## Optimal use of the information provided by indirect measurements of atmospheric vertical profiles

Optics Express, Vol. 17, Issue 7, pp. 4944-4958 (2009)

http://dx.doi.org/10.1364/OE.17.004944

Acrobat PDF (193 KB)

### Abstract

A procedure for the optimal utilization and representation of the information retrieved from atmospheric observations is discussed. We show that the “measurement-space solution” exclusively contains the information present in the observations, but is not suitable for a representation of the retrieved profile in the form of a graph. The new method of the “null-space regularization”, which provides in a complement space an external constraint and makes the solution compliant with this representation, is presented. In this way the measurement and the constraint are separately determined and are given in a vertical grid that can be freely chosen as fine as desired. The method is applied to simulated measurements and is used to define a procedure for data fusion.

© 2009 Optical Society of America

## 1. Introduction

1. C. D. Rodgers, *Inverse Methods for Atmospheric Sounding: Theory and Practice*, Vol. 2 of Series on Atmospheric, Oceanic and Planetary Physics (World Scientific, 2000). [CrossRef]

3. B. Carli, P. Raspollini, M. Ridolfi, and B. M. Dinelli, “Discrete representation and resampling in limb-sounding measurements,” Appl. Opt. **40**, 1261–1268 (2001). [CrossRef]

1. C. D. Rodgers, *Inverse Methods for Atmospheric Sounding: Theory and Practice*, Vol. 2 of Series on Atmospheric, Oceanic and Planetary Physics (World Scientific, 2000). [CrossRef]

1. C. D. Rodgers, *Inverse Methods for Atmospheric Sounding: Theory and Practice*, Vol. 2 of Series on Atmospheric, Oceanic and Planetary Physics (World Scientific, 2000). [CrossRef]

11. S. Ceccherini, B. Carli, E. Pascale, M. Prosperi, P. Raspollini, and B. M. Dinelli, “Comparison of measurements made with two different instruments of the same atmospheric vertical profile,” Appl. Opt. **42**, 6465–6473 (2003). [CrossRef] [PubMed]

12. J. Joiner and A. M. da Silva, “Efficient methods to assimilate remotely sensed data based on information content,” Q. J. R. Meteorol. Soc. **124**, 1669–1694 (1998). [CrossRef]

13. T. von Clarmann and U. Grabowski, “Elimination of hidden a priori information from remotely sensed profile data,” Atmos. Chem. Phys. **7**, 397–408 (2007). [CrossRef]

12. J. Joiner and A. M. da Silva, “Efficient methods to assimilate remotely sensed data based on information content,” Q. J. R. Meteorol. Soc. **124**, 1669–1694 (1998). [CrossRef]

## 2. Theory

### 2.1 Terminology and notation

*Inverse Methods for Atmospheric Sounding: Theory and Practice*, Vol. 2 of Series on Atmospheric, Oceanic and Planetary Physics (World Scientific, 2000). [CrossRef]

**y**of

*m*elements and the vertical profile of the unknown atmospheric parameter with a vector

**x**of

*n*elements corresponding to a predefined altitude grid. The forward model is a function

**F**(

**x**) that provides the value of the observations when the profile

**x**is known. Therefore, the relationship between the vectors

**x**and

**y**is

**ε**is the vector containing the experimental errors of the observations, characterized by a variance-covariance matrix (VCM)

**S**given by the mean value of the product of

_{y}**ε**times its transposed.

**y**and

**x**. In this case

**F**(

**x**) can be expanded up to the first order around a specific value of

**x**, identified by

**x**and referred to as the

_{0}*linearization point*, and Eq. (1) becomes equal to:

**K**is the Jacobian matrix (that is the partial derivatives of

**F**(

**x**) with respect to the elements of

**x**) calculated at

**x**.

_{0}**y**-

**F**(

**x**) are the scalar products between

_{0}**x**-

**x**and the rows of

_{0}**K**plus the error components. It follows that the knowledge of

**y**-

**F**(

**x**) determines the knowledge of the component of

_{0}**x**-

**x**that lies in the space generated by the rows of

_{0}**K**. In agreement with [11

11. S. Ceccherini, B. Carli, E. Pascale, M. Prosperi, P. Raspollini, and B. M. Dinelli, “Comparison of measurements made with two different instruments of the same atmospheric vertical profile,” Appl. Opt. **42**, 6465–6473 (2003). [CrossRef] [PubMed]

**y**are the observations and

**x**are the aimed measurements, we shall refer to this space as the

*measurement space*. Notice that this is not the terminology adopted in [1

*Inverse Methods for Atmospheric Sounding: Theory and Practice*, Vol. 2 of Series on Atmospheric, Oceanic and Planetary Physics (World Scientific, 2000). [CrossRef]

*row space*and the expression

*measurement space*is instead used to indicate the space of

*m*dimensions to which the observations

**y**belong.

**x**-

**x**that has to be applied to

_{0}**x**in order to determine

_{0}**x**. Since the correction is made in the measurement space and no information is acquired about the components of

**x**-

**x**in the orthogonal space that is complementary to the measurement space, also the determination of

_{0}**x**is to be considered as only made in the measurement space. However, this statement requires some further explanation. In fact, if

**x**has a component in the orthogonal space, when

_{0}**x**is combined with

_{0}**x**-

**x**to give

_{0}**x**also

**x**acquires this component. However such a component cannot be considered to be a result of the retrieval process because it is a contingent quantity that depends on the procedure rather than on the observations. On the basis of these considerations it is correct to say that since

**x**-

**x**is determined in the measurement space, also

_{0}**x**is determined in this space. The orthogonal complement space to the measurement space is referred to as the

*null space*[1

*Inverse Methods for Atmospheric Sounding: Theory and Practice*, Vol. 2 of Series on Atmospheric, Oceanic and Planetary Physics (World Scientific, 2000). [CrossRef]

**S**

_{y}^{-1/2}is characterized by a VCM that is the unity matrix.

### 2.2 Representation of the profile

**x**is a vector of n elements and belongs, therefore, to the space ℝ

^{n}which can be decomposed in the direct sum of the measurement space and of the null space. If we indicate with

*p*the dimension of the measurement space then the dimension of the null space is

*n*-

*p*. Since each vector of ℝ

^{n}can be decomposed as the sum of a vector belonging to the measurement space and a vector belonging to the null space, we can write:

**x**and

_{a}**x**, respectively, belong to the measurement space and to the null space. They can be expressed as:

_{b}**V**is a matrix whose columns are an orthonormal basis of the measurement space,

**W**is a matrix whose columns are an orthonormal basis of the null space and

**a**and

**b**are the projections of

**x**on these orthornormal bases:

^{T}denotes transposed matrices.

### 2.3 Estimation of the components in the measurement space

**x**in the measurement space is the only quantity that can be derived from the observations. In order to find

**x**from Eq. (5) we need to identify

_{a}**V**and

**a**. To this purpose we perform the SVD of

**S**

_{y}^{-1/2}

**K**:

**U**is a matrix of dimension

*mxp*whose columns (referred to as left singular vectors) are an orthonormal basis of the space generated by the columns of

**S**

_{y}^{-1/2}

**K**,

**Λ**is a nonsingular diagonal matrix of dimension

*pxp*and

**V**is a matrix of dimension

*nxp*whose columns (referred to as right singular vectors) are an orthonormal basis of the space generated by the rows of

**S**

_{y}^{-1/2}

**K**. Since

**S**

_{y}^{-1/2}is a nonsingular matrix the space generated by the rows of

**S**

_{y}^{-1/2}

**K**coincides with the space generated by the rows of

**K**, therefore the columns of

**V**are an orthonormal basis of the measurement space and, among all the possible orthonormal bases of the measurement space, it can be chosen for representing

**x**with Eq. (5). We can now determine the components of

_{a}**x**relative to this orthonormal basis.

_{a}**Λ**

^{-1}

**U**

*and using Eq. (7), after some rearrangements, we obtain that*

^{T}**â**, i.e. the estimation of

**a**deduced from the observations, is given by:

**â**as the estimation of

**a**and is characterized by the diagonal VCM:

**â**are independent of each other and are characterized by variances given by the inverse of the squared singular values of

**S**

_{y}^{-1/2}

**K**. Therefore, components corresponding to large singular values are well determined while components corresponding to small singular values are poorly determined.

**x**in terms of

_{a}**V**and

**a**, determined respectively by Eq. (9) and by Eq. (10), will be referred to as the

*measurement-space solution*(MSS). This solution is obtained exploiting all the information coming from the observations without any a-priori information. It is, therefore, the optimal quantity to be used for further post-retrieval processing. This conclusion is consistent with the results obtained by [12

12. J. Joiner and A. M. da Silva, “Efficient methods to assimilate remotely sensed data based on information content,” Q. J. R. Meteorol. Soc. **124**, 1669–1694 (1998). [CrossRef]

*n*is chosen with the desirable redundancy, components of the MSS that correspond to small singular values are poorly determined and make this solution ill-conditioned. On the other hand, if the poorly determined components are removed, the size of the null space grows at the expenses of the measurement space and, as it will be shown in Section 3, the retrieved profile acquires a rather unphysical shape.

*Inverse Methods for Atmospheric Sounding: Theory and Practice*, Vol. 2 of Series on Atmospheric, Oceanic and Planetary Physics (World Scientific, 2000). [CrossRef]

### 2.4 Estimation of the components in the null space

**x**is determined by the observations and

_{a}**x**is given by Eq. (6) with an arbitrary value of the components

_{b}**b**. The problem is here how to choose these components in the null space. When further processing of the data is planned it is adequate to provide the measurement information in a sub-space and taking advantage of this opportunity it is better to avoid any choice about

**x**. Indeed, the further processing may provide some estimation about the null-space components and any constraint (even the condition

_{b}**b**=0) does introduce a bias. However, if a graphical representation of the profile is requested different considerations apply. In a graphical representation the result is to be presented in a complete space. The representation of the

**x**profile with the MSS corresponds to selecting the value zero for

**b**. This is a particular choice among the infinitive possible ones, which does not necessarily provide the best graphical representation, as it will be shown in Section 3.

**a**) and determining in the null space (that now includes the null space enlarged with the poorly measured components rejected from the measurement space) the value of

**b**that minimizes the oscillations. To this purpose, we minimize the quantity:

**b**, where

**L**is the first-derivative matrix [7

_{1}7. T. Steck, “Methods for determining regularization for atmospheric retrieval problems,” Appl. Opt. **41**, 1788–1797 (2002). [CrossRef] [PubMed]

**x**. In these equations the matrices

**V**and

**W**do not necessarily coincide with the matrices used in Sections 2.2 and 2.3; they are now, respectively, the matrices containing the bases of the well measured subspace of the measurement space and of the new complementary null-space. Setting equal to zero the derivatives of the quantity in Eq. (13) with respect the components of

**b**and substituting for

**a**our estimation

**â**it results that the estimation of

**b**is given by:

**R**=

**L**

_{1}

^{T}

**L**. This calculation of the components of

_{1}**x**in the new null space allows obtaining the smoothest profile compatible with the observations and corresponds to a regularization of the profile; accordingly, it will be referred to as the

*null- space regularization*(NSR).

### 2.5 Solution and its characterization

**x**̂ of the profile

**x**that makes use of both the MSS and the NSR is obtained from Eqs. (4)–(6) and (14) and is equal to:

**â**is given by Eq. (10). As it will be shown in Section 3, the solution

**x**̂ given by Eq. (15) provides a useful graphical representation of the retrieved profile and it will be referred to as

*regularized measurement-space solution*(RMSS). The equations that provide the RMSS can be used to determine its VCM and its averaging kernel matrix [1

*Inverse Methods for Atmospheric Sounding: Theory and Practice*, Vol. 2 of Series on Atmospheric, Oceanic and Planetary Physics (World Scientific, 2000). [CrossRef]

**x**̂, given by Eq. (15), with respect to the true profile

**x**[1

*Inverse Methods for Atmospheric Sounding: Theory and Practice*, Vol. 2 of Series on Atmospheric, Oceanic and Planetary Physics (World Scientific, 2000). [CrossRef]

## 3. Application of the method to a simulated case

14. 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]

15. M. Ridolfi, B. Carli, M. Carlotti, T. v. 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]

16. 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]

**x**. The tangent altitudes for which the spectra are simulated are those corresponding to the MIPAS measurement scenario adopted between July 2002 and March 2004, that is with a 3 km step between 6 and 42 km, with a 5 km step between 42 and 52 km and with an 8 km step between 52 and 68 km (for a total of 17 tangent altitudes). The simulated observations

**y**are obtained adding a realistic random noise to the radiances calculated with the forward model. The microwindow approach, described in [17

17. A. Dudhia, V. L. Jay, and C. D. Rodgers, “Microwindow selection for high-spectral-resolution sounders,” Appl. Opt. **1**, 3665–3673 (2002). [CrossRef]

**S**.

_{y}**x**and of a linearization point

**x**close enough to the true profile in such a way that the linear approximation of the forward model is appropriate. The predefined grid can be chosen as fine as wished and, thanks to this freedom, it can be determined on the basis of the application rather than according to the vertical resolution of the measurements. In this example we have chosen a vertical grid of 1 km step between 0 and 100 km. The linearization point

_{0}**x**has been obtained interpolating at the predefined grid the ozone profile retrieved by the operational retrieval at the tangent altitudes grid. Accordingly, we have calculated the forward model and the Jacobian in the linearization point (respectively

_{o}**F**(

**x**) and

_{0}**K**). With these quantities we have all the ingredients needed to apply the method described in Section 2.

**S**

_{y}^{-1/2}

**K**(a matrix of dimension 4231×101) determines 98 singular values different from zero that indicate a measurement space of dimension

*p*=98. The large rank of the matrix comes out from the SVD operation; however only few components correspond to large singular values (we recall that, according to Eq. (12), components corresponding to small singular values are determined with a large error). All the 98 measured components with their errors can be considered with no harm in successive data assimilation or data fusion operations (because in these applications the components will be weighted according to their errors), however if we want to make a graphical representation of the profile it is preferable not to include components with large errors because they introduce large uncertainties in the values of the profile. In this case, as suggested in Subsection 2.4, it is preferable to reduce the dimension

*p*of the measurement space and to consider the components affected by a large error as belonging to the null space.

**X**̂ profile (red line) obtained with the procedure described in Section 2 in the case of considering only the 18 largest singular values. In the same panel the two components

**X̂**(blue line) and

_{a}**X̂**(green line) in the measurement space and in the null space, respectively, of

_{b}**X̂**are shown, illustrating at each altitude the relative importance of MSS and NSR contributions. In panel (b) the differences between

**X̂**and the

*true profile*

**x**

_{true}used to generate the observations (red line) and between

**X̂**and

_{a}**x**

_{true}(blue line) are shown on an expanded scale.

**X̂**profile obtained with the MSS method is characterized by oscillations and the addition of the

_{a}**X̂**component obtained with the NSR method is able to remove them and to decrease significantly the difference between the retrieved profile and the true profile.

_{b}**X̂**is predominant at altitudes larger than 68 km (the highest measured tangent altitude), below 6 km (the lowest measured tangent altitude) and at some altitudes between 52 and 68 km where the step of the measured tangent altitudes is the largest (8 km).

_{b}*Inverse Methods for Atmospheric Sounding: Theory and Practice*, Vol. 2 of Series on Atmospheric, Oceanic and Planetary Physics (World Scientific, 2000). [CrossRef]

**X̂**obtained with the procedure described in Section 2 in the case of considering the 10 (blue line) and the 30 (red line) largest singular values.

**S**given by Eq. (16), while the smoothing errors have been calculated as the absolute value of the differences between the profile obtained with the procedure described in Section 2, where

_{x}**a**is calculated projecting directly the true profile on the basis represented by the columns of

**V**, and the true profile.

**X̂**profile is shown in Fig. 1. It is interesting to notice that the value 18 is larger than the number of measured tangent altitudes, which is equal to 17, but not significantly different. This confirms that also in the operational retrieval [15

15. M. Ridolfi, B. Carli, M. Carlotti, T. v. 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]

16. 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]

**X̂**and

**x**

_{true}(black line) is compared with the difference between the profile obtained by the operational retrieval interpolated on the predefined grid (that in our case coincides with the linearization point

**x**) and

_{0}**x**

_{true}(red line). The two methods (RMSS and operational retrieval) have comparable quality even if the first better reproduces the values at high altitude and the feature at 24 km while the second does better reproduce the values below 20 km. The differences are due to the fact that the two methods provide a slightly different compromise between vertical resolution and precision; however the general agreement shows that the NSR has provided an efficient regularization at least as good as the constraint of a limited number of retrieval points.

## 4. Application to data fusion

3. B. Carli, P. Raspollini, M. Ridolfi, and B. M. Dinelli, “Discrete representation and resampling in limb-sounding measurements,” Appl. Opt. **40**, 1261–1268 (2001). [CrossRef]

**x**, so that it is not necessary to maintain memory of them.

_{0}**x**. The results of this section can be easily extended to an arbitrary number of independent measurements.

^{n}. The two MSSs are characterized by the matrices

**V**and

_{1}**V**that identify the measurement spaces (of dimensions

_{2}*p*and

_{1}*p*respectively) and by the two vectors

_{2}**â**and

_{1}**â**(with their VCM

_{2}**S**and

_{a1}**S**). Each MSS provides the elements of the relationship shown in Eq. (10). We can write the two relationships in the compact form:

_{a2}**Q**below the rows of the matrix (vector)

**P**and

**α**and

_{a1}**α**contain the errors with which

_{a2}**a**=

_{1}**V**

_{1}^{T}

**x**and

**a**=

_{2}**V**

_{2}^{T}

**x**are estimated by

**â**and

_{1}**â**.

_{2}**x**in the space generated by the columns of

**V**and of

_{1}**V**. This space is in the union space of the two measurement spaces and the profile that determined by these measurements is the result of the data fusion. This problem is equal to that faced in Section 2 consisting in finding a representation of the profile

_{2}**x**of ℝ

^{n}when the component in a subspace is measured; accordingly, also the same procedure is followed in order to determine the solution. We can represent the profile

**x**in the form of the summation of a vector of the union space of the two measurement spaces and of a vector of the orthogonal complement space to this space (which coincides with the intersection space of the two null spaces of the original measurements). The vector in the union space is the MSS of the data fusion problem and its identification implies the determination of the elements of the following relationship:

**â**

_{12}is the estimation of the components of

**x**in the basis of the union space represented by the matrix

**V**, and

_{12}**ε**is its error.

_{a12}**U**is a matrix of dimension (

_{12}*p*+

_{1}*p*)x×

_{2}*p*, with

_{12}*p*≤(

_{12}*p*+

_{1}*p*),

_{2}**Λ**is a nonsingular diagonal matrix of dimension

_{12}*p*×

_{12}*p*and

_{12}**V**is a matrix of dimension

_{12}*n*×

*p*whose columns are an orthonormal basis of the space generated by the rows of

_{12}**S**

_{a1}^{-1/2}and

**S**

_{a2}^{-1/2}are nonsingular matrices, the space generated by the rows of

**V**and of

_{1}**V**. Consequently the columns of

_{2}**V**are an orthonormal basis of the union space of the two measurement spaces.

_{12}**V**can be used to determine the vector

_{12}**â**

_{12}. Substituting Eq. (22) in Eq. (21) and multiplying both terms on the left by

**Λ**

_{12}^{-1}

**U**

_{12}*we obtain:*

^{T}**ε**is given by:

_{a12}**V**obtained from Eq. (22) provide the MSS of

_{12}**x**in the union space of the two measurements. The undetermined component of

**x**lies in the orthogonal complement space of the union space, and can be determined following the procedure described in subsection 2.4. Finally the complete expression of the solution and its characterisation can be obtained as described in subsection 2.5.

**V**exclusively includes the two original measurement spaces, the MSS in the union space has been obtained by utilizing all the information provided by the two sets of observations and without any a-priori information. This characteristic makes the proposed method for data fusion optimal and equivalent to the simultaneous analysis of the two sets of observations.

_{12}## 5. Conclusions

*measurement-space solution*(MSS) which gives the retrieved profile in terms of its components in the subspace defined by the rows of the jacobian matrix. The determination of the MSS coincides with the method proposed by Joiner et al. [12

**124**, 1669–1694 (1998). [CrossRef]

*null-space regularization*(NSR) method for the calculation of the null-space component which allows to obtain the smoothest profile compatible with the observations. This is a regularization and is different from the other regularizations since it does not affect the measured component and only determines the one that has not been measured. The fact that two components (the MSS and the NSR) are determined respectively in the measurement space and in the null space provides a clear and easily traceable distinction between the measurement and the constraint. Simulations have shown that MSS plus NSR, that is referred to as the

*regularized measurement-space solution*(RMSS), provides a good graphical representation of the retrieved profile.

## References and links

1. | C. D. Rodgers, |

2. | S. Twomey, |

3. | B. Carli, P. Raspollini, M. Ridolfi, and B. M. Dinelli, “Discrete representation and resampling in limb-sounding measurements,” Appl. Opt. |

4. | A. Tikhonov, “On the solution of incorrectly stated problems and method of regularization,” Dokl. Akad. Nauk. SSSR |

5. | S. Twomey, “On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature,” J. Assoc. Comput. Mach. |

6. | B. Schimpf and F. Schreier, “Robust and efficient inversion of vertical sounding atmospheric high-resolution spectra by means of regularization,” J. Geophys. Res. |

7. | T. Steck, “Methods for determining regularization for atmospheric retrieval problems,” Appl. Opt. |

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

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

10. | C. D. Rodgers and B. J. Connor, “Intercomparison of remote sounding instruments,” J. Geophys. Res. |

11. | S. Ceccherini, B. Carli, E. Pascale, M. Prosperi, P. Raspollini, and B. M. Dinelli, “Comparison of measurements made with two different instruments of the same atmospheric vertical profile,” Appl. Opt. |

12. | J. Joiner and A. M. da Silva, “Efficient methods to assimilate remotely sensed data based on information content,” Q. J. R. Meteorol. Soc. |

13. | T. von Clarmann and U. Grabowski, “Elimination of hidden a priori information from remotely sensed profile data,” Atmos. Chem. Phys. |

14. | 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. |

15. | M. Ridolfi, B. Carli, M. Carlotti, T. v. 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. |

16. | 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. |

17. | A. Dudhia, V. L. Jay, and C. D. Rodgers, “Microwindow selection for high-spectral-resolution sounders,” Appl. Opt. |

**OCIS Codes**

(000.3860) General : Mathematical methods in physics

(010.1280) Atmospheric and oceanic optics : Atmospheric composition

(100.3190) Image processing : Inverse problems

(120.0280) Instrumentation, measurement, and metrology : Remote sensing and sensors

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: November 12, 2008

Revised Manuscript: December 13, 2008

Manuscript Accepted: January 16, 2009

Published: March 16, 2009

**Citation**

Simone Ceccherini, Piera Raspollini, and Bruno Carli, "Optimal use of the information provided by indirect measurements of atmospheric vertical profiles," Opt. Express **17**, 4944-4958 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-7-4944

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### References

- C. D. Rodgers, Inverse Methods for Atmospheric Sounding: Theory and Practice, Vol. 2 of Series on Atmospheric, Oceanic and Planetary Physics (World Scientific, 2000). [CrossRef]
- S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, New York, 1977).
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