Abstract

The design of spectroscopic measurements of the atmosphere with the limb-scanning technique for the retrieval of constituent altitude profiles requires choosing instrumental, observational, and retrieval parameters. An approach to this problem is discussed, and the mathematical tools that make it possible to study the trade-off between the two conflicting requirements of optimum vertical resolution and small error of the profile are derived. As a first illustrative application, implementation of the mathematical tools in the design of measurements to be carried out from a satellite platform is shown; a set of parameters that provide a satisfactory compromise between the vertical resolution and the uncertainties of the retrieved profile has been identified. The mathematical model discussed can simulate the results obtained with retrieval techniques that are based on the global inversion of the kernel that relates the observations and the unknown profile. As a second application, a comparison between the a priori estimate of the uncertainties provided by the mathematical tools and the results of the global analysis of data collected with a balloon-borne experiment is shown.

© 1994 Optical Society of America

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References

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  1. M. Carlotti, “Global-fit approach to the analysis of limb-scanning atmospheric measurements,” Appl. Opt. 27, 3250–3254(1988).
    [CrossRef] [PubMed]
  2. W. Menke, Geophysical Data Analysis: Discrete Inverse Theory (Academic, San Diego, Calif., 1984).
  3. C. D. Rodgers, “Characterization and error analysis of profiles retrieved from remote sounding measurements,” J. Geophys. Res. 95, 5587–5595 (1990).
    [CrossRef]
  4. B. Carli, M. Carlotti, “Far-infrared and microwave spectroscopy of the Earth’s atmosphere,” in The Spectroscopy of the Earth’s Atmosphere and Interstellar Medium, K. N. Rao, A. Weber, eds. (Academic, San Diego, Calif., 1992), pp. 1–95.
  5. G. E. Backus, J. F. Gilbert, “Uniqueness in the inversion of inaccurate gross Earth data,” Philos. Trans. R. Soc. London Ser. A 266, 123–192 (1970).
    [CrossRef]
  6. R. E. Kalman, “Algebraic aspects of the generalized inverse of a rectangular matrix,” in Proceedings of Advanced Seminar on Generalized Inverse and Applications, M. Z. Nashed, ed. (Academic, San Diego, Calif., 1976), p. 111.
  7. M. Carlotti, A. Barbis, B. Carli, “Stratospheric ozone vertical distribution from far-infrared balloon spectra and statistical analysis of the errors,” J. Geophys. Res. 94, 16365–16371 (1989).
    [CrossRef]

1990 (1)

C. D. Rodgers, “Characterization and error analysis of profiles retrieved from remote sounding measurements,” J. Geophys. Res. 95, 5587–5595 (1990).
[CrossRef]

1989 (1)

M. Carlotti, A. Barbis, B. Carli, “Stratospheric ozone vertical distribution from far-infrared balloon spectra and statistical analysis of the errors,” J. Geophys. Res. 94, 16365–16371 (1989).
[CrossRef]

1988 (1)

1970 (1)

G. E. Backus, J. F. Gilbert, “Uniqueness in the inversion of inaccurate gross Earth data,” Philos. Trans. R. Soc. London Ser. A 266, 123–192 (1970).
[CrossRef]

Backus, G. E.

G. E. Backus, J. F. Gilbert, “Uniqueness in the inversion of inaccurate gross Earth data,” Philos. Trans. R. Soc. London Ser. A 266, 123–192 (1970).
[CrossRef]

Barbis, A.

M. Carlotti, A. Barbis, B. Carli, “Stratospheric ozone vertical distribution from far-infrared balloon spectra and statistical analysis of the errors,” J. Geophys. Res. 94, 16365–16371 (1989).
[CrossRef]

Carli, B.

M. Carlotti, A. Barbis, B. Carli, “Stratospheric ozone vertical distribution from far-infrared balloon spectra and statistical analysis of the errors,” J. Geophys. Res. 94, 16365–16371 (1989).
[CrossRef]

B. Carli, M. Carlotti, “Far-infrared and microwave spectroscopy of the Earth’s atmosphere,” in The Spectroscopy of the Earth’s Atmosphere and Interstellar Medium, K. N. Rao, A. Weber, eds. (Academic, San Diego, Calif., 1992), pp. 1–95.

Carlotti, M.

M. Carlotti, A. Barbis, B. Carli, “Stratospheric ozone vertical distribution from far-infrared balloon spectra and statistical analysis of the errors,” J. Geophys. Res. 94, 16365–16371 (1989).
[CrossRef]

M. Carlotti, “Global-fit approach to the analysis of limb-scanning atmospheric measurements,” Appl. Opt. 27, 3250–3254(1988).
[CrossRef] [PubMed]

B. Carli, M. Carlotti, “Far-infrared and microwave spectroscopy of the Earth’s atmosphere,” in The Spectroscopy of the Earth’s Atmosphere and Interstellar Medium, K. N. Rao, A. Weber, eds. (Academic, San Diego, Calif., 1992), pp. 1–95.

Gilbert, J. F.

G. E. Backus, J. F. Gilbert, “Uniqueness in the inversion of inaccurate gross Earth data,” Philos. Trans. R. Soc. London Ser. A 266, 123–192 (1970).
[CrossRef]

Kalman, R. E.

R. E. Kalman, “Algebraic aspects of the generalized inverse of a rectangular matrix,” in Proceedings of Advanced Seminar on Generalized Inverse and Applications, M. Z. Nashed, ed. (Academic, San Diego, Calif., 1976), p. 111.

Menke, W.

W. Menke, Geophysical Data Analysis: Discrete Inverse Theory (Academic, San Diego, Calif., 1984).

Rodgers, C. D.

C. D. Rodgers, “Characterization and error analysis of profiles retrieved from remote sounding measurements,” J. Geophys. Res. 95, 5587–5595 (1990).
[CrossRef]

Appl. Opt. (1)

J. Geophys. Res. (2)

M. Carlotti, A. Barbis, B. Carli, “Stratospheric ozone vertical distribution from far-infrared balloon spectra and statistical analysis of the errors,” J. Geophys. Res. 94, 16365–16371 (1989).
[CrossRef]

C. D. Rodgers, “Characterization and error analysis of profiles retrieved from remote sounding measurements,” J. Geophys. Res. 95, 5587–5595 (1990).
[CrossRef]

Philos. Trans. R. Soc. London Ser. A (1)

G. E. Backus, J. F. Gilbert, “Uniqueness in the inversion of inaccurate gross Earth data,” Philos. Trans. R. Soc. London Ser. A 266, 123–192 (1970).
[CrossRef]

Other (3)

R. E. Kalman, “Algebraic aspects of the generalized inverse of a rectangular matrix,” in Proceedings of Advanced Seminar on Generalized Inverse and Applications, M. Z. Nashed, ed. (Academic, San Diego, Calif., 1976), p. 111.

B. Carli, M. Carlotti, “Far-infrared and microwave spectroscopy of the Earth’s atmosphere,” in The Spectroscopy of the Earth’s Atmosphere and Interstellar Medium, K. N. Rao, A. Weber, eds. (Academic, San Diego, Calif., 1992), pp. 1–95.

W. Menke, Geophysical Data Analysis: Discrete Inverse Theory (Academic, San Diego, Calif., 1984).

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Figures (13)

Fig. 1
Fig. 1

Example of three vectors that make a base of the space f used in the case of the layer model defined by the four altitudes h1, h2, h3, and h4.

Fig. 2
Fig. 2

Sensitivity functions of OH limb-scanning measurements for 55 observation geometries, in the case of infinitesimal spectral resolution and field of view, at the frequency that provides maximum amplitude. P/P0 is the fractional pressure relative to sea level. The curves are plotted down to a minimum altitude below which the sensitivity drops to zero.

Fig. 3
Fig. 3

Sensitivity functions of OH limb-scanning measurements for the same observation geometries of Fig. 2, in the case of 0.004 cm−1 spectral resolution and infinitesimal field of view, at the frequency that provides maximum amplitude.

Fig. 4
Fig. 4

Effect of the finite field of view. The thin curves are a selection of six sensitivity functions from Fig. 3, plotted with a factor of 3 of amplification in the signal scale. The thick curves are the functions at the same values of (ν, h, θ) in the case of a field of view of 0.06°.

Fig. 5
Fig. 5

Overall set of the sensitivity functions of an OH limb-scanning measurement in the case of finite spectral resolution and field of view. Three frequencies, displaced by 0.0, 0.004, and 0.008 cm−1 from the line center, respectively, are reported for each of the observation geometries of Fig. 2. The sensitivity scale has a factor of 10 of amplification with respect to Fig. 2.

Fig. 6
Fig. 6

Three rows of the transfer matrix in the case of the inversion of ideal OH measurements made with infinitesimal spectral resolution and field of view. The rows correspond at altitudes that are (from the bottom) approximately 18, 55, and 92 km, respectively.

Fig. 7
Fig. 7

MTF’s corresponding to the rows of the transfer matrix reported in Fig. 6.

Fig. 8
Fig. 8

Three rows of the transfer matrix, at the same altitudes as the rows of the transfer matrix reported in Fig. 6, in the case of the inversion of OH measurements made with 0.004 cm−1 spectral resolution and a field of view of 0.06°.

Fig. 9
Fig. 9

MTF’s corresponding to the rows of the transfer matrix reported in Fig. 8.

Fig. 10
Fig. 10

Percent error values of the retrieved OH profile in the case of measurements made with 0.004 cm−1 spectral resolution, a field of view of 0.06° and σ(%) = 0.5% for a profile segmentation with 1.5-km layers.

Fig. 11
Fig. 11

Percent error values of the retrieved OH profile for measurements, as in Fig. 10, and a profile segmentation defined by layers broadened with respect to Fig. 10. An amplification factor of 20 is applied in they-axis scale with respect to Fig. 10.

Fig. 12
Fig. 12

Vertical resolution of the retrieved OH profile for the profile segmentations of Fig. 10 (lower curve) and Fig. 11 (upper curve), respectively.

Fig. 13
Fig. 13

Percent error values of the retrieved OH profile in the case in which the same profile segmentation is adopted as in Fig. 11 but the sensitivity functions that peak at altitudes outside the altitude interval 27–89 km are removed from the matrix K.

Tables (2)

Tables Icon

Table 1 Predicted and Calculated Errors (%) for Three Profile Segmentations

Tables Icon

Table 2 Relevant Parameters and Computing Time for the Calculation of the Sensitivity Functions

Equations (30)

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S ( x , q z ) = S ( x , q ˜ z ) + 0 [ S ( x , q z ) ln ( q z ) ] q ˜ z [ ln ( q z ) ln ( q ˜ z ) ] d z .
N ( x ) = 0 K ( x , z ) y z d z ,
N ( x ) = S ( x , q z ) S ( x , q ˜ z ) ,
K ( x , z ) = [ S ( x , q z ) ln ( q z ) ] q ˜ z ,
y z = [ ln ( q z ) ln ( q ˜ z ) ] .
n = K y ,
y ˆ = D n .
y ˆ = D K y = T y ,
T = D K
( MTF ) i = 1 [ t i ] ,
[ cov y ˆ ] = σ 2 [ D D T ] ,
V = D D T
Δ y ˆ i = σ ( ν i i ) 1 / 2 .
Δ y ˆ i = Δ [ ln ( q i ) ln ( q ˜ i ) ] = Δ q i q i + Δ q ˜ i q ˜ i ,
Δ y ˆ i = Δ q i q i .
Δ q i = σ q i ( ν i i ) 1 / 2 .
c i j = ν i j ( ν i i ν j j ) 1 / 2 .
K # = K T ( K K T ) 1 .
K @ = H T ( H K T K H T ) 1 H K T ,
Δ q i ( % ) = σ ( % ) ( ν i i ) 1 / 2 ,
S ( ν ) = + S ( ν ) I ( ν ν ) d ν + I ( ν ν ) d ν ,
S ( θ ) = π/ 2 + π/ 2 S ( θ ) A ( θ θ ) d θ π/ 2 + π/ 2 A ( θ θ ) d θ ,
A X = X A = I ,
( i ) A X A = A , ( ii ) X A X = X , ( iii ) A X = X T A T , ( iv ) XA = A T X T ,
A # = A T ( B T A C T ) 1 B T ,
A # = ( A T A ) 1 A T ,
A # = A T ( A A T ) 1 .
( i ) A X A = A , ( ii ) X A X = X .
A @ = L T ( M T A L T ) 1 M T ,
det ( L T C ) 0 , det ( M T B ) 0 ,

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