Abstract

We present a general analysis of the error budget in the spectral inversion of atmospheric radiometric measurements. By focussing on the case of an occultation experiment, we simplify the problem through a reduced number of absorbers in a linearized formalism. However, our analysis is quite general and applies to many other situations. For a spectrometer having an infinite spectral resolution, we discuss the origin of systematic and random errors. In particular, the difficult case of aerosols is investigated and several inversion techniques are compared. We underline the importance of carefully simulating the spectral inversion as a function of the target constituent to be retrieved, and the required accuracy level.

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References

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  1. E. Kyrola, E. Shivola, Y. Kotivuori, M. Tikka, and T. Tuomi, "Inverse Theory for Occultation Measurements: 1. Spectral Inversion," J. Geophys. Res. 98, 7367-7381 (1993).
    [CrossRef]
  2. D. E. Flittner, B. M. Herman, K. J. Thome, J. M. Simpson, and J. A. Reagan, "Total Ozone and Aerosol Optical Depths Inferred from Radiometric Measurements in the Chappuis Absorption Band," J. Atmos. Sci. 50, 1113-1121 (1993).
    [CrossRef]
  3. M. King, "Sensitivity of Constrained Linear Inversions to the Selection of the Lagrange Multiplier," J. Atmos. Sci. 39, 1356-1369 (1982).
    [CrossRef]
  4. W. P. Chu, M. P. McCormick, J. Lenoble, C. Brogniez, and P. Pruvost, "SAGE II Inversion Algorithm," J. Geophys. Res. 94, 8839-8351 (1989).
    [CrossRef]
  5. J. L. Bertaux, G. Megie, T. Widemann, E. Chasse.ere, R. Pellinen, E. Kyrola, S. Korpela, and P. Simon, "Monitoring of ozone trend by stellar occultations: the GOMOS instrument," Advances in Space Research 11, 237-242 (1991).
    [CrossRef]
  6. D. Fussen, F. Vanhellemont, and C. Bingen, "Evolution of stratospheric aerosols in the post-Pinatubo period measured by the occultation radiometer experiment ORA," Atmos. Env. 35, 5067-5078 (2001).
    [CrossRef]
  7. S. Twomey, "Comparison of Constrained Linear Inversion and an Iterative Nonlinear Algorithm Applied to the Indirect Estimation of Particule Size Distributions," J. Comput. Phys. 18, 188-200 (1975).
    [CrossRef]
  8. G. H. Golub and C. F. Van Loan, "Matrix Computations," (The Johns Hopkins University Press 1996).
  9. U. Platt, "Air monitoring by Spectroscopic Techniques, Chapter 2," (John Wiley and Sons 1994).
  10. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, "Numerical Recipes in FORTRAN, Second Edition," (Cambridge University Press, Cambridge 1992).
  11. M. U. Bromba and H. Ziegler, "Applications Hints for Savitzky-Golay Smoothing Filters," Analytical Chemistry 53, 1583-1586 (1981).
    [CrossRef]

Other

E. Kyrola, E. Shivola, Y. Kotivuori, M. Tikka, and T. Tuomi, "Inverse Theory for Occultation Measurements: 1. Spectral Inversion," J. Geophys. Res. 98, 7367-7381 (1993).
[CrossRef]

D. E. Flittner, B. M. Herman, K. J. Thome, J. M. Simpson, and J. A. Reagan, "Total Ozone and Aerosol Optical Depths Inferred from Radiometric Measurements in the Chappuis Absorption Band," J. Atmos. Sci. 50, 1113-1121 (1993).
[CrossRef]

M. King, "Sensitivity of Constrained Linear Inversions to the Selection of the Lagrange Multiplier," J. Atmos. Sci. 39, 1356-1369 (1982).
[CrossRef]

W. P. Chu, M. P. McCormick, J. Lenoble, C. Brogniez, and P. Pruvost, "SAGE II Inversion Algorithm," J. Geophys. Res. 94, 8839-8351 (1989).
[CrossRef]

J. L. Bertaux, G. Megie, T. Widemann, E. Chasse.ere, R. Pellinen, E. Kyrola, S. Korpela, and P. Simon, "Monitoring of ozone trend by stellar occultations: the GOMOS instrument," Advances in Space Research 11, 237-242 (1991).
[CrossRef]

D. Fussen, F. Vanhellemont, and C. Bingen, "Evolution of stratospheric aerosols in the post-Pinatubo period measured by the occultation radiometer experiment ORA," Atmos. Env. 35, 5067-5078 (2001).
[CrossRef]

S. Twomey, "Comparison of Constrained Linear Inversion and an Iterative Nonlinear Algorithm Applied to the Indirect Estimation of Particule Size Distributions," J. Comput. Phys. 18, 188-200 (1975).
[CrossRef]

G. H. Golub and C. F. Van Loan, "Matrix Computations," (The Johns Hopkins University Press 1996).

U. Platt, "Air monitoring by Spectroscopic Techniques, Chapter 2," (John Wiley and Sons 1994).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, "Numerical Recipes in FORTRAN, Second Edition," (Cambridge University Press, Cambridge 1992).

M. U. Bromba and H. Ziegler, "Applications Hints for Savitzky-Golay Smoothing Filters," Analytical Chemistry 53, 1583-1586 (1981).
[CrossRef]

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Figures (7)

Fig. 1.
Fig. 1.

Slant path optical thicknesses (using GOMOS cross sections) at a tangent altitude of 20 km for climatological values. Crosses: air, diamonds: ozone, circles: nitrogen dioxide, Plus signs: aerosol, full bold line: total.

Fig. 2.
Fig. 2.

Top: different possible aerosol wavelength dependences as γ is varied from -0.1 to 0.1. Bottom: Relative systematic errors on retrieved optical thicknesses for n = 2. Crosses: air, circles: ozone, plus signs: nitrogen dioxide, full bold line: aerosol

Fig. 3.
Fig. 3.

Evolution of bias and random errors when the aerosol polynomial degree n is varied. Full lines refer to bias error and dashed lines refer to random error when the detector sensitivity is S=100 (circles), S=1000 (diamonds) and S=10000 (squares).

Fig. 4.
Fig. 4.

Isopleths for bias (left column) and random (right column) errors when c 1 and c 2 are varied. Dashed white isopleths represent zero bias error and full white circles correspond to minimal random error on these isopleths.

Fig. 5.
Fig. 5.

Evolution of bias (dashed), random (dot-dashed) and total (full) errors versus the regularization parameter ρ for different aerosol polynomial order n.

Fig. 6.
Fig. 6.

Evolution of bias (dashed), random (dot-dashed) and total (full) errors versus the parameter χ when using the merit function M 2 (Eqn. 31)

Fig. 7.
Fig. 7.

Isopleths of total error et when the Savitzsky-Golay number of nodes nSG and approximating polynomial degree mSG are varied. Notice the presence of a clear minimum.

Equations (31)

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T ( h , λ ) = exp ( τ ( h , λ ) )
τ ( λ ) = a N 2 τ N 2 ( λ ) + a O 3 τ O 3 ( λ ) + a N O 2 τ NO 2 ( λ ) + τ A ( λ )
R ( λ ) = x N 2 τ N 2 ( λ ) + x O 3 τ O 3 ( λ ) + x N O 2 τ NO 2 ( λ ) + [ x 0 τ 0 + x 1 τ 1 ( λ ) + x n τ n ( λ ) ]
Δ T = T S
Δ τ = 1 TS
M ( x N 2 , x n ) = λ 1 λ 2 ( R ( λ ) τ ( λ ) Δ τ ( λ ) ) 2
M x N 2 = M x O 3 = = M x n = 0
C x = y
C = ( τ N 2 τ N 2 τ N 2 τ O 3 τ N 2 τ NO 2 τ N 2 τ 0 τ N 2 τ n τ O 3 τ N 2 τ O 3 τ O 3 τ O 3 τ NO 3 τ O 2 τ 0 τ O 3 τ n τ NO 2 τ N 2 τ NO 2 τ O 3 τ NO 2 τ NO 2 τ NO 2 τ 0 τ NO 2 τ n τ 0 τ N 2 τ 0 τ O 3 τ 0 τ NO 2 τ 0 τ 0 τ 0 τ n τ n τ N 2 τ n τ O 3 τ n τ NO 2 τ n τ 0 τ n τ n )
τ i τ j = λ 1 λ 2 τ i ( λ ) τ j ( λ ) Δτ ( λ ) 2 d λ = λ 1 λ 2 τ i ( λ ) τ j ( λ ) T ( λ ) S d λ
y T = ( τ τ N 2 τ τ O 3 τ τ NO 2 τ τ 0 τ τ n )
C = U Λ U T
x = C 1 y = U Λ 1 U T y
e r i = ( C 1 ) ii { i = N 2 , O 3 , NO 2 }
e r A ( λ ) = ( τ 0 τ n ( λ ) ) ( C 1 ) i = 0 . . n , i = 0 . . n ( τ 0 τ n ( λ ) ) T
r i = e r i a i { i = N 2 , O 3 , NO 2 }
r A = ( λ 1 λ 2 ( e r A ( λ ) τ A ( λ ) ) 2 d λ λ 2 λ 1 ) 1 2
δ y T = ( Δ τ A τ N 2 Δ τ A τ O 3 Δ τ A τ NO 2 Δ τ A τ 0 Δ τ A τ n )
δ x = C 1 δ y
b i = δ x i a i { i = N 2 , O 3 , NO 2 }
e b A ( λ ) = τ A ( λ ) x T ( i = 0 . . n ) * ( τ 0 τ n ( λ ) ) T
b A = ( λ 1 λ 2 ( e b A ( λ ) τ A ( λ ) ) 2 d λ λ 2 λ 1 ) 1 2
τ A ( λ ) = exp ( ln ( λ 1 ) γ λ 2 )
M ( x N 2 , x n ) = λ 1 λ 2 ( R ( λ ) τ ( λ ) Δ τ ( λ ) ) 2 F ( λ )
F ( λ ) = exp ( ( λ c 1 c 2 ) 2 )
L = r ( c 1 , c 2 ) + ξ b ( c 1 , c 2 )
r c 1 + ξ b c 1 = 0
r c 2 + ξ b c 2 = 0
b ( c 1 , c 2 ) = 0
M 1 ( x N 2 , x n ) = M ( x N 2 , x n ) + ρ λ 1 λ 2 ( d 2 τ A ( λ ) d λ 2 ) 2
M 2 ( x N 2 , x n ) = ( 1 x ) λ 1 λ 2 ( R ( λ ) τ ( λ ) Δ τ ( λ ) ) 2 + x λ 1 λ 2 ( R′ ( λ ) τ′ ( λ ) Δ τ′ ( λ ) ) 2

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