region,” Appl. Spect. 44, 1654–1658 (1990).
[CrossRef]

Baroncelli, A.

Bonasoni, P.

F. Evangelisti, A. Baroncelli, P. Bonasoni, G. Giovanelli, F. Ravegnani, “Differential optical absorption spectrometer for measurement of tropospheric pollutants,” Appl. Opt. 34, 2737–2744 (1995).
[CrossRef] [PubMed]

P. Bonasoni, F. Evangelisti, G. Giovanelli, F. Ravegnani, “Remote sensing DOAS system for measurements of atmospheric trace gases,” Phys. Med. 9, 304–307 (1993).

G. Giovanelli, P. Bonasoni, F. Evangelisti, F. Ravegnani, “Ozone ground-based measurements by the GASCOD near UV and visible DOAS system,” in Proceedings of the Quadrennial Ozone Symposium 1992, R. D. Hudson, ed. (Goddard Space Flight Center, Greenbelt, Md., 1992), pp. 707–710.

Brassington, D. J.

Di Paolo, R.

D. Schinca, R. Di Paolo, F. Videla, E. Rodríguez, J. Tocho, Medidas en Planta de SO2 y NO2 en la Región de La Plata, (Asociación Física Argentina, Argentina, 1999).

Edner, H.

H. Axelsson, A. Eilard, A. Emanuelsson, B. Galle, H. Edner, P. Ragnarson, H. Kloo, “Measurement of aromatic hydrocarbons with the DOAS technique,” Appl. Spect. 49, 1254–1260 (1995).
[CrossRef]

H. Edner, P. Ragnarson, S. Spännare, S. Svanberg, “Differential optical absorption spectroscopy (DOAS) system for urban atmospheric pollution monitoring,” Appl. Opt. 32, 327–333 (1993).
[CrossRef] [PubMed]

H. Axelsson, H. Edner, B. Galle, P. Ragnarson, M. Rudin, “Differential optical absorption spectroscopy (DOAS) measurements of ozone in the 280–290 nm wavelength region,” Appl. Spect. 44, 1654–1658 (1990).
[CrossRef]

Eilard, A.

H. Axelsson, A. Eilard, A. Emanuelsson, B. Galle, H. Edner, P. Ragnarson, H. Kloo, “Measurement of aromatic hydrocarbons with the DOAS technique,” Appl. Spect. 49, 1254–1260 (1995).
[CrossRef]

Emanuelsson, A.

H. Axelsson, A. Eilard, A. Emanuelsson, B. Galle, H. Edner, P. Ragnarson, H. Kloo, “Measurement of aromatic hydrocarbons with the DOAS technique,” Appl. Spect. 49, 1254–1260 (1995).
[CrossRef]

Evangelisti, F.

F. Evangelisti, A. Baroncelli, P. Bonasoni, G. Giovanelli, F. Ravegnani, “Differential optical absorption spectrometer for measurement of tropospheric pollutants,” Appl. Opt. 34, 2737–2744 (1995).
[CrossRef] [PubMed]

P. Bonasoni, F. Evangelisti, G. Giovanelli, F. Ravegnani, “Remote sensing DOAS system for measurements of atmospheric trace gases,” Phys. Med. 9, 304–307 (1993).

G. Giovanelli, P. Bonasoni, F. Evangelisti, F. Ravegnani, “Ozone ground-based measurements by the GASCOD near UV and visible DOAS system,” in Proceedings of the Quadrennial Ozone Symposium 1992, R. D. Hudson, ed. (Goddard Space Flight Center, Greenbelt, Md., 1992), pp. 707–710.

Galle, B.

H. Axelsson, A. Eilard, A. Emanuelsson, B. Galle, H. Edner, P. Ragnarson, H. Kloo, “Measurement of aromatic hydrocarbons with the DOAS technique,” Appl. Spect. 49, 1254–1260 (1995).
[CrossRef]

H. Axelsson, H. Edner, B. Galle, P. Ragnarson, M. Rudin, “Differential optical absorption spectroscopy (DOAS) measurements of ozone in the 280–290 nm wavelength region,” Appl. Spect. 44, 1654–1658 (1990).
[CrossRef]

Giovanelli, G.

F. Evangelisti, A. Baroncelli, P. Bonasoni, G. Giovanelli, F. Ravegnani, “Differential optical absorption spectrometer for measurement of tropospheric pollutants,” Appl. Opt. 34, 2737–2744 (1995).
[CrossRef] [PubMed]

P. Bonasoni, F. Evangelisti, G. Giovanelli, F. Ravegnani, “Remote sensing DOAS system for measurements of atmospheric trace gases,” Phys. Med. 9, 304–307 (1993).

G. Giovanelli, P. Bonasoni, F. Evangelisti, F. Ravegnani, “Ozone ground-based measurements by the GASCOD near UV and visible DOAS system,” in Proceedings of the Quadrennial Ozone Symposium 1992, R. D. Hudson, ed. (Goddard Space Flight Center, Greenbelt, Md., 1992), pp. 707–710.

Herzberg, G.

G. Herzberg, Electronic Spectra of Polyatomic Molecules (Van Nostrand Reinhold, New York, 1966).

Kloo, H.

H. Axelsson, A. Eilard, A. Emanuelsson, B. Galle, H. Edner, P. Ragnarson, H. Kloo, “Measurement of aromatic hydrocarbons with the DOAS technique,” Appl. Spect. 49, 1254–1260 (1995).
[CrossRef]

Molina, L. T.

L. T. Molina, M. J. Molina, “Absolute absorption cross sections of ozone in the 185- to 350-nm wavelength range,” J. Geophys. Res. 91, 14501–14508 (1986).
[CrossRef]

Molina, M. J.

L. T. Molina, M. J. Molina, “Absolute absorption cross sections of ozone in the 185- to 350-nm wavelength range,” J. Geophys. Res. 91, 14501–14508 (1986).
[CrossRef]

Pätz, H.

U. Platt, D. Perner, H. Pätz, “Simultaneous measurements of atmospheric CH2O, O3, and NO2 by differential optical absorption,” J. Geophys. Res. 84, 6329–6335 (1979).
[CrossRef]

Perner, D.

U. Platt, D. Perner, “Direct measurements of atmospheric CH2O, HNO2, O3, NO2 and SO2 by differential optical absorption in the near UV,” J. Geophys. Res. 85, 7453–7458 (1980).
[CrossRef]

D. Perner, U. Platt, “Detection of nitrous acid in the atmosphere by differential optical absorption,” Geophys. Res. Lett. 6, 917–920 (1979).
[CrossRef]

U. Platt, D. Perner, H. Pätz, “Simultaneous measurements of atmospheric CH2O, O3, and NO2 by differential optical absorption,” J. Geophys. Res. 84, 6329–6335 (1979).
[CrossRef]

U. Platt, D. Perner, “Measurements of atmospheric trace gases by long path differential UV/visible absorption spectroscopy,” in Optical and Laser Remote Sensing Techniques, D. K. Killinger, A. Mooradian, eds., (Springer-Verlag, Berlin, 1983), pp. 97–105.
[CrossRef]

Platt, U.

U. Platt, D. Perner, “Direct measurements of atmospheric CH2O, HNO2, O3, NO2 and SO2 by differential optical absorption in the near UV,” J. Geophys. Res. 85, 7453–7458 (1980).
[CrossRef]

D. Perner, U. Platt, “Detection of nitrous acid in the atmosphere by differential optical absorption,” Geophys. Res. Lett. 6, 917–920 (1979).
[CrossRef]

U. Platt, D. Perner, H. Pätz, “Simultaneous measurements of atmospheric CH2O, O3, and NO2 by differential optical absorption,” J. Geophys. Res. 84, 6329–6335 (1979).
[CrossRef]

U. Platt, D. Perner, “Measurements of atmospheric trace gases by long path differential UV/visible absorption spectroscopy,” in Optical and Laser Remote Sensing Techniques, D. K. Killinger, A. Mooradian, eds., (Springer-Verlag, Berlin, 1983), pp. 97–105.
[CrossRef]

U. Platt, “Differential optical absorption spectroscopy,” in Monitoring by Spectroscopy Techniques, M. Sigrist, ed. (Wiley, New York, 1994), pp. 27–84.

Ragnarson, P.

H. Axelsson, A. Eilard, A. Emanuelsson, B. Galle, H. Edner, P. Ragnarson, H. Kloo, “Measurement of aromatic hydrocarbons with the DOAS technique,” Appl. Spect. 49, 1254–1260 (1995).
[CrossRef]

H. Edner, P. Ragnarson, S. Spännare, S. Svanberg, “Differential optical absorption spectroscopy (DOAS) system for urban atmospheric pollution monitoring,” Appl. Opt. 32, 327–333 (1993).
[CrossRef] [PubMed]

H. Axelsson, H. Edner, B. Galle, P. Ragnarson, M. Rudin, “Differential optical absorption spectroscopy (DOAS) measurements of ozone in the 280–290 nm wavelength region,” Appl. Spect. 44, 1654–1658 (1990).
[CrossRef]

Ravegnani, F.

F. Evangelisti, A. Baroncelli, P. Bonasoni, G. Giovanelli, F. Ravegnani, “Differential optical absorption spectrometer for measurement of tropospheric pollutants,” Appl. Opt. 34, 2737–2744 (1995).
[CrossRef] [PubMed]

P. Bonasoni, F. Evangelisti, G. Giovanelli, F. Ravegnani, “Remote sensing DOAS system for measurements of atmospheric trace gases,” Phys. Med. 9, 304–307 (1993).

G. Giovanelli, P. Bonasoni, F. Evangelisti, F. Ravegnani, “Ozone ground-based measurements by the GASCOD near UV and visible DOAS system,” in Proceedings of the Quadrennial Ozone Symposium 1992, R. D. Hudson, ed. (Goddard Space Flight Center, Greenbelt, Md., 1992), pp. 707–710.

Rodríguez, E.

D. Schinca, R. Di Paolo, F. Videla, E. Rodríguez, J. Tocho, Medidas en Planta de SO2 y NO2 en la Región de La Plata, (Asociación Física Argentina, Argentina, 1999).

Rudin, M.

H. Axelsson, H. Edner, B. Galle, P. Ragnarson, M. Rudin, “Differential optical absorption spectroscopy (DOAS) measurements of ozone in the 280–290 nm wavelength region,” Appl. Spect. 44, 1654–1658 (1990).
[CrossRef]

Schinca, D.

D. Schinca, J. Tocho, “Development of a DOAS system based on a cross dispersion spectrograph,” in Water, Ground, and Air Pollution Monitoring and Remediation, T. Vo-Dinh, R. Spellicy, eds., Proc. SPIE4199, 77–85 (2001).
[CrossRef]

D. Schinca, R. Di Paolo, F. Videla, E. Rodríguez, J. Tocho, Medidas en Planta de SO2 y NO2 en la Región de La Plata, (Asociación Física Argentina, Argentina, 1999).

Spännare, S.

Svanberg, S.

Tocho, J.

D. Schinca, R. Di Paolo, F. Videla, E. Rodríguez, J. Tocho, Medidas en Planta de SO2 y NO2 en la Región de La Plata, (Asociación Física Argentina, Argentina, 1999).

D. Schinca, J. Tocho, “Development of a DOAS system based on a cross dispersion spectrograph,” in Water, Ground, and Air Pollution Monitoring and Remediation, T. Vo-Dinh, R. Spellicy, eds., Proc. SPIE4199, 77–85 (2001).
[CrossRef]

Videla, F.

D. Schinca, R. Di Paolo, F. Videla, E. Rodríguez, J. Tocho, Medidas en Planta de SO2 y NO2 en la Región de La Plata, (Asociación Física Argentina, Argentina, 1999).

Appl. Opt. (3)

Appl. Spect. (2)

H. Axelsson, A. Eilard, A. Emanuelsson, B. Galle, H. Edner, P. Ragnarson, H. Kloo, “Measurement of aromatic hydrocarbons with the DOAS technique,” Appl. Spect. 49, 1254–1260 (1995).
[CrossRef]

H. Axelsson, H. Edner, B. Galle, P. Ragnarson, M. Rudin, “Differential optical absorption spectroscopy (DOAS) measurements of ozone in the 280–290 nm wavelength region,” Appl. Spect. 44, 1654–1658 (1990).
[CrossRef]

Geophys. Res. Lett. (1)

D. Perner, U. Platt, “Detection of nitrous acid in the atmosphere by differential optical absorption,” Geophys. Res. Lett. 6, 917–920 (1979).
[CrossRef]

J. Geophys. Res. (3)

U. Platt, D. Perner, “Direct measurements of atmospheric CH2O, HNO2, O3, NO2 and SO2 by differential optical absorption in the near UV,” J. Geophys. Res. 85, 7453–7458 (1980).
[CrossRef]

U. Platt, D. Perner, H. Pätz, “Simultaneous measurements of atmospheric CH2O, O3, and NO2 by differential optical absorption,” J. Geophys. Res. 84, 6329–6335 (1979).
[CrossRef]

L. T. Molina, M. J. Molina, “Absolute absorption cross sections of ozone in the 185- to 350-nm wavelength range,” J. Geophys. Res. 91, 14501–14508 (1986).
[CrossRef]

Phys. Med. (1)

P. Bonasoni, F. Evangelisti, G. Giovanelli, F. Ravegnani, “Remote sensing DOAS system for measurements of atmospheric trace gases,” Phys. Med. 9, 304–307 (1993).

Other (7)

U. Platt, D. Perner, “Measurements of atmospheric trace gases by long path differential UV/visible absorption spectroscopy,” in Optical and Laser Remote Sensing Techniques, D. K. Killinger, A. Mooradian, eds., (Springer-Verlag, Berlin, 1983), pp. 97–105.
[CrossRef]

U. Platt, “Differential optical absorption spectroscopy,” in Monitoring by Spectroscopy Techniques, M. Sigrist, ed. (Wiley, New York, 1994), pp. 27–84.

G. Herzberg, Electronic Spectra of Polyatomic Molecules (Van Nostrand Reinhold, New York, 1966).

D. Schinca, J. Tocho, “Development of a DOAS system based on a cross dispersion spectrograph,” in Water, Ground, and Air Pollution Monitoring and Remediation, T. Vo-Dinh, R. Spellicy, eds., Proc. SPIE4199, 77–85 (2001).
[CrossRef]

D. Schinca, R. Di Paolo, F. Videla, E. Rodríguez, J. Tocho, Medidas en Planta de SO2 y NO2 en la Región de La Plata, (Asociación Física Argentina, Argentina, 1999).

E. Lloyd, ed., Handbook of Applicable Mathematics, (Wiley, New York, 1984), Vol. VI, Part A, Chap. 8.

G. Giovanelli, P. Bonasoni, F. Evangelisti, F. Ravegnani, “Ozone ground-based measurements by the GASCOD near UV and visible DOAS system,” in Proceedings of the Quadrennial Ozone Symposium 1992, R. D. Hudson, ed. (Goddard Space Flight Center, Greenbelt, Md., 1992), pp. 707–710.

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Figures (11)

Fig. 1
Fig. 1

Part of the raw spectra (solid curve) with interpolating third-degree polynomial (dashed curve). Inset: Typical raw spectra showing the SO2 absorption modulation in the 300-nm region.

Fig. 2
Fig. 2

Experimental setup.

Fig. 3
Fig. 3

(a) Typical absorbance SO2 spectrum. The labeled areas are used for concentration determination. (b) Differential absorption cross section taken from Ref. 12 convoluted with instrument function.

Fig. 4
Fig. 4

Linear regression corresponding to Eq. (6). Its slope is proportional to the concentration.

Fig. 5
Fig. 5

(a) Differential absorbance curves for different concentration ranges. The numbers in the top right corner are the concentration measured by the electrochemical sensor. (b) Linear regression for the differential absorbance curves of part (a). The confidence interval is shown for each case. The numbers in the top left corner are the retrieved concentration values.

Fig. 6
Fig. 6

Comparison between concentration values measured with electrochemical sensor and those retrieved by the integral method for single-component atmosphere.

Fig. 7
Fig. 7

(a) Differential absorbance curves for low concentration range. (b) Relation between DOAS and electrochemical sensor concentrations for the low concentration range.

Fig. 8
Fig. 8

(a) Differential absorbance spectrum of NO2 in the 295–310 nm range. The number in the top left corner is the concentration measured by the electrochemical sensor. (b) Linear regression for the differential absorbance curve of part (a). The number in the top left corner is the retrieved concentration.

Fig. 9
Fig. 9

Measured total differential absorbance (solid curve) and calculated differential absorbance (dotted curve) by use of the concentration coefficients determined by the integral method for different mixing ratios of SO2 and NO2 (see text). (a) SO2, 105 ppm; NO2, 161 ppm. (b) SO2, 23 ppm; NO2, 216 ppm. (c) SO2, 8 ppm; NO2, 231 ppm.

Fig. 10
Fig. 10

Linear regression of the values calculated applying Eq. (5) to the resultant spectrum (inset) for the mixtures (a) SO2, 105 ppm; NO2, 161 ppm. (b) SO2, 23 ppm; NO2, 216 ppm. (c) SO2, 8 ppm; NO2, 231 ppm.

Fig. 11
Fig. 11

Measured total differential absorbance (solid curve) and calculated differential absorbance (dotted curve) by use of the standard least-squares method for different mixing ratios of SO2 and NO2 (see text). (a) SO2, 105 ppm; NO2, 161 ppm. (b) SO2, 23 ppm; NO2, 216 ppm. (c) SO2, 8 ppm; NO2, 231 ppm.

Equations (11)

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

Iλ=I0λexp-Lcσλ,
c=logI0λIλσλL=AλσλL,
σiλ=σi0λ+σiλ,
Aλ=logI0λ/Iλ=cLσiλ,
λiλf Aλdλ=cL λiλf σλdλ.
i=1K Ai=cL i=1K σi+εK,
L j=1m cjλiλf σjλdλ,
λ0λfabsλdλ-Lc1λ0λf σ1λdλ+c2λ0λf σ2λdλ=ελf,
C=STS-1STA,
si1=λ0λk σ1λdλ, si2=λ0λk σ2λdλ k=1..n,
ai=λ0λkabsλdλ.

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