## Optimal spectral inversion of atmospheric radiometric measurements in the near-UV to near-IR range: A case study

Optics Express, Vol. 10, Issue 1, pp. 70-82 (2002)

http://dx.doi.org/10.1364/OE.10.000070

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### 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.

© Optical Society of America

## 1 Introduction

1. E. Kyrola, E. Shivola, Y. Kotivuori, M. Tikka, and T. Tuomi, “Inverse Theory for OccultationMeasurements: 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 InversionAlgorithm,” J. Geophys. Res. **94**, 8839–8351 (1989). [CrossRef]

5. J. L. Bertaux, G. Megie, T. Widemann, E. Chassefiere, 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]

## 2 Spectral inversion for an ideal occultation radiometer

*h*is the tangent altitude of the measured ray of light and

*τ*(λ) is the slant path optical thickness due to absorption or scattering. It is well known that refractive effects of the Earth’s atmosphere have to be taken into account below altitudes of about 40 km. In particular, the diverging atmospheric lens induces a dilution of the incoming flux (for light sources smaller than the instrument field of view) and a shift in the tangent altitude. The optical path length increase due to ray bending is however very small and, as most of the extinction occurs in the neighbourhood of the tangent point, we can approximate the slant path optical thickness by taking the logarithm of the dilution corrected transmittance at the physical (refracted) tangent altitude.

*μ*m), the true optical thickness (unknown to the experimenter) may be described by:

*τ*

_{N2},

*τ*

_{O3},

*τ*

_{NO2}respectively stand for slant path optical thicknesses of air, ozone and nitrogen dioxide. For a standard atmosphere,

*a*

_{N2},

*a*

_{O3}and

*a*

_{NO2}are set to 1. The individual optical thicknesses will not be translated into trace gas columns and it should be kept in mind that this step could introduce supplementary uncertainties if the absorption or scattering cross sections vary along the line-of-sight.

*τ*

_{A}(λ) is the aerosol optical thickness of which the wavelength dependence is always smooth, due to integration of the Mie cross section over the particle size distribution. In order to simplify the discussion hereafter, we have deliberately removed contributions from water vapour, A and B oxygen bands as well as other minor constituents. In Fig. 1, we have represented the partial slant path optical thicknesses for a tangent altitude of 20 km and from climatological values (

*a*

_{N2}=

*a*

_{O3}=

*a*

_{NO2}= 1). The λ

^{-1}aerosol wavelength dependence of

*τ*

_{A}(λ) in a moderate volcanic situation [6

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]

^{-4}) are clearly mutual competitors both in magnitude and in spectral shape. Considering that the natural smoothness of aerosol optical thickness suggests its representation by a low order (

*n*) polynomial, the spectral inversion consists of retrieving the partial optical thicknesses by using a model

*R*(λ) defined by (we remove the variable h for the sake of readiness):

*τ*

_{i}(λ) ∝ (λ - λ

_{0})

^{i}and λ

_{0}is an arbitrary wavelength reference (the

*x*’s are the retrieved parameters). The experimental data will always be measured at a defined noise level depending on the light source (star, planet or Sun) and the detector intrinsic sensitivity. For the sake of simplicity we will only consider here shot noise, which is proportional to the square root of the signal for a Poisson distribution. If we call

*S*the detector sensitivity (the dynamic range expressed in photons or electrons per wavelength unit), we may write for the error Δ

*T*on transmittance (Eq. 1):

1. E. Kyrola, E. Shivola, Y. Kotivuori, M. Tikka, and T. Tuomi, “Inverse Theory for OccultationMeasurements: 1. Spectral Inversion,” J. Geophys. Res. **98**, 7367–7381 (1993). [CrossRef]

*T*occasionally higher than 1) and saturation regimes (with possible negative

*T*and troubles in computing

*τ*(λ)). These are regions where the information content of the measurement is questionable and problems can be partially cured by appropriate screening and filtering. We will assume that Eq. 5 is valid during the complete occultation even if more accurate approximations could be used.

*M*:

_{1}=0.2

*μ*m and λ

_{2}=1

*μ*m as default values. The minimization of

*M*is obtained by imposing

*x*⃗is the column vector of unknowns and

*C*is the design or covariance matrix:

*y*⃗ contains the respective projections of the experimental

*τ*(λ) onto the basis functions

*τ*

_{i}(λ) as

*C*being a real symmetric matrix, it can be diagonalized by using an orthogonal transformation

*U*:

*C*

^{-1}. Therefore, the expected random error

*e*

_{r}on N

_{2},O

_{3},NO

_{2}is:

*C*

^{-1}matrix and hence by the smallest eigenvalues of the covariance matrix

*C*. A first consequence is that any aerosol that can be expressed as a quasi-linear combination of the spectral dependence of the other constituents will makes

*C*almost singular and will induce huge random errors in the solution. A trivial example could be a λ

^{-4}dependence that would make aerosol and Rayleigh scattering physically undistinguishable by the optical experiment. A second consequence is that increasing the polynomial degree

*n*necessarily implies more and more small eigenvalues in the spectrum of

*C*resulting in enhanced random error. On the other hand, increasing the spectral range [λ

_{1}, λ

_{2}] or the detector sensitivity

*S*will increase the elements of

*C*and ameliorate the random error.

*τ*

_{A}the difference between the true aerosol and a pure polynomial expansion, this difference will project onto all the basis functions, producing a

*δy*⃗

*δx*⃗

*C*(small eigenvalues), these elements will be amplified and mixed. This means that the aerosol error will cause a bias error

*e*

_{b}for all retrieved constituents and this kind of bias has been often disregarded in the literature. For N

_{2}, O

_{3}and NO

_{2}, the relative bias errors are

*δy*⃗ will generally increase. In Fig. 2, we have computed the systematic error produced by different ”true” aerosols generated from

*γ*= 0. Clearly, when

*γ*is varied, within the range of realistic values corresponding to already observed volcanic modes, the systematic error of all constituents evolves considerably. Notice that the ozone errors are always quite small due to the pronounced spectral signature of the ozone cross section.

*τ*

_{A}is to increase the polynomial degree

*n*but this will increase the random error. Hence, we are led to the conclusion that there must exist an optimal

*n*that realizes a trade-off between the random and bias errors. This is illustrated in Fig. 3 where this optimal value is found to be different from constituent to constituent. An important consequence of this analysis is that a global inversion of all constituents together cannot be optimal and that the trade-off should depend on the geophysical objectives (e. g. an optimal accuracy for independent ozone retrievals). This has not been noticed by [1

1. E. Kyrola, E. Shivola, Y. Kotivuori, M. Tikka, and T. Tuomi, “Inverse Theory for OccultationMeasurements: 1. Spectral Inversion,” J. Geophys. Res. **98**, 7367–7381 (1993). [CrossRef]

## 3 Exploring alternative solutions to global spectral inversion

### 3.1 Do microwindows help?

*τ*(λ) which acts as a spectral range cut-off when transmittance gets too weak, avoiding oversensitivity to the UV part. In view of the above-mentioned influence of the constituent non-orthogonality, it is interesting to consider the possibility of selecting only a subrange of pixels to perform the optimal inversion of a particular constituent. A simple way to investigate this possibility is the inclusion of a filter function

*F*(λ) into the merit function as

*F*can be given a simple gaussian dependence in order to simulate a single spectral window:

*c*

_{1}and

*c*

_{2}respectively play the role of center and width of the window. In Fig. 4, we present the isopleths of bias and random errors for the four retrieved constituents when both parameters are varied. As expected, the random error (right column) increases when a smaller window is used (low

*c*

_{2}values) but not uniformly: air and nitrogen dioxide are better retrieved toward the UV range while ozone and aerosol are favored by the center of the spectral range. Interesting patterns emerge from the bias error in the left column. Air retrieval is more effective in the UV-part in opposition to aerosol for which no clear improvement can be obtained with respect to full scale window (

*c*

_{2}=∞). Ozone and nitrogen dioxide exhibit very structured domains. In particular, isolines of zero error bias exist that can remove possible systematic errors caused by the aerosol incomplete description. Indeed, if we respectively call

*e*

_{r}(

*c*

_{1},

*C*

_{2}) and

*e*

_{b}(

*c*

_{1},

*C*

_{2}) the random and bias error as function of the spectral window, a natural strategy could be to minimize

*∊*

_{r}under the constraint

*∊*

_{b}= 0. This can be easily accomplished by constructing the Lagrangian function

*L*as

_{2}a solution at the location (

*c*

_{2}=∞. Although the random error is somewhat larger, the benefit of having a zero bias is evident when data will be averaged or binned in a set of different measurements because the random error will also decrease by a factor equal to the square root of the number of measurements. For only one measurement however, the total error

*γ*in Eqn. 23) and of the polynomial order

*n*. The same procedure can be applied for an optimal ozone retrieval.

*F*(λ) is no longer a simple Gaussian but is allowed to take multiple maxima over the whole spectral range and fully optimized by simulated annealing. For

*n*= 2, a weak amelioration is observed with respect to the Gaussian window, and the optimal windows are mainly centered at locations where some constituents have the same order of magnitude (for instance, there is a maximum around λ = 0.5

*μm*where Rayleigh, aerosol and ozone extinctions have similar contributions to the optical thickness). When using a

*n*= 3 polynomial, the optimal window is the full window (

*F*(λ) = 1) and no more progress can be expected from this approach.

### 3.2 Regularization

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]

*M*

_{1}from Eqn. 24 as:

*n*is. The latter equation consists of the sum of two quadratic forms and leads to a modified linear system similar to Eqn. 8. The relative contribution of the constraint with respect to

*M*is controlled by

*ρ*, whose optimal value can be determined by generalized cross validation [8]. However, it is interesting to investigate the outcome of the constrained inversion for a large range of

*ρ*values. The results are presented in Fig. 5 for aerosol bias, random and total errors when

*n*is varied from 2 to 7. Clearly, when

*n*becomes larger, there is a competition between smoothing (a factor of random error reduction) that increases with

*ρ*and accuracy (that reduces bias error) when

*ρ*is very small. The total error shows a clear minimum when

*n*≥ 4 but the important point is that the value of the total error at this minimum is not better than for

*n*= 2 or

*n*= 3. We deduce that constrained linear inversion does not improve the aerosol retrieval.

*M*

_{2}as:

*χ*is an adjustable parameter in the range [0,1]. Inversion results are presented in Fig. 6. As expected, the use of the higher frequencies (

*χ*≃ 1) increases accuracy and noise sensitivity. Notice, however, the drastic reduction of the bias error on the NO

_{2}retrieval which is the basis of the differential absorption spectroscopy (DOAS) method[9].

### 3.3 Filtering

## 4 Conclusions

*n*in order to diminish the bias error but in doing so we always increase the noise sensitivity. In the near-UV to near-IR wavelength range that we have considered here, the number of aerosol parameters that can be retrieved from radiometric measurements can hardly exceed 4 or 5, depending on the signal to noise ratio of the spectrometer. This should be kept in mind when trying to retrieve bimodal particle size distributions (requiring at least 6 parameters) from the wavelength dependence of the aerosol optical thickness.

## Acknowledgements

## References and links

1. | E. Kyrola, E. Shivola, Y. Kotivuori, M. Tikka, and T. Tuomi, “Inverse Theory for OccultationMeasurements: 1. Spectral Inversion,” J. Geophys. Res. |

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. |

3. | M. King, “Sensitivity of Constrained Linear Inversions to the Selection of the Lagrange Multiplier,” J. Atmos. Sci. |

4. | W. P. Chu, M. P. McCormick, J. Lenoble, C. Brogniez, and P. Pruvost, “SAGE II InversionAlgorithm,” J. Geophys. Res. |

5. | J. L. Bertaux, G. Megie, T. Widemann, E. Chassefiere, R. Pellinen, E. Kyrola, S. Korpela, and P. Simon, “Monitoring of ozone trend by stellar occultations: the GOMOS instrument,” Advances in Space Research |

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. |

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. |

8. | G. H. Golub and C. F. Van Loan, “Matrix Computations,” (The Johns Hopkins University Press1996). |

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 |

**OCIS Codes**

(010.1100) Atmospheric and oceanic optics : Aerosol detection

(010.1280) Atmospheric and oceanic optics : Atmospheric composition

**ToC Category:**

Research Papers

**History**

Original Manuscript: November 30, 2001

Published: January 14, 2002

**Citation**

Didier Fussen, Filip Vanhellemont, and Christine Bingen, "Optimal spectral inversion of atmospheric radiometric measurements in the near-UV to near-IR range: A case study," Opt. Express **10**, 70-82 (2002)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-1-70

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### References

- 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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