## Iterative approach to self-adapting and altitude-dependent regularization for atmospheric profile retrievals |

Optics Express, Vol. 19, Issue 27, pp. 26696-26709 (2011)

http://dx.doi.org/10.1364/OE.19.026696

Acrobat PDF (823 KB)

### Abstract

In this paper we present the IVS (Iterative Variable Strength) method, an altitude-dependent, self-adapting Tikhonov regularization scheme for atmospheric profile retrievals. The method is based on a similar scheme we proposed in 2009. The new method does not need any specifically tuned minimization routine, hence it is more robust and faster. We test the self-consistency of the method using simulated observations of the Michelson Interferometer for Passive Atmospheric Sounding (MIPAS). We then compare the new method with both our previous scheme and the scalar method currently implemented in the MIPAS on-line processor, using both synthetic and real atmospheric limb measurements. The IVS method shows very good performances.

© 2011 OSA

## 1. Introduction

1. 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]

2. 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]

3. 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, “Mipas level 2 operational analysis,” Atmos. Chem. Phys. **6**, 5605–5630 (2006). [CrossRef]

_{2}O, O

_{3}, HNO

_{3}, CH

_{4}, N

_{2}O and NO

_{2}.

_{4}and N

_{2}O VMR [4

4. S. Payan, C. Camy-Peyret, H. Oelhaf, G. Wetzel, G. Maucher, C. Keim, M. Pirre, N. Huret, A. Engel, M. C. Volk, H. Kuellmann, J. Kuttippurath, U. Cortesi, G. Bianchini, F. Mencaraglia, P. Raspollini, G. Redaelli, C. Vigouroux, M. De Mazière, S. Mikuteit, T. Blumenstock, V. Velazco, J. Notholt, E. Mahieu, P. Duchatelet, D. Smale, S. Wood, N. Jones, C. Piccolo, V. Payne, A. Bracher, N. Glatthor, G. Stiller, K. Grunow, P. Jeseck, Y. Te, and A. Butz, “Validation of version-4.61 methane and nitrous oxide observed by mipas,” Atmos. Chem. Phys. **9**, 413–442 (2009). [CrossRef]

6. S. Ceccherini, “Analytical determination of the regularization parameter in the retrieval of atmospheric vertical profiles,” Opt. Lett. **30**, 2554–2556 (2005). [CrossRef] [PubMed]

7. 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]

- LS - The Least Squares solution, see equation (2).
- OE - The Optimal Estimation solution, see equation (3).
- LM - The Levenberg-Marquardt solution, see equation (3) and subsequent explanations.
- EC - The solution regularized with the Error Consistency method, see [6].
6. S. Ceccherini, “Analytical determination of the regularization parameter in the retrieval of atmospheric vertical profiles,” Opt. Lett.

**30**, 2554–2556 (2005). [CrossRef] [PubMed] - VS - The solution regularized with the Variable Strength method, see Eq. (11) and [7].
7. 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] - IVS - The solution regularized with the Iterative Variable Strength method, the new approach introduced in this paper, see Section 3.

## 2. Theory

**y**=

**f**(

**x**) be the forward problem, where

**y**is the

*m*–dimensional vector of the observations with error covariance matrix

**S**

_{y},

**f**is the forward model, function of the

*n*–dimensional atmospheric state vector

**x**, whose components represent the unknown profile at altitudes

*z*=

*z*,

_{j}*j*= 1,...,

*n*. The Tikhonov solution is the state vector

**x**

_{t}minimizing the following cost function:

*χ*

^{2}and represents the cost function minimized in the least–squares (LS) approach. The vector

**x**

_{s}is an a-priori estimate of the solution,

**L**is a

*h*×

*n*matrix operator, usually approximating a linear combination of the

*i*–

*th*order vertical derivatives (

*i*= 0,1,2). The

*h*×

*h*matrix

**Λ**is diagonal, positive semi-definite and drives the strength of the regularization. Note that normally

*h*≤

*n*. Assuming that

**Λ**

*as the regularization strength at*

_{jj}*z*=

*z̃*,

_{j}*j*= 1,...,

*h*. The altitudes

*z̃*are determined by the choice of

_{j}**L**, and may not necessarily coincide with the retrieval grid

*z*,

_{j}*j*= 1,...,

*n*. Thus we may speak of a vertical profile of

**Λ**.

**Λ**=

*λ*

**I**. There are many different methods to determine

*λ*, a good review may be found in [8

8. A. Doicu, T. Trautmann, and F. Schreier, *Numerical regularization for atmospheric inverse problems* (Springer, 2010). [CrossRef]

9. S. S. Kulawik, G. Osterman, D. B. A. Jones, and K. W. Bowman, “Calculation of altitude-dependent tikhonov constraints for tes nadir retrievals,” IEEE Trans. Geosci. Remote Sens. **44**, 1334–1342 (2006). [CrossRef]

10. K. W. Bowman, C. Rodgers, S. S. Kulawik, J. Worden, E. Sarkissian, G. Osterman, T. Steck, M. Lou, A. Eldering, M. Shephard, H. Worden, M. Lampel, S. Clough, P. Brown, C. Rinsland, M. Gunson, and R. Beer, “Tropospheric emission spectrometer: Retrieval method and error analysis,” IEEE Trans. Geosci. Remote Sens. **44**, 1297–1307 (2006). [CrossRef]

11. J. Steinwagner and G. Schwarz, “Shape-dependent regularization for the retrieval of atmospheric state parameter profiles,” Appl. Opt. **45**, 1000–1009 (2006). [CrossRef] [PubMed]

**Λ**=

*λ*

**S**

*and*

_{h}**S**

*is a diagonal matrix containing the reciprocal of the a-priori estimation of the profile. The VS method was introduced and tested on MIPAS measurements in [7*

_{h}**9**, 1883–1897 (2009). [CrossRef]

**Λ**as the result of an iterative process. We choose to apply the regularization a-posteriori to obtain a better convergence rate, as reported in [12

12. S. Ceccherini, C. Belotti, B. Carli, P. Raspollini, and M. Ridolfi, “Technical note: Regularization performances with the error consistency method in the case of retrieved atmospheric profiles,” Atmos. Chem. Phys. **7**, 1435–1440 (2007). [CrossRef]

**9**, 1883–1897 (2009). [CrossRef]

*χ*

^{2}via a Gauss-Newton iterative method: where

*p*is the iteration count and

**K**

*is the*

_{p}*m*×

*n*Jacobian matrix of

**f**in

**x**

*. Let*

_{p}*k*be the iteration count at convergence, and

**x**

_{LS}≡

**x**

_{k}_{+1}the LS solution. The covariance matrix of

**x**

_{LS}is

*p*≤

*k*the matrix

13. C. D. Rodgers, *Inverse Methods for Atmospheric Sounding: Theory and Practice* (Atmospheric, Oceanic and Planetary Physics, World Scientific, 2000). [CrossRef]

**x**

_{a}is an a-priori estimate of the profile

**x**with covariance matrix

**S**

_{a}. The LM solution with damping factor

*α*can also be represented with Eq. (3), by setting

**x**

_{a}=

**x**

*and*

_{p}**M**, diag(

**M**) is a matrix having the same elements as

**M**on the main diagonal and 0 elsewhere. With the OE/LM modifications the matrix to be inverted is

*k*be the iteration count at convergence, thus

**x**

_{OE}≡

**x**

_{k}_{+1}, the unregularized solution. Let

**A**

_{OE}be the averaging kernel of

**x**

_{OE}and

**S**

_{OE}its measurement error covariance matrix. The ESA retrieval algorithm uses the LM solution, and

**A**

_{OE}and

**S**

_{OE}can be calculated with alternative algorithms of different sophistication as explained in [12

12. S. Ceccherini, C. Belotti, B. Carli, P. Raspollini, and M. Ridolfi, “Technical note: Regularization performances with the error consistency method in the case of retrieved atmospheric profiles,” Atmos. Chem. Phys. **7**, 1435–1440 (2007). [CrossRef]

14. S. Ceccherini and M. Ridolfi, “Technical note: Variance-covariance matrix and averaging kernels for the levenberg-marquardt solution of the retrieval of atmospheric vertical profiles,” Atmos. Chem. Phys. **10**, 3131–3139 (2010). [CrossRef]

**A**

_{OE}and

**S**

_{OE}.

**x**

_{Λ}as the Gauss-Newton iterate for the minimization of: starting from

**x**

*. Thus*

_{k}**x**

_{Λ}is given by: Both

**x**

_{a}and

**x**

_{s}are estimates of the solution, however usually

**x**

_{a}constrains the values of the profile, while

**x**

_{s}constrains its derivatives. Therefore two different symbols are used. If we extract the term

**A**

_{Λ}and the measurement error covariance matrix

**S**

_{Λ}of

**x**

_{Λ}can be written as: Since in practical cases it is always difficult to find reliable a-priori profile estimates, in this paper we always select

**x**

_{s}=

**0**. With this choice Eq. (6) can be simplified as: Vertical resolution is a measure of the dispersion of the signal, usually calculated via the averaging kernel

**A**

_{Λ}. There are many practical ways of estimating the vertical resolution, we use: where

*z*,

_{j}*j*= 1,...,

*n*are the altitudes, and

*z*

_{0}=

*z*

_{1}+(

*z*

_{1}–

*z*

_{2}),

*z*

_{n+1}=

*z*

_{n}+(

*z*–

_{n}*z*

_{n−1}). Formula (10) is a variation of the full width half height (FWHH) estimation proposed in [13

13. C. D. Rodgers, *Inverse Methods for Atmospheric Sounding: Theory and Practice* (Atmospheric, Oceanic and Planetary Physics, World Scientific, 2000). [CrossRef]

**A**

_{Λ}|

*in order to penalize the negative lobes of the averaging kernel. Note that rows of the averaging kernel not peaking at the diagonal element are penalized by Eq. (10), which in this case provides an overestimate of the FWHH. When*

_{ii}**A**

_{Λ}=

**I**, Eq. (10) provides the vertical step Δ

*z*= (

_{i}*z*

_{i−1}−

*z*

_{i+1})/2 of the retrieval grid.

## 3. Altitude-dependent regularization

**9**, 1883–1897 (2009). [CrossRef]

**Λ**-profile is determined as the minimizer of the following target function: where the bar over a vector stands for the average of the vector elements, and a superscript

^{+}stands for the positive part of a function. The constants

*w*and

_{e}*w*are tunable parameters. The first term of formula (11) represents the error of the regularized profile. The other terms are penalization terms which take effect if the regularized profile is not compatible with the observations (second term), or if the vertical resolution is degraded beyond a pre-defined margin (third term). Loosely speaking, the minimum of (11) is obtained for the largest possible

_{r}**Λ**-profile that keeps small enough the penalization terms.

*ψ*

_{VS}. In the IVS approach we define a

**Λ**-profile

*λ*(

*z*) on a vertical grid so fine that we can consider it a continuous function. We start with a large

*λ*

^{(0)}(

*z*) =

*λ*

_{max}constant profile and decrease it iteratively until the following requirements are fulfilled: where

*l*is the iteration count and

**Λ**

*=*

_{jj}*λ*

^{(l)}(

*z̃*),

_{j}*j*= 1,...,

*h*. Condition (12) ensures that, on average, the regularized profile lies within a fraction

*w*of the error bars of the unregularized profile. It pursues the same objective of the second term of

_{e}*ψ*

_{VS}(

**Λ**), as shown by Eq. (13) of [7

**9**, 1883–1897 (2009). [CrossRef]

*ψ*

_{VS}(

**Λ**).

*λ*

_{min}. Let

*J*⊂ {1,...,

*n*} the set of indices of the altitudes

*z*for which

_{j}*λ*

^{(l)}(

*z*) >

_{j}*λ*

_{min}and: If the requirements (12–13) are not met,

*J*is not empty and we decrease

*λ*

^{(l)}(

*z*). The decreased profile

*λ*

^{(l+1)}(

*z*) is calculated as: where

*T*is the triangular shaped function: and 0 <

*r*< 1,

*δ*> 0 are constants. The parameter

_{j}*r*drives the speed of the attenuation of the

**Λ**-profile, in our implementation we used

*r*= 0.99. Furthermore we set

*δ*= 3Δ

_{j}*z*on the basis of the following considerations. The

_{j}**x**

_{Λ}profile is obtained from

**x**

_{OE}via Eq. (9). For any standard choice of

**L**,

**L**

^{T}**ΛL**is at most a pentadiagonal matrix, i.e. a matrix having non-zero elements (

*i, j*) only for |

*i*–

*j*|

*≤*2. Moreover the matrix

**Λ**

*is mostly localized in the altitude range*

_{jj}*z*∈ [

_{j}*z*– 3Δ

_{j}*z*,

_{j}*z*+ 3Δ

_{j}*z*]. Note that

_{j}*λ*

^{(l+1)}(

*z*) <

*λ*

^{(l)}(

*z*) if and only if

*z*∈ (

*z*–

_{j}*δ*,

_{j}*z*+

_{j}*δ*) for some

_{j}*j*∈

*J*.

## 4. Implementation details

2. 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]

3. 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, “Mipas level 2 operational analysis,” Atmos. Chem. Phys. **6**, 5605–5630 (2006). [CrossRef]

*h*=

*n*– 2, with the operator

**L**defined so that: Let

*z̃*

_{j−1}= (

*z*

_{j−1}+ 2

*z*+

_{j}*z*

_{j+1})/4,

*j*= 2,...,

*n*– 1. Thus the definition of

**L**given by Eq. (18) corresponds to

*j*= 1,...,

*h*, i.e. to the second derivative operator

**L**

_{2}. In the case of a constant Δ

*z*we have

_{j}*z̃*

_{j−1}=

*z*,

_{j}*j*= 2,...,

*n*− 1.

*λ*

_{min}= 10

^{−2}, and

*λ*

_{max}= 10. A notable exception is the regularization of H

_{2}O profiles where we set

*λ*

_{max}= 10

^{3}, and implemented a damping scheme which rapidly reduces

*λ*

^{(0)}(

*z*) from

*λ*

_{max}to

*λ*

_{min}below the tropopause. The aim of this modification is to avoid introducing a systematic bias in the H

_{2}O profile in this altitude range. Any regularization in this region, even if within the error bars, would introduce a bias around or below the knee of the profile. This effect is due either to the large second derivative values necessary to model the profile knee around the tropopause (when

**L**=

**L**

_{2}) or to the large values of the profile and its first derivative below the tropopause (when

**L**=

**L**

_{0}or

**L**=

**L**

_{1}).

**Λ**-profile. For this reason, the

**Λ**-profile of the VS method was defined in our tests on a coarse grid of 9 points and then interpolated to the grid

*z̃*,

_{j}*j*= 1,...,

*n*. Finally we set (

*w*,

_{e}*w*) = (1, 5) for all retrieval targets, according to the results obtained when testing the VS method on real observations [7

_{r}**9**, 1883–1897 (2009). [CrossRef]

**L**=

**L**

_{1}.

## 5. Self-consistency test with synthetic observations from a single limb scan

_{3}retrieval based on synthetic observations already used to assess the VS method [7

**9**, 1883–1897 (2009). [CrossRef]

*reference*atmosphere model of [15

15. J. J. Remedios, R. J. Leigh, A. M. Waterfall, D. P. Moore, H. Sembhi, I. Parkes, J. Greenhough, M. P. Chipperfield, and D. Hauglustaine, “Mipas reference atmospheres and comparisons to v4.61/v4.62 mipas level 2 geophysical data sets,” Atmos. Chem. Phys. Discuss. **7**, 9973–10017 (2007). [CrossRef]

_{3}profile was modified with a sharp bump in the 18–24 km altitude range, reflecting the double–peak sometimes observed in the real O

_{3}profiles in pre ozone hole conditions (see [16]).

17. A. Dudhia, “Mipas-related section of the web site of the Oxford university, department of physics,” http://www.atm.ox.ac.uk/group/mipas/ (2008).

*α*= 10

^{−3}) in order to limit its regularization effect. Because of the artificially amplified errors, the LM term is however necessary to guarantee the convergence of the minimization sequence.

**S**

_{OE}(dashed blue). The same panel shows also the differences between the LM (solid blue), VS (solid green), EC (solid orange), IVS (solid purple) solutions and the reference profile, i.e. the actual errors.

*w*= 1 of the LM error bars as required by the algorithms. Note that the relatively large errors obtained in this test retrieval are mainly due to the artificial amplification of the measurement noise that we applied above 40 km. Therefore the results of this test, while useful to assess the consistency of the IVS method, should not be considered as representative of the real MIPAS performance.

_{e}## 6. Results of retrievals from a simulated full MIPAS orbit

**Λ**aims at a reduction of Ω̄

_{2}, at the expenses of an increase of

*E*only as an additional parameter to evaluate the quality of a regularization scheme.

13. C. D. Rodgers, *Inverse Methods for Atmospheric Sounding: Theory and Practice* (Atmospheric, Oceanic and Planetary Physics, World Scientific, 2000). [CrossRef]

*n*of points of the retrieved profile, since this latter can vary from scan to scan due to cloud contamination. We then take the arithmetic mean

*σ*) of the difference between the retrieved and the true profile at each retrieval grid point along the orbit.

15. J. J. Remedios, R. J. Leigh, A. M. Waterfall, D. P. Moore, H. Sembhi, I. Parkes, J. Greenhough, M. P. Chipperfield, and D. Hauglustaine, “Mipas reference atmospheres and comparisons to v4.61/v4.62 mipas level 2 geophysical data sets,” Atmos. Chem. Phys. Discuss. **7**, 9973–10017 (2007). [CrossRef]

*χ*

^{2}. The thresholds were adjusted to keep the convergence error smaller than 10% of the error due to the measurement noise. A more comprehensive description of the convergence criteria may be found in the technical note [19

19. M. Ridolfi and L. Sgheri, “Improvement of the orm convergence criteria: test of performance and implementation details,” http://www.fci.unibo.it/~ridolfi/hak/tnconvcrit.pdf (2011).

*σ*for each target parameter and retrieval method. For reference, we also report the standard deviation of the LS profiles. Since the pure LS method does not always converge, we calculate the

*σ*of the LS profiles (

*σ*

_{LS}) as the average of

*z*and the scans of the orbit.

_{j}_{2}O retrieval. The

*σ*of the LS method (

*σ*

_{LS}) is calculated by binning

_{2}O profiles for a sample scan.

_{2},

*E*and

_{2}is defined. The last row of the table contains the efficiency

*E*averaged over the retrieval targets.

_{2}is achieved at the expenses of a marginal increase of the

_{2}quantifier is dominated by the contributions of the steep part of the profile below the tropopause, where the profile is only marginally regularized because of the damping we applied to

*λ*

^{(0)}(

*z*). Nevertheless the profile oscillations in the stratosphere are well smoothed out by the VS and IVS methods as illustrated in Fig. 3, even if this effect is not properly factored by Ω̄

_{2}.

## 7. Retrieval from real MIPAS data

_{2}O retrieval in the EC case has been carried out with the LM method. The difference in the H

_{2}O results with respect to the full LM case is mainly due to the method used to retrieve the temperature.

_{2}of the LM retrieved profiles is generally smaller in the synthetic case. This is due to the combination of two causes. First, synthetic measurements do not include systematic model errors which are present in the real observations. Second, the reference model atmosphere of the synthetic test retrieval is probably smoother than the actual atmosphere sounded by MIPAS in the selected orbits. The only exception is H

_{2}O. In this case the presence of clouds in the sample of real measurements makes the average bottom altitude of the profiles higher than in case of orbit 15451, used to model the synthetic measurements. As already mentioned, in the case of H

_{2}O the Ω̄

_{2}quantifier is dominated by the contributions of the profile below the tropopause, hence the smaller value obtained with real measurements.

## 8. Conclusions

**9**, 1883–1897 (2009). [CrossRef]

## Acknowledgments

## References and links

1. | 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. |

2. | 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. |

3. | 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, “Mipas level 2 operational analysis,” Atmos. Chem. Phys. |

4. | S. Payan, C. Camy-Peyret, H. Oelhaf, G. Wetzel, G. Maucher, C. Keim, M. Pirre, N. Huret, A. Engel, M. C. Volk, H. Kuellmann, J. Kuttippurath, U. Cortesi, G. Bianchini, F. Mencaraglia, P. Raspollini, G. Redaelli, C. Vigouroux, M. De Mazière, S. Mikuteit, T. Blumenstock, V. Velazco, J. Notholt, E. Mahieu, P. Duchatelet, D. Smale, S. Wood, N. Jones, C. Piccolo, V. Payne, A. Bracher, N. Glatthor, G. Stiller, K. Grunow, P. Jeseck, Y. Te, and A. Butz, “Validation of version-4.61 methane and nitrous oxide observed by mipas,” Atmos. Chem. Phys. |

5. | P. Raspollini, B. Carli, S. Ceccherini, and M. Ridolfi, “The new measurement scenario of mipas,” in “ASSFTS 14 Workshop, Firenze,” (2009). |

6. | S. Ceccherini, “Analytical determination of the regularization parameter in the retrieval of atmospheric vertical profiles,” Opt. Lett. |

7. | M. Ridolfi and L. Sgheri, “A self-adapting and altitude-dependent regularization method for atmospheric profile retrievals,” Atmos. Chem. Phys. |

8. | A. Doicu, T. Trautmann, and F. Schreier, |

9. | S. S. Kulawik, G. Osterman, D. B. A. Jones, and K. W. Bowman, “Calculation of altitude-dependent tikhonov constraints for tes nadir retrievals,” IEEE Trans. Geosci. Remote Sens. |

10. | K. W. Bowman, C. Rodgers, S. S. Kulawik, J. Worden, E. Sarkissian, G. Osterman, T. Steck, M. Lou, A. Eldering, M. Shephard, H. Worden, M. Lampel, S. Clough, P. Brown, C. Rinsland, M. Gunson, and R. Beer, “Tropospheric emission spectrometer: Retrieval method and error analysis,” IEEE Trans. Geosci. Remote Sens. |

11. | J. Steinwagner and G. Schwarz, “Shape-dependent regularization for the retrieval of atmospheric state parameter profiles,” Appl. Opt. |

12. | S. Ceccherini, C. Belotti, B. Carli, P. Raspollini, and M. Ridolfi, “Technical note: Regularization performances with the error consistency method in the case of retrieved atmospheric profiles,” Atmos. Chem. Phys. |

13. | C. D. Rodgers, |

14. | S. Ceccherini and M. Ridolfi, “Technical note: Variance-covariance matrix and averaging kernels for the levenberg-marquardt solution of the retrieval of atmospheric vertical profiles,” Atmos. Chem. Phys. |

15. | J. J. Remedios, R. J. Leigh, A. M. Waterfall, D. P. Moore, H. Sembhi, I. Parkes, J. Greenhough, M. P. Chipperfield, and D. Hauglustaine, “Mipas reference atmospheres and comparisons to v4.61/v4.62 mipas level 2 geophysical data sets,” Atmos. Chem. Phys. Discuss. |

16. | A. V. Nemuc and R. L. Dezafra, “Ground based measurements of stratospheric ozone in antarctica,” Rom. Rep. Phys. |

17. | A. Dudhia, “Mipas-related section of the web site of the Oxford university, department of physics,” http://www.atm.ox.ac.uk/group/mipas/ (2008). |

18. | P. R. Bevington and D. K. Robinson, |

19. | M. Ridolfi and L. Sgheri, “Improvement of the orm convergence criteria: test of performance and implementation details,” http://www.fci.unibo.it/~ridolfi/hak/tnconvcrit.pdf (2011). |

**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: September 14, 2011

Revised Manuscript: November 3, 2011

Manuscript Accepted: November 3, 2011

Published: December 14, 2011

**Citation**

Marco Ridolfi and Luca Sgheri, "Iterative approach to self-adapting and altitude-dependent regularization for atmospheric profile retrievals," Opt. Express **19**, 26696-26709 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-27-26696

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### References

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- S. Ceccherini, C. Belotti, B. Carli, P. Raspollini, and M. Ridolfi, “Technical note: Regularization performances with the error consistency method in the case of retrieved atmospheric profiles,” Atmos. Chem. Phys.7, 1435–1440 (2007). [CrossRef]
- C. D. Rodgers, Inverse Methods for Atmospheric Sounding: Theory and Practice (Atmospheric, Oceanic and Planetary Physics, World Scientific, 2000). [CrossRef]
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- J. J. Remedios, R. J. Leigh, A. M. Waterfall, D. P. Moore, H. Sembhi, I. Parkes, J. Greenhough, M. P. Chipperfield, and D. Hauglustaine, “Mipas reference atmospheres and comparisons to v4.61/v4.62 mipas level 2 geophysical data sets,” Atmos. Chem. Phys. Discuss.7, 9973–10017 (2007). [CrossRef]
- A. V. Nemuc and R. L. Dezafra, “Ground based measurements of stratospheric ozone in antarctica,” Rom. Rep. Phys.57, 445–452 (2005).
- A. Dudhia, “Mipas-related section of the web site of the Oxford university, department of physics,” http://www.atm.ox.ac.uk/group/mipas/ (2008).
- P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 3rd ed. (McGraw–Hill, 2003).
- M. Ridolfi and L. Sgheri, “Improvement of the orm convergence criteria: test of performance and implementation details,” http://www.fci.unibo.it/~ridolfi/hak/tnconvcrit.pdf (2011).

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