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Optics Express

  • Editor: Michael Duncan
  • Vol. 10, Iss. 1 — Jan. 14, 2002
  • pp: 70–82
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Optimal spectral inversion of atmospheric radiometric measurements in the near-UV to near-IR range: A case study

Didier Fussen, Filip Vanhellemont, and Christine Bingen  »View Author Affiliations


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

In remote sensing experiments, atmospheric constituents are retrieved from measurements of a spatially integrated optical property. Remote probing can be carried out either in a passive or in an active way and using many spectral regions. From the near-ultraviolet (UV) to the near-infrared (IR) domain of the spectrum, the total column amount of a specific absorbing gas can be inferred from ground-based measurements combined with the Langley method. On the other hand, space-borne instruments using the occultation technique offer three major advantages: they possess a natural self-calibration property, they allow an accurate retrieval of vertical profiles of number density, with a possible quasi-global coverage.

However, all these techniques face a common problem: measured vertical or slant-path optical thicknesses generally result, at any wavelength, from additive light extinctions (absorption or scattering) by different atmospheric constituents. The optical thickness separation into its different contributions is often called the spectral inversion as opposed to the vertical inversion where the optical thickness is decomposed into the contribution of individual atmospheric layers.

In the literature, many methods have been proposed [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]

],[2

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

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

],[4

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]

] to solve this problem in a pragmatic way. Although these retrieval schemes turned out to be reasonably successful, they are often obscure in their genesis and do not say very much about the fundamental limitations of the measurement. Even more, there are few assertions about the possible superiority of a particular method and the error budget is mostly discussed only with respect to random errors.

The purpose of the present paper is to discuss the problems associated with the spectral inversion of an occultation experiment. For instance, the GOMOS experiment [5

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]

](on board Envisat) is a clear example that illustrates the difficulties encountered although the analysis can also be easily extended to other cases, including ground-based experiments or even reflected solar radiation in the Earth’s atmosphere. Therefore we will first reduce the problem to a simplified formalism by means of reasonable approximations. All essential features that will be explained hereafter are necessarily present in more sophisticated and (possibly) exact inversion schemes. In section 2, we will discuss the issue of bias and random errors and we will suggest a criterion for the optimal description of the aerosol optical thickness. Finally, in section 3, we will extend the analysis to the problem of microwindows and filtering. No definitive optimal solution will be proposed to the reader. Instead, we will underline the importance of carefully simulating the spectral inversion as a function of the target constituent to be retrieved, and the required accuracy level.

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.

2 Spectral inversion for an ideal occultation radiometer

According to Beer-Lambert law, the atmospheric transmittance measured by an ideal monochromatic occultation radiometer working at wavelength λ can be written:

T(h,λ)=exp(τ(h,λ))
(1)

where 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.

If we restrict our analysis to the near UV - near IR range (λ ∈ [0.2,1.0] μm), the true optical thickness (unknown to the experimenter) may be described by:

τ(λ)=aN2τN2(λ)+aO3τO3(λ)+aNO2τNO2(λ)+τA(λ)
(2)

where 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]

] is considered as our standard case. It is clear that ozone is characterized by two strong spectral signatures in the Huggins (λ < 340 nm) and Chappuis (λ ≃ 600 nm) bands, and this constituent should normally be the easiest to retrieve. Nitrogen dioxide is not very absorbing but possesses a useful differential structure superimposed on a broad continuum centered at about λ ≃ 400 nm. Finally, aerosol and Rayleigh scattering (∝ λ-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):

R(λ)=xN2τN2(λ)+xO3τO3(λ)+xNO2τNO2(λ)+[x0τ0+x1τ1(λ)+xnτn(λ)]
(3)

where τi (λ) ∝ (λ - λ0)i and λ0 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 ΔT on transmittance (Eq. 1):

ΔT=TS
(4)

which leads, by first order propagation of errors, to

Δτ=1TS
(5)

Here we fully agree with [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]

] that the true a priori distribution of the model parameters is only gaussian in a first approximation, which may cause difficulties in low absorption (when noise makes 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.

It is useful to consider the problem for an infinite resolution spectrometer because it is the limiting case of modern instruments where spectral resolutions of 1 nanometer can be achieved nowadays. Therefore the matching between the modelled and measured optical thicknesses will be evaluated by means of the following merit function M:

M(xN2,xn)=λ1λ2(R(λ)τ(λ)Δτ(λ))2
(6)

with λ1=0.2 μm and λ2 =1 μm as default values. The minimization of M is obtained by imposing

MxN2=MxO3==Mxn=0
(7)

which leads to the linear system

Cx=y
(8)

where x⃗is the column vector of unknowns and C is the design or covariance matrix:

C=(τN2τN2τN2τO3τN2τNO2τN2τ0τN2τnτO3τN2τO3τO3τO3τNO3τO2τ0τO3τnτNO2τN2τNO2τO3τNO2τNO2τNO2τ0τNO2τnτ0τN2τ0τO3τ0τNO2τ0τ0τ0τnτnτN2τnτO3τnτNO2τnτ0τnτn)
(9)

whose elements are the inner products:

τiτj=λ1λ2τi(λ)τj(λ)Δτ(λ)2dλ=λ1λ2τi(λ)τj(λ)T(λ)Sdλ
(10)

The column vector y⃗ contains the respective projections of the experimental τ(λ) onto the basis functions τi (λ) as

yT=(ττN2ττO3ττNO2ττ0ττn)
(11)

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

C=UΛUT
(12)

where Λ is the diagonal matrix of eigenvalues and the solution of the spectral inversion reads

x=C1y=UΛ1UTy
(13)

whereas the covariance matrix of the solution is simply C -1. Therefore, the expected random error er on N2,O3,NO2 is:

eri=(C1)ii{i=N2,O3,NO2}
(14)

The expected random error error on the aerosol optical thickness is

erA(λ)=(τ0τn(λ))(C1)i=0..n,i=0..n(τ0τn(λ))T
(15)

The relative random errors are defined by:

ri=eriai{i=N2,O3,NO2}
(16)

and

rA=(λ1λ2(erA(λ)τA(λ))2dλλ2λ1)12
(17)

Apart from random errors, the spectral inversion may also suffer from systematic or bias errors. Indeed, if the true unknown aerosol is not completely described by a polynomial expansion, there will also subsist an incompleteness error which cannot be cured by a statistics of a large number of inversions as it is possible for random errors. If we call Δτ A the difference between the true aerosol and a pure polynomial expansion, this difference will project onto all the basis functions, producing a δy

δyT=(ΔτAτN2ΔτAτO3ΔτAτNO2ΔτAτ0ΔτAτn)
(18)

which, in turn, will induce a positive or negative bias δx

δx=C1δy
(19)

It is clear that all elements have, a priori, non-zero values. Furthermore, depending on the ill-conditioning of C (small eigenvalues), these elements will be amplified and mixed. This means that the aerosol error will cause a bias error eb for all retrieved constituents and this kind of bias has been often disregarded in the literature. For N2, O3 and NO2, the relative bias errors are

bi=δxiai{i=N2,O3,NO2}
(20)

and for aerosols, we have

ebA(λ)=τA(λ)xT(i=0..n)*(τ0τn(λ))T
(21)

and

bA=(λ1λ2(ebA(λ)τA(λ))2dλλ2λ1)12
(22)

In particular, this can lead to the naive idea that increasing the spectral range may surely ameliorate the spectral inversion. Mostly, the contrary will occur because the elements of δy⃗ will generally increase. In Fig. 2, we have computed the systematic error produced by different ”true” aerosols generated from

τA(λ)=exp(ln(λ1)γλ2)
(23)

which reduces to the default aerosol for 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.

A possible way to decrease Δτ 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]

] because they performed their spectral inversion by assuming that the aerosol wavelength dependence is known a priori, which is not the case in reality.

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

3 Exploring alternative solutions to global spectral inversion

3.1 Do microwindows help?

Looking at Eqn. 6, it is worth investigating the role played by Δτ(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

M(xN2,xn)=λ1λ2(R(λ)τ(λ)Δτ(λ))2F(λ)
(24)

and, for instance, F can be given a simple gaussian dependence in order to simulate a single spectral window:

F(λ)=exp((λc1c2)2)
(25)

where 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 er (c 1,C 2) and eb (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

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).
L=r(c1,c2)+ξb(c1,c2)
(26)

and numerically solving the nonlinear system of equations

rc1+ξbc1=0
(27)
rc2+ξbc2=0
(28)
b(c1,c2)=0
(29)

For example, one gets for NO2 a solution at the location (c1*, c2*) = (0.450,0.148) where (b*,r*) = (0,49.2) [%] instead of (12.5,30.2) [%] when 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 t=b2+r2 may be the most important quantity to consider. It is worth mentioning that the parameters of the optimal spectral windows were found to be quasi-independent of the aerosol spectral shape (the value of γ in Eqn. 23) and of the polynomial order n. The same procedure can be applied for an optimal ozone retrieval.

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.

In order to ameliorate the aerosol retrieval, we have also implemented a multi-window retrieval, where 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 = 0.5μm 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.

Fig. 5. Evolution of bias (dashed), random (dot-dashed) and total (full) errors versus the regularization parameter ρ for different aerosol polynomial order n.

3.2 Regularization

A possible solution to increase the accuracy of the inversion, with simultaneous control of the sensitivity to random errors, could be achieved by a constrained linear inversion [7

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]

]. In this regularization technique, we construct a more elaborate merit function M 1 from Eqn. 24 as:

M1(xN2,xn)=M(xN2,xn)+ρλ1λ2(d2τA(λ)dλ2)2
(30)

where the second term of the left-hand side constrains the aerosol optical thickness to be smooth whatever the polynomial order 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

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

]. 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.

An other approach is to consider the possibility of exploiting the low and high frequency components of the measured optical thickness, i.e. we construct the merit function M 2 as:

M2(xN2,xn)=(1x)λ1λ2(R(λ)τ(λ)Δτ(λ))2+xλ1λ2(R′(λ)τ′(λ)Δτ′(λ))2
(31)

where the symbol prime refers to the first derivative (in practice, the differentiation is performed numerically) with respect to wavelength and 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 NO2 retrieval which is the basis of the differential absorption spectroscopy (DOAS) method[9

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

].

3.3 Filtering

On the other hand, we know that the total atmospheric optical thickness is a relatively smooth function of wavelength on which measurement noise is superimposed.

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)

However, it is important to realize that, until now, we have investigated the inversion error budget by using the true aerosol optical thickness and comparing it to the retrieved value. In the real world, it would then be necessary to have a good approximation of the solution to estimate, via a table look-up procedure, the optimal filter parameters.

4 Conclusions

This work has proven that the optimal retrieval of a particular atmospheric constituent may require a specific strategy instead of a global inversion.

For aerosols, we conclude the above-mentioned approaches by the following consideration: the retrieval faces an apparently untractable problem: it is necessary to increase 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.

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.

Acknowledgements

This work was partly performed within projects ”SADE (Prodex 6)” and “Measurement, understanding and climatology of stratospheric aerosols (grant MO/35/004)” funded by the SSTC/DWTC service of the Belgian Government.

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. 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]

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 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 53, 1583–1586 (1981). [CrossRef]

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

  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]

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