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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 9 — May. 6, 2013
  • pp: 11465–11474
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On the choice of retrieval variables in the inversion of remotely sensed atmospheric measurements

Marco Ridolfi and Luca Sgheri  »View Author Affiliations


Optics Express, Vol. 21, Issue 9, pp. 11465-11474 (2013)
http://dx.doi.org/10.1364/OE.21.011465


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Abstract

In this paper we introduce new variables that can be used to retrieve the atmospheric continuum emission in the inversion of remote sensing measurements. This modification tackles the so-called sloppy model problem. We test this approach on an extensive set of real measurements from the Michelson Interferometer for Passive Atmospheric Sounding. The newly introduced variables permit to achieve a more stable inversion and a smaller value of the minimum of the cost function.

© 2013 OSA

1. Introduction

For the retrieval of atmospheric constituents and state variables such as pressure and temperature from remote sensing spectroscopic measurements it is necessary to model the radiative transfer through an inhomogeneous medium. The inhomogeneouos atmosphere is usually discretized in small elements which are assumed to be homogeneous. The law governing absorption in an homogeneous medium is the Beer-Lambert law
ddxI(x)=KσI(x),
(1)
where I(x) is the intensity at spatial coordinate x, and Kσ is a frequency-dependent constant. Equation (1) has the general solution I(x) = cexp(−Kσx). As a consequence the radiance reaching the instrument is modeled as a linear combination of exponential terms. Disregarding scattering, which is justified in the mid-to-far infrared spectral region in clear sky conditions, the radiance S(σ) at frequency σ reaching the instrument for a fixed observation geometry is described by the radiative transfer equation [1

1. T. von Clarmann, M. Hpfner, B. Funke, M. López-Puertas, A. Dudhia, V. Jay, F. Schreier, M. Ridolfi, S. Ceccherini, B. J. Kerridge, J. Reburn, R. Siddans, and J.-M. Flaud, “Modelling of atmospheric mid-infrared radiative transfer: The AMIL2DA algorithm intercomparison experiment,” J. Quant. Spectrosc. Radiat. Transfer 78(3–4), 381–407 (2003) [CrossRef] .

]
S(σ)=B0(σ)τ(σ,0,sobs)+0sobsB(σ,T(s))sτ(σ,s,sobs)ds.
(2)
Here s is the coordinate along the line of sight, sobs is the coordinate of the observing instrument, τ(σ, s1, s2) is the spectral transmittance along the line of sight from position s1 to position s2, B0(σ) is the background radiance and B(σ, T(s)) is the source function which, under the assumption of local thermodynamic equilibrium, becomes the Planck function, which depends on the kinetic temperature T(s) and the frequency σ. In the limb scanning geometry, the term B0(σ) is assumed to be zero. A common choice is to discretize the atmosphere with a series of spherical homogeneous layers. In this case, let l = 1,...,N be the layers crossed by a given line of sight, and ple and Tle the equivalent pressure and temperature of layer l[2

2. A. R. Curtis, “Discussion of a statistical model for water vapor absorption,” Q. J. R. Meteorol. Soc. 78, 638–640 (1952) [CrossRef] .

, 3

3. W. L. Godson, “The evaluation of infrared radiative fluxes due to atmospheric water vapour,” Q. J. R. Meteorol. Soc. 79, 367–379 (1953) [CrossRef] .

]. Equation (2) may then be rewritten as
S(σ)=l=1NB(σ,Tle)(1τσ,l)j=1l1τσ,j,
(3)
where τσ,l is the transmittance through layer l at frequency σ. The transmittance in the layer l has the following expression [4

4. M. Ridolfi, B. Carli, M. Carlotti, T. von Clarmann, B. M. Dinelli, A. Dudhia, J.-M. Flaud, M. Höpfner, P. E. Morris, P. Raspollini, G. Stiller, and R. J. Wells, “Optimized forward model and retrieval scheme for MIPAS near-real-time data processing,” Appl. Opt. 39, 1323–1340 (2000) [CrossRef] .

, 5

5. P. Raspollini, C. Belotti, A. Burgess, B. Carli, M. Carlotti, S. Ceccherini, B. M. Dinelli, A. Dudhia, J.-M. Flaud, B. Funke, M. Höpfner, M. López-Puertas, V. Payne, C. Piccolo, J. J. Remedios, M. Ridolfi, and R. Spang, “MI-PAS level 2 operational analysis,” Atmos. Chem. Phys. 6, 5605–5630 (2006) [CrossRef] .

]
τσ,l=exp(mkσ,m(ple,Tle)cl,m)=mexp(kσ,m(ple,Tle)cl,m)mτσ,l,m,
(4)
where m is the gas index, kσ,m is the cross-section of that gas at frequency σ, which depends only on ple and Tle. Finally cl,m is the gas column in the layer along the line of sight, which depends on the gas Volume Mixing Ratio (VMR) xm(z) in the following way:
cl,m=lxm(z)η(z)dsdz(z)dz,
(5)
where z is the altitude, η(z) is the number density and ds/dz is the linear element along the line of sight. Let zi be the retrieval grid. We may use the values xi,m = xm(zi) as retrieval variables, and model the VMR with a piecewise linear curve with nodes zi. It follows that cl,m is a linear function of xi,m so that the entire spectrum will be a linear combination of exponential terms ej of the form exp(∑iI(j)dixi,m) for some constants di, and the sum is extended on subsets I(j) of indexes.

The inversion of the measured spectra is obtained by optimizing the values of the retrieval variables. In the weighted least-squares approach the cost function χ2 is defined as 2 norm of the differences between observed and simulated radiances (the so-called residuals) weighted with the inverse of the error covariance matrix of the measurements.

While the xi,m have a physical meaning, their selection as retrieval variables makes the forward model a sloppy model, as described in [6

6. M. K. Transtrum, B. B. Machta, and J. P. Sethna, “Geometry of nonlinear least squares with applications to sloppy models and optimization,” Phys. Rev. E 83, 036701 (2011) [CrossRef] .

]. The main drawback of such a model is that the model manifold includes wide regions where the residuals of the reconstructed spectrum are large, and long, narrow and tortuous valleys where the residuals are small. As a consequence many different sets of retrieval variables may lead to similar χ2 values close to the minimum.

2. Theory

2.1. The continuum variables

Conventionally the cross-sections kσ,m appearing in Eq. (4) are computed including only the contributions of transitions not farther than 25 cm−1 from σ[4

4. M. Ridolfi, B. Carli, M. Carlotti, T. von Clarmann, B. M. Dinelli, A. Dudhia, J.-M. Flaud, M. Höpfner, P. E. Morris, P. Raspollini, G. Stiller, and R. J. Wells, “Optimized forward model and retrieval scheme for MIPAS near-real-time data processing,” Appl. Opt. 39, 1323–1340 (2000) [CrossRef] .

, 5

5. P. Raspollini, C. Belotti, A. Burgess, B. Carli, M. Carlotti, S. Ceccherini, B. M. Dinelli, A. Dudhia, J.-M. Flaud, B. Funke, M. Höpfner, M. López-Puertas, V. Payne, C. Piccolo, J. J. Remedios, M. Ridolfi, and R. Spang, “MI-PAS level 2 operational analysis,” Atmos. Chem. Phys. 6, 5605–5630 (2006) [CrossRef] .

]. In restricted spectral intervals, such as the microwindows (MWs) used in our retrievals (less than 3 cm−1) [7

7. P. Raspollini, B. Carli, M. Carlotti, S. Ceccherini, A. Dehn, B.M. Dinelli, A. Dudhia, J.-M. Flaud, M. López-Puertas, F. Niro, J.J. Remedios, M. Ridolfi, H. Sembhi, L. Sgheri, and T. von Clarmann, “Ten years of MIPAS measurements with ESA Level 2 processor V6 – Part I: retrieval algorithm and diagnostics of the products,” Atmos. Meas. Tech. Discuss. 6, 461–518 (2013) [CrossRef] .

], the contribution of the lines beyond the 25 cm−1 threshold is modeled with a frequency-independent term, the so-called continuum, which is normally MW and altitude dependent. This term may also account for unmodeled contributions from aerosols or residual instrument calibration errors. In the forward model this term is usually implemented as an additional gas, with VMR equal to 1. As a consequence the columns of the continuum depend only on the observation geometry and the density of the layers. The retrieval variables model the continuum cross-section vertical profile. This profile can be approximated with a curve with nodes zi (the retrieval grid) and values ki = k(zi). The discretization used for the radiative transfer calculation (3) may be finer than the retrieval grid. However, for each layer l the continuum kl in that layer will depend on at most two values ki and ki−1 with consecutive indeces. Thus we have
kl,c=al,iki+bl,iki1,
(6)
where 0 ≤ al,i, bl,i ≤ 1 are constants determined by the interpolation law, for instance a linear interpolation in pressure.

The air columns can be explicitly calculated, but again we obtain a sloppy model because the contribution of the continuum in the productory (4) is
τl,cexp(kl,ccl,air)=exp((al,iki+bl,iki1)cl,air).
(7)
We can uniformly bound the air columns with a constant Caircl,air for any l and any line of sight. In our implementation we choose Cair = 1025 cm−2. Then we define the new variables:
ξi=exp(kiCair),
(8)
so that
τl,c=ξial,icl,air/Cairξi1bl,icl,air/Cair.
(9)
Note that while the acceptable values for the ki are ki ≥ 0, the range for the new variables is 0 ≤ ξi ≤ 1. This is clearly an improvement because the sensitivity of the forward model to ki vanishes for large enough values of ki because of the exponential dependence. The new variables ξi are polynomially connected with τl,c, that represents the optical transparency of layer l due to the continuum. Thus they are tightly linked to the measured spectra.

For the minimization required by the inversion, most techniques, such as the widely used Gauss-Newton method, require the derivatives of the simulated spectrum with respect to the retrieval variables. The derivatives with respect to the ki continuum variables are calculated as
S(σ)kj=lS(σ)τl,cτl,ckj,
(10)
where the summation is extended to all the layers depending on the value kj. Because of (6), for each l there are only two non vanishing derivatives, τl,cki and τl,cki1. With the new approach the derivatives become
S(σ)ξj=jS(σ)τl,cτl,cξj.
(11)
and from (9) the only non vanishing derivatives are
τl,cξi=al,icl,airCairτl,cξi,τl,cξi1=bl,icl,airCairτl,cξi1.
(12)

2.2. The VMR variables

In principle this approach can be extended also to the VMR variables, however there is one important difference. In Eq. (4) the quantity holding the dependence on the continuum retrieval variables is the cross-section kl. On the other hand, the cross-sections kl,m(σ)kσ,m(ple,Tle) do not depend on the VMR retrieval variables, the dependence being via the gas columns cl,m.

If the VMR profile is modeled with a piecewise-linear function with nodes zi and values xi = x(zi), then there are constants
Al,i=lzi1zzi1ziη(z)dsdz(z)dz,Bl,i=lzzizi1ziη(z)dsdz(z)dz,
(13)
such that
cl,m=Al,ixi+Bl,ixi1.
(14)
As in the continuum case we may define new variables ζi = exp (−xiCm) for some gas-dependent constant Cm, so that:
τσ,l,m=ζikl,m(σ)Al,i/Cmζi1kl,m(σ)Bl,i/Cm.
(15)

The derivatives of the spectrum with respect to these new variables are:
S(σ)ζj=lS(σ)τσ,l,mτσ,l,mζj.
(16)
and from (15) the only non vanishing terms are
τσ,l,mζi=Al,ikl,m(σ)Cmτσ,l,mζi,τσ,l,mζi1=Bl,ikl,m(σ)Cmτσ,l,mζi1.
(17)

3. Test of the method on MIPAS measurements

3.1. Tests on a set of 12 MIPAS orbits

To assess the improvements of the change of variables described in Section 2, we first tested the retrieval on a limited set of real observations. We selected 12 orbits of MIPAS measurements from the year 2007. The measurements cover the four seasons to ensure that the test includes sufficient atmospheric variability. For this test we compared the NOM, CONT and POLY approaches, both without regularization and with the IVS regularization (NOM_R, CONT_R, POLY_R).

Fig. 1 Probability density of the log10 of the spectral condition number of the GN approximation of the Hessian of the cost function, at first and last iteration. Retrieval of temperature (left panel) and CFC-11 (right panel)

The results are summarized in Table 1. For each test and each target parameter, the table reports the quantifiers described above. The last block of the table contains the percentage variation of the quantifiers with respect to the NOM approach, averaged over the retrieval targets.

Table 1. Average retrieval performances in a sample of 12 MIPAS orbits.

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Generally, all the retrieval targets exhibit similar behaviors, with two notable exceptions. The NO2 VMR is retrieved only above 24 km, while the continuum parameters are fitted only below 30 km. Thus, only a small number of continuum parameters are retrieved, in an altitude range where the opacity of the atmosphere is small. Hence the sensitivity of the retrieval to continuum parameters is good also in the NOM approach. In the retrieval of N2O5, the selected MWs span a limited spectral range (approximately 27 cm−1), therefore a single continuum profile is retrieved to make the inversion better conditioned. Hence the effect of the CONT approach on these two species is small.

From the last block of Table 1 we see that the CONT approach is able to reduce the final χ¯R2 of about 3% and the number of iterations of about 15% with respect to the NOM approach, at the expenses of a 5% increase of the Ω̄2. The regularizing effect of the LM method in this case is weaker due to the smaller initial value of the LM damping parameter. This is confirmed by the 5% increase in DoF/n¯. In the CONT_R approach the regularization is able to reduce the amplitude of the oscillations to the level of the NOM_R approach, while maintaining the 3% reduction of the χ¯R2.

We tested the statistical significance of the reduction in the number of iterations and in the χ¯R2 with a per-target Welch’s t-test. Given the relatively large sample size, all the improvements of the CONT_R with respect to the NOM_R method are significant with a probability threshold of 0.01, except for the aforementioned NO2 and N2O5 species.

The outcome of the POLY and POLY_R tests shows no advantage with respect to the CONT and CONT_R approaches. We attribute this result to the fact that the cross-sections kl,m(σ) of the retrieved gas depend on the frequency within each MW. On the other hand the cross-sections kl,c for the continuum are constant within each MW. The same argument also applies to the transparencies τσ,l,m and τl,c. In the NOM approach a sudden large increase of a continuum parameter may easily lead to a fully opaque layer τl,c ≈ 0 for all the frequencies within the involved MW. As a consequence, any emission from a layer beneath l is blocked by the fully opaque layer. On the other hand, a large increase in the VMR parameter hardly leads to a full MW with τσ,l,m ≈ 0, due to the dependence on σ. Also in the CONT approach we can get an opaque layer with τl,c ≈ 0. However, while in the NOM case we have τl,ckj0 when kj → ∞, in the CONT case τl,cξj when ξj → 0, because the exponents in Eq. (9) are less than 1. Consequently, in this case the sensitivity to the continuum parameters is not lost in the CONT approach.

3.2. Tests on a whole month of orbits

Table 2. Average retrieval performances in a sample of 1 month of orbits.

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In Fig. 2 we report the behavior of the χR2 versus the latitude of the limb scans. The graph is obtained by binning the χR2 in 5 deg. latitude intervals, and then averaging the data within each bin. The four panels refer to different retrieval targets. We do not report all target species, because they all show very similar behaviors. We note that the reduction of χR2 obtained by the CONT_R approach is visible at all latitudes. For the reasons explained in Section 3.1, the change of variable shows no advantage in the retrieval of N2O5, see bottom right panel of the figure.

Fig. 2 Behavior of normalized χ2 versus latitude for the NOM_R and CONT_R approaches for the October 2007 orbits. Each panel refers to the target indicated in the plot.

Despite the smaller χR2 achieved, we found that the average differences between the NOM_R and CONT_R profiles show no evident systematic features above the noise error.

4. Conclusions

In this paper we propose a different set of retrieval variables for the atmospheric continuum. Tests with real MIPAS observations show that a more stable retrieval is obtained with the new variables. As a consequence we obtain a reduction in the number of iterations, a smaller minimum of the cost function, and slightly less oscillating profiles. The same modification can be applied also to the VMR retrieval variables. Our tests show however that – at least in the present formulation – we get no real advantage connected with this additional change.

The technique can be applied to any inverse problem whenever the forward model depends exponentially on the inversion variables. Its effectiveness however should be evaluated case by case on the basis of numerical tests.

Acknowledgments

We thank the IFAC-CNR institute in Firenze for making available computing and technical facilities through associate contract to M.R., in the frame of the research activity TA.P06.002. This study was supported by the ESA-ESRIN contract 21719/08/I-OL.

References and links

1.

T. von Clarmann, M. Hpfner, B. Funke, M. López-Puertas, A. Dudhia, V. Jay, F. Schreier, M. Ridolfi, S. Ceccherini, B. J. Kerridge, J. Reburn, R. Siddans, and J.-M. Flaud, “Modelling of atmospheric mid-infrared radiative transfer: The AMIL2DA algorithm intercomparison experiment,” J. Quant. Spectrosc. Radiat. Transfer 78(3–4), 381–407 (2003) [CrossRef] .

2.

A. R. Curtis, “Discussion of a statistical model for water vapor absorption,” Q. J. R. Meteorol. Soc. 78, 638–640 (1952) [CrossRef] .

3.

W. L. Godson, “The evaluation of infrared radiative fluxes due to atmospheric water vapour,” Q. J. R. Meteorol. Soc. 79, 367–379 (1953) [CrossRef] .

4.

M. Ridolfi, B. Carli, M. Carlotti, T. von Clarmann, B. M. Dinelli, A. Dudhia, J.-M. Flaud, M. Höpfner, P. E. Morris, P. Raspollini, G. Stiller, and R. J. Wells, “Optimized forward model and retrieval scheme for MIPAS near-real-time data processing,” Appl. Opt. 39, 1323–1340 (2000) [CrossRef] .

5.

P. Raspollini, C. Belotti, A. Burgess, B. Carli, M. Carlotti, S. Ceccherini, B. M. Dinelli, A. Dudhia, J.-M. Flaud, B. Funke, M. Höpfner, M. López-Puertas, V. Payne, C. Piccolo, J. J. Remedios, M. Ridolfi, and R. Spang, “MI-PAS level 2 operational analysis,” Atmos. Chem. Phys. 6, 5605–5630 (2006) [CrossRef] .

6.

M. K. Transtrum, B. B. Machta, and J. P. Sethna, “Geometry of nonlinear least squares with applications to sloppy models and optimization,” Phys. Rev. E 83, 036701 (2011) [CrossRef] .

7.

P. Raspollini, B. Carli, M. Carlotti, S. Ceccherini, A. Dehn, B.M. Dinelli, A. Dudhia, J.-M. Flaud, M. López-Puertas, F. Niro, J.J. Remedios, M. Ridolfi, H. Sembhi, L. Sgheri, and T. von Clarmann, “Ten years of MIPAS measurements with ESA Level 2 processor V6 – Part I: retrieval algorithm and diagnostics of the products,” Atmos. Meas. Tech. Discuss. 6, 461–518 (2013) [CrossRef] .

8.

H. Fischer, M. Birk, C. Blom, B. Carli, M. Carlotti, T. von Clarmann, L. Delbouille, A. Dudhia, D. Ehhalt, M. Endemann, J. M. Flaud, R. Gessner, A. Kleinert, R. Koopman, J. Langen, M. López-Puertas, P. Mosner, H. Nett, H. Oelhaf, G. Perron, J. Remedios, M. Ridolfi, G. Stiller, and R. Zander, “MIPAS: an instrument for atmospheric and climate research,” Atmos. Chem. Phys. 8, 2151–2188 (2008) [CrossRef] .

9.

M. Lampton, “Damping-undamping strategies for the Levenberg-Marquardt nonlinear least- squares method,” Comput. Phys. 11, 110–115 (1997) [CrossRef] .

10.

M. Ridolfi and L. Sgheri, “A self-adapting and altitude-dependent regularization method for atmospheric profile retrievals,” Atmos. Chem. Phys. 9, 1883–1897 (2009) [CrossRef] .

11.

M. Ridolfi and L. Sgheri, “Iterative approach to self-adapting and altitude-dependent regularization for atmospheric profile retrievals,” Opt. Express , 19, 26696–26709 (2011) [CrossRef] .

12.

P. R. Bevington and D. K. Robinson, Data reduction and error analysis for the physical sciences, 3rd ed. (McGraw–Hill, 2003).

13.

C. D. Rodgers, Inverse methods for atmospheric sounding: Theory and practice Atmospheric, Oceanic and Planetary Physics, (World Scientific, 2000).

OCIS Codes
(000.3860) General : Mathematical methods in physics
(010.1280) Atmospheric and oceanic optics : Atmospheric composition
(100.3190) Image processing : Inverse problems
(280.4991) Remote sensing and sensors : Passive remote sensing

ToC Category:
Atmospheric and Oceanic Optics

History
Original Manuscript: February 8, 2013
Revised Manuscript: April 5, 2013
Manuscript Accepted: April 8, 2013
Published: May 3, 2013

Citation
Marco Ridolfi and Luca Sgheri, "On the choice of retrieval variables in the inversion of remotely sensed atmospheric measurements," Opt. Express 21, 11465-11474 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-9-11465


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References

  1. T. von Clarmann, M. Hpfner, B. Funke, M. López-Puertas, A. Dudhia, V. Jay, F. Schreier, M. Ridolfi, S. Ceccherini, B. J. Kerridge, J. Reburn, R. Siddans, and J.-M. Flaud, “Modelling of atmospheric mid-infrared radiative transfer: The AMIL2DA algorithm intercomparison experiment,” J. Quant. Spectrosc. Radiat. Transfer78(3–4), 381–407 (2003). [CrossRef]
  2. A. R. Curtis, “Discussion of a statistical model for water vapor absorption,” Q. J. R. Meteorol. Soc.78, 638–640 (1952). [CrossRef]
  3. W. L. Godson, “The evaluation of infrared radiative fluxes due to atmospheric water vapour,” Q. J. R. Meteorol. Soc.79, 367–379 (1953). [CrossRef]
  4. M. Ridolfi, B. Carli, M. Carlotti, T. von Clarmann, B. M. Dinelli, A. Dudhia, J.-M. Flaud, M. Höpfner, P. E. Morris, P. Raspollini, G. Stiller, and R. J. Wells, “Optimized forward model and retrieval scheme for MIPAS near-real-time data processing,” Appl. Opt.39, 1323–1340 (2000). [CrossRef]
  5. P. Raspollini, C. Belotti, A. Burgess, B. Carli, M. Carlotti, S. Ceccherini, B. M. Dinelli, A. Dudhia, J.-M. Flaud, B. Funke, M. Höpfner, M. López-Puertas, V. Payne, C. Piccolo, J. J. Remedios, M. Ridolfi, and R. Spang, “MI-PAS level 2 operational analysis,” Atmos. Chem. Phys.6, 5605–5630 (2006). [CrossRef]
  6. M. K. Transtrum, B. B. Machta, and J. P. Sethna, “Geometry of nonlinear least squares with applications to sloppy models and optimization,” Phys. Rev. E83, 036701 (2011). [CrossRef]
  7. P. Raspollini, B. Carli, M. Carlotti, S. Ceccherini, A. Dehn, B.M. Dinelli, A. Dudhia, J.-M. Flaud, M. López-Puertas, F. Niro, J.J. Remedios, M. Ridolfi, H. Sembhi, L. Sgheri, and T. von Clarmann, “Ten years of MIPAS measurements with ESA Level 2 processor V6 – Part I: retrieval algorithm and diagnostics of the products,” Atmos. Meas. Tech. Discuss.6, 461–518 (2013). [CrossRef]
  8. H. Fischer, M. Birk, C. Blom, B. Carli, M. Carlotti, T. von Clarmann, L. Delbouille, A. Dudhia, D. Ehhalt, M. Endemann, J. M. Flaud, R. Gessner, A. Kleinert, R. Koopman, J. Langen, M. López-Puertas, P. Mosner, H. Nett, H. Oelhaf, G. Perron, J. Remedios, M. Ridolfi, G. Stiller, and R. Zander, “MIPAS: an instrument for atmospheric and climate research,” Atmos. Chem. Phys.8, 2151–2188 (2008). [CrossRef]
  9. M. Lampton, “Damping-undamping strategies for the Levenberg-Marquardt nonlinear least- squares method,” Comput. Phys.11, 110–115 (1997). [CrossRef]
  10. M. Ridolfi and L. Sgheri, “A self-adapting and altitude-dependent regularization method for atmospheric profile retrievals,” Atmos. Chem. Phys.9, 1883–1897 (2009). [CrossRef]
  11. M. Ridolfi and L. Sgheri, “Iterative approach to self-adapting and altitude-dependent regularization for atmospheric profile retrievals,” Opt. Express, 19, 26696–26709 (2011). [CrossRef]
  12. P. R. Bevington and D. K. Robinson, Data reduction and error analysis for the physical sciences, 3rd ed. (McGraw–Hill, 2003).
  13. C. D. Rodgers, Inverse methods for atmospheric sounding: Theory and practice Atmospheric, Oceanic and Planetary Physics, (World Scientific, 2000).

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